Miyuki Koiso^{1}

^{1}Kyushu University, Institute of Mathematics for Industry, Japan

We study a variational problem for surfaces in the euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered surface, which was introduced to model the surface tension of a small crystal. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wulff shape. The Wulff shape and equilibrium surfaces of this energy for volume-preserving variations are generalizations of the round sphere and constant mean curvature surfaces, respectively. However, they are not smooth in general. In this talk, we give a suitable formulation of piecewise-smooth surfaces and discuss geometry of equilibrium surfaces. Especially, we give recent studies on this subject and open questions. Moreover, we apply the theory to capillary problems.

Fei Zhang^{1}
, Xinping Zhou^{2}

^{1}Huazhong University of Science and Technology, Department of Mechanics, China

^{2}Huazhong University of Science and Technology, Department of Mechanics, China

A model dealing with the conditions of equilibrium and initial stability of floating bodies in two dimensions with an arbitrary cross-section has been proposed recently (J. Fluid Mech. 847:489-519, 2018). The method is then applied to the configuration of an elliptical cylinder floating on an infinite liquid bath. The hydrostatics and stability of the cylinder with surface tension effects are investigated by two dimensionless parameters: the eccentricity of the elliptical cross-section $e$, the bond number $Bo$. It is shown that the prolate cylinder with a greater eccentricity $e$ can bear a greater pressing force and a greater moment and therefore is more stable. It is known that the bond number $Bo$ measures the importance of gravitational forces compared to surface tension forces. In general, the dimensionless forces and the dimensionless moments of the mesoscale cylinders ($Bo \sim 1$) are found to be varied continuously between the cases of the macroscale ($Bo \gg 1$) and microscale ($Bo \to 0$) cylinders with other parameters remaining unchanged. However, the incline angle of the cylinder corresponding to the maximum moment does not obey this rule. The maximum incline angle occurs at an intermediate bond number.

This research was partly supported by the National Natural Science Foundation of China (no. 11672116).

Yoshihito Kohsaka^{1}

^{1}Kobe University, Maritime Sciences, Japan

The stability of the steady states for the geometric evolution equation $$ V=-\Delta_{\Gamma_t}H $$ will be studied. Here $V$ is the normal velocity of the evolving surfaces $\Gamma_t$, $H$ is the mean curvature of $\Gamma_t$, and $\Delta_{\Gamma_t}$ is the Laplace-Beltrami operator on $\Gamma_t$. The geometric evolution equation mentioned above is called the surface diffusion equation, which was first derived by Mullins (1957, J. Appl. Phys.) to model the development of surface grooves at the grain boundaries of a heated polycrystal. The surface diffusion equation is obtained as the $H^{-1}$-gradient flow of the area functional for the evolving surfaces $\Gamma_t$, so that the surface diffusion equation has a variational structure that the area of the surface decreases whereas the volume of the region enclosed by the surface is preserved. This provides the constant mean curvature surfaces (CMC surfaces) as the steady states for this equation. Our goal is to derive the criteria of the stability of the axisymmetric CMC surfaces via the linearized stability analysis to the surface diffusion equation.

Giorgio NORDO^{1}

^{1}Messina University, MIFT - Department of Mathematics and Computer Science, Physical Sciences and Earth Sciences, Italy

Very recently, the idea of studying structures equipped with two or more soft topologies has been considered by several researchers. Soft bitopological spaces were introduced and studied, in 2014, by Ittanagi as a soft counterpart of the notion of bitopological space. and, independently, in 2015, by Naz, Shabir and Ali. In 2017, Hassan too introduced the concept of soft tritopological spaces and gave some first results. The notion of $N$-topological space related to ordinary topological spaces was instead introduced and studied, in 2011, by Tawfiq and Majeed. In this paper we introduce the concept of Soft $N$-Topological Space as generalization both of the concepts of Soft Topological Space and $N$-Topological Space and we investigate such class of spaces and their basic properties with particular regard to their subspaces, the parameterized families of crisp topologies generated by them and some new separation axioms called $N$-wise soft $T_0$, $N$-wise soft $T_1$, and $N$-wise soft $T_2$.

René Verreault^{1}

^{1}Université du Québec à Chicoutimi, Fundamental Sciences, Canada

The linear anisotropy of a real Foucault pendulum has been thoroughly studied by Kamerlingh Onnes (KO) in his dissertation using perturbation calculation. A simulation of KO’s original pendulum has been analysed with the anisosphere. KO theory shows that, under certain conditions on the ratio of linear to circular anisotropy, precession velocity and precession angle show a cyclic behaviour. The azimuth of the major axis of the elliptical orbit oscillates, depending on initial conditions, about the azimuth of longest or shortest swinging period, respectively the slow or fast eigenaxis. It is shown that the cyclic behaviour of the ellipse major axis correspond to a circular phase curve that does not enclose a pole on the anisosphere model. The centre of the phase circle is one of the orthogonal elliptical eigenstates of the pendulum. The oscillation of the major axis of the successive elliptical orbits constitutes by itself a second level oscillation mode of the pendulum of the same nature as that of a torsional balance. The extremely small recall torque of that mode is indicative of an extremely shallow gravitational potential well.

The author is indebted to M. Thomas Goodey and M. Dimitrie Olenici for their dedicated efforts in testing the anisosphere concepts with the help of two exceptional instruments at their Romanian pendularium in Horodnic de Sus.

René Verreault^{1}

^{1}Université du Québec à Chicoutimi, Fundamental Sciences, Canada

Applying celestial body perturbation theory to the Foucault pendulum in his doctor dissertation, Nobel laureate Kamerlingh Onnes showed that the essential features of a Foucault pendulum are its inherent circular and linear anisotropies. A spherical differential 2D sub-space can be defined, where the group of the points of a spherical surface with respect to the operation rotation about a diametral axis is isomorphic with the group of sequential states of oscillation of a 2D pendulum with respect to the operation translation in time. Any Foucault pendulum is then characterized by two elliptical eigenstates M and N which are represented by the poles of that rotation axis on the so-called anisosphere. Moreover, those eigenstates move drastically within a meridian plane when nonlinear restoring torque giving rise to Airy precession occurs. The concept of anisosphere constitutes a very powerful tool for analysing and optimizing actual Foucault pendulum implementations.

The author is profoundly indebted to Mr Thomas Goodey and Mr Dimitrie Olenici for their dedicated efforts in testing the anisosphere concepts with the help of two high performance instruments at their Romanian pendularium in Horodnic de Sus.

Hovik Matevossian^{1}

^{1}Russian Academy of Sciences, Federal Research Center "Computer Science and Control", Russian Federation

We study the asymptotic behavior of solutions of elliptic boundary value problems at infinity under the assumption that generalized solutions of these problems have the finite weighted Dirichlet (energy) integral. Moreover, we study the properties of generalized solutions of boundary value problems for the biharmonic Steklov and Steklov--type problems in unbounded domains with a compact and non-compact boundaries (in particular, the exterior of a compact set, half-space, the domain with conical points) under the assumption that the Dirichlet (energy) integral with weight $|x|^a$ is finite for such solutions. Admitting different boundary conditions, we used the variation principle and depending on the value of the parameter $a$, we obtained uniqueness (or non-uniqueness) theorems for these solutions or present exact formulas for the dimension of the space of solutions.

Hovik Matevossian^{1}
, Mikhail Nikabadze^{2}
, Armine Ulukhanyan^{3}

^{1}Russian Academy of Sciences, Federal Research Center "Computer Science and Control", Russian Federation

^{2}Lomonosov Moscow State University, Department of Mechanics and Mathematics, Afghanistan

^{3}Bauman Moscow State Technical University, Mathematical Physics, Russia

It is known that the eigenvalues of the tensor and the tensor-block matrix are invariant quantities. Therefore, in this work, our goal is to find the expression for the velocities of wave propagation of certain media through the eigenvalues of the material tensors. In particular, we consider materials with the anisotropy symbol \{1.5\} and \{5.1\}, as well as isotropic materials, and for them we determine the expressions for the velocities of wave propagation. In addition, we obtained expressions for the velocities of wave propagation for materials of cubic syngony with the anisotropy symbol \{1,2,3\} (the matrix of the elastic modulus tensor components has three independent components), hexagonal system (transversal isotropy) with anisotropy symbol \{1,1,2,2\} (the matrix of the elastic modulus tensor components has five independent components), trigonal system with anisotropy symbol \{1,1,2,2\} (the matrix of the elastic modulus tensor components has six independent components), tetragonal system with anisotropy symbol \{1,1,1,2,1\} (the matrix of the elastic modulus tensor components has six independent components). We also obtained the expressions for the velocities of wave propagation for a micro-polar medium with the anisotropy symbol \{1.5.3\} and \{5.1.3\}, and for an isotropic micro-polar material.

Tokuei Sako^{1}

^{1}Nihon University, College of Science and Technology, Japan

Correlated dynamics of a few electrons confined initially in a low-dimensional nanostructure and then tunneling through the potential barrier by an external triggering electric pulse has been studied by solving the time-dependent Schroedinger equation directly. The transient current has been calculated as a flux of the probability density of electrons absorbed by an artificial complex potential placed at the downstream side of the computational domain. Numerous simulation has been performed by changing the central frequency and pulse width of the incident electric field and also the initial configuration of the confined electrons, that have led us to identify two distinct pathways for the transient current generation. The results have been rationalized on the basis of the nodal patterns of the probability density distributions.

This study has been supported in part by the Grants-in-Aid for Scientific Research (No.17H02725 and No.18K19065) from JSPS and by Nihon University CST Research Grants (2018).

Artem Arutyunov^{1}
, Kirill Troshin^{2}
, Alexey Nikitin ^{3}
, Andrey Beliaev ^{4}
, Vladimir Arutyunov^{5}

^{1}Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics (CMC), Russia

^{2}Semenov Institute of Chemical Physics of Russian Academy of Sciences (ICP RAS), , Russian Federation

^{3}Semenov Institute of Chemical Physics of Russian Academy of Sciences (ICP RAS), , Russian Federation

^{4}Semenov Institute of Chemical Physics of Russian Academy of Sciences (ICP RAS), , Russian Federation

^{5}Semenov Institute of Chemical Physics of Russian Academy of Sciences (ICP RAS), , Russian Federation

Associated petroleum gases (APG) is a mixture of light alkanes from C1 to C5 and higher dissolved in oil, which releases from it as a result of pressure drop at oil extraction on surface. The main component of APG, as well as natural gas, is methane. However, the presence of significantly higher concentrations of heavy methane homologues with a low detonation resistance makes APG unsuitable for direct use as fuel for gas piston engines that form the basis of a distributed power generation grid in the range up to 5 MW. Since the profitability of all methods of extraction or chemical transformation of these heavy components decreases sharply with the increasing of the depth of purification or conversion of these impurities, it is very important to determine the necessary cleaning depth, sufficient to achieve the required detonation resistance. One of the most informative and demonstrative parameters that determine the detonation characteristics of gas mixtures is the delay in their self-ignition. Unfortunately, in the world literature there are practically no systematic data on delays of self-ignition even for binary mixtures of methane with its heavier homologues, not to mention the more complex mixtures corresponding to the compositions of real APG. The paper presents the results of computer simulation of self-ignition delays for binary and complex methane-alkane-air mixtures with alkanes C2-C6. For modeling, the NUI Galway (Natural Gas II) kinetic model and its reduced version were used. The model of the Calorimetric Bomb Reactor (CBR, V = const) was used. Results were compared with specially conducted experiments in Closed Bomb Reactor and demonstrated very well correlation. Therefore, the elaborated reduced model can be used for the calculation of self-ignition delays for complex methane-alkane-air mixtures.

Valeriy Kiryushin^{1}

^{1}Moscow Technological University (MIREA), Institute of Cybernetics, Russia

Stationary flows of viscous fluids with a static or moving contact line are considered. Contact line separates three phases: vapor, liquid and solid. No-slip condition on the solid-fluid surface and ordinary conditions with account constant surface tension for the fluid-vapor interface are supposed to be fulfilled. The flows presented in the report are induced by some physical mechanism concentrated in the very small region near the contact line. Such contact line is the origin of the flow and treated as a hydrodynamics singularity. As an example the flow in a two- dimensional viscous fluid drop which rests or steadily moves along a horizontal solid surface is considered. Motions of this type can be observed in experiments if the solid-fluid surface wettability is non-uniform. A sequence of solutions for the velocity field and the free surface shape with the successively increasing applicability region near the static or moving contact lines is obtained. At first stage the solution of the problem is found in the case when the distortion of the free surface of the drop during motion can be neglected. The problem is then reformulated using functions of a complex variable and expanded variables are introduced. In the new variables a more accurate solution of the same problem is found, with a much more narrow inapplicability region near the contact lines. Asymptotic behavior of the flow near the contact lines is discussed.

Chengwei Zhu^{1}
, Xinping Zhou^{2}

^{1}Huazhong University of Science and Technology, Department of Mechanics, China

^{2}Huazhong University of Science and Technology, Department of Mechanics, China

The equilibrium interface of gas and liquid between two partially immersed vertical parallel plates is determined uniquely by its contact angle and the distance between two plates, with gravity directed downward into the liquid. However, the free surface between a straight plate and a curved plate or two curved plates exists two or more equilibrium configurations. Only existing one equilibrium interface is some special cases. Some of these equilibrium configurations are unstable. The exotic plate that admits a continuum of distinct equilibrium free surfaces (between a straight plate and it) is investigated. These equilibrium surfaces all bound with plates and have the same contact angle at the boundary. The governing differential equations are clarified and simplified by a class of parameters, which make the equations appropriate for numerical integration. The profiles of the exotic plates are depicted graphically for a range of gravity accelerations, locations of initial points in initial equilibrium surface, and contact angles.

Natalya Shipulya^{1}

^{1}Saint-Petersburg National Research University of Information Technologies, Mechanics and Optics, Department of Instrument Technology, Russia

In this work we present the results of analytical modeling and 3D computer simulation of microfluidic comparator, which based on capillary effect. Microfluidic comparator is internal element of the component base of the microfluidic pressure sensor, which we considered earlier (doi:10.1088/1742-6596/738/1/012071). The advantage of such a device is in operating principle. Microfluidic comparator does not have mechanical control valves and other elements that create additional measurement errors. Also, mechanical elements impose limitations on operational requirements. The proposed system has high-quality operability in the extended range of ambient temperatures; this is stable with increased vibrations and hit. We investigated the behavior of this device for various working fluids and gases. Analytical calculations are supported by numerical simulation results. Finally, microfluidic differential pressure sensor with microfluidic comparator has improved characteristics in comparison with analogues, and this opens up new opportunities for using it in the markets of medical and aerospace instrument.

The research was partially supported by FASIE.

Hanzhe Chen^{1}

^{1}University of Waterloo, Applied Mathematics, Canada

The study of floating objects can be traced back to Archimedes' time. With the consideration of surface tension, the ability and stability of flotation become more complicated. We consider a ball floating on an unbounded reservoir and assume the floating conguration is radially symmetric. The fluid interface can be determined by the Young-Laplace equation. We propose two methods to determine the fluid interface: the shooting method and the iterative method. Especially, through the iterative method, the approximating solution has high accuracy only after two iterative steps. In analysis of stability, both the force and the energy approaches will be applied. The relation between the total force $F_T$ and the total energy $E_T$ can also be established. We will give an example with two force balanced points, both of which are stable. General results on the number and stability of equilibrium congurations are anticipated.

pierre gaillard^{1}

^{1}université de Bourgogne, mathématiques, France

We construct solutions to the Johnson equation in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions called solutions of order N depend on 2N-1 parameters. They can be written as a quotient of 2 polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters. We explicitly construct the expressions until the order 5 and we study the patterns of their modulus in the plane (x,y) and their evolution according to time and parameters.

Antonio Gesualdo^{1}
, M Guadagnuolo^{2}
, F Penta^{3}

^{1}University of Naples Federico II, Dept. of Structures for Engineering and Architecture, Italy

^{2}University of Campania Luigi Vanvitelli, Department of Architecture and Industrial Design, Italy

^{3}University of Naples Federico II, Department of Industrial Engineering, Italy

The global interest in renewable energy sources has increased the attention to the manufacturing of wind turbine towers, since they are largely diffused in seismic areas too. Different types of towers have been produced in recent years. Among them, the truss structures assure a reduced mass and the modular characteristics necessary for easy transportation. Reduced costs of production, installation and maintenance are typical of these structures. Nonlinear dynamics is an efficient framework to analyze structures subjected to variable actions, i.e. to assess the seismic safety of wind turbine towers in case of earthquake actions. This study outlines a procedure to evaluate the post-elastic behavior of truss towers for wind turbines. Rigid-plastic behaviour is taken into account to develop approximate solutions for the problem of a tower modeled as a vertical cantilever beam and subjected to harmonic base motion. A comparison with the results of a finite element model is proposed.

Acknowledgements to PRIN 2015 "Advanced mechanical modeling of new materials and structures for the solution of 2020 Horizon challenges" (Miur - Ministry of Education, Universitiy and Research)

Sergey Pikulin^{1}

^{1}CC RAS, Applied Mathematical Physics, Russia

The quasiclassical model by Thomas and Fermi [1] of the charge distribution inside a heavy cooled atom leads to the nonlinear equation $\Delta \Phi=\Phi^{3/2}$ in $R^3$, which gets reduced in the radially symmetric case to the Emden --- Fowler equation being a singular ordinary differential equation (with respect to the screening coefficient $\Psi$, where $\Phi=:\Psi/r$): $$ \frac{d^2 \Psi}{dr^2} - {r}^{\nu}\, \Psi^{1+\sigma}(r)=0, \qquad r \in (0, +\infty), \qquad \Psi(0)=1, \qquad \Psi(+\infty)=0, $$ whith the parameters value~$\nu=-1/2$, $\sigma=1/2$. The main result of the talk is the following parametric representation of the solution of the problem for the Emden --- Fowler equation formulated above which holds when the value of $\nu$ is rational but not integer, $\nu=m/n$, $m,n \in \mathbf{N}$: $$ r(t) = C_1\, t^{1/\alpha} \, (1-t)^n \, {H}(t), \qquad y(t) = C_2\, t^{-\beta/\alpha} \, {H}^{-\beta}(t), \qquad t \in [0,1], $$ where $\beta,\alpha$ are calculated explicitly in terms of parameters $\nu, \sigma$, $C_1, C_2 = \mathrm{const}>0$, the function ${H}(t)$ is positive and analytic on the whole unit segment~$t \in [0,1]$ including the endpoints $t=0$ (corresponding to the limit as $r\to+\infty$, $\Psi \to 0$), and $t=1$ ($r=0$, $\Psi=1$). In the particular case of the Thomas --- Fermi problem, the method of calculation the solution based on this representation combines the high accuracy of the result with relatively small computational complexity and allows one to effectively find $\Psi(r)$ and its derivative~$\Psi'(r)$ at an arbitrary point of the ray with a prescribed accuracy. The obtained representation is based on an analytic property of solutions of the auxiliary Abel equation of the II kind obtained from the original Emden --- Fowler equation, namely, the partial passage of Painlev\'e test in the (fixed) node singular point of this auxiliary equation. The results of the talk are prepared for the publication [2].

The work is partially supported by Russian Foundation of Basic Research (RFBR project 16-01-00781).

Grigory Fedotenkov^{1}
, Olga Peshcherikova^{2}

^{1}Moscow Aviation Institute (National Research University), General engineering training, Russia

^{2}Moscow Aviation Institute (National Research University), Mathematical Modeling , Russia

The paper considers plain nonstationary contact problems for perfectly rigid dies translating into an elastic half-space. As a general case, the boundary of the contact area is assumed to be moving one. The method of functionally invariant solutions is used as a tool of analytic solution and study of distribution of contact stresses. By making use this method, the formulations of the problems are reduced to Riemann-Hilbert problem. It is shown that the distribution of stresses near the boundaries of the region differs for four ranges of contact speed: superseismic, first transseismic, second transseismic and subseismic. It is also shown that in the case when the given stresses or displacements are arbitrary continuous functions, the problem reduces to the considered way with help of approximating the given functions by homogeneous polynomials. The authors found and studied the analytic solution of the problems of indentation with conical and parabolic dies.

The authors would like to acknowledge the financial support of the Russian Founda-tion for Basic Research (project 16-08-00260 A).

Eugenia Echkina^{1}
, Maysuradze Archil^{2}

^{1}Moscow State University, Computational Mathematics and Cybernetics, Russia

^{2}Lomonosov Moscow State University, Computational Mathematics and Cybernetics, Russia

Numerical modeling of significantly nonlinear spatial processes results in need to analyze non-stationary three-dimensional vector fields which structure is a priori unknown. The high precision of calculations necessary for such researches leads to increase in the sizes of grids, used when modeling. From the practical point of view it means impossibility of the direct analysis of numerical results, as brings graphic (visual) data presentation to the forefront, In article an opportunity to use implicit surfaces for visualization of power lines of the vector field and to apply hierarchical splitting space for effective realization of the offered method which is further development of a method of a stream of spheres is discussed.

Andrei Zemskov^{1}
, Dmitrii V. Tarlakovskii^{2}

^{1}Moscow Aviation Institute (National Research University), Department 311 "Applied software and mathematical methods", Russia

^{2}Lomonosov Moscow State University, Institute of Mechanics , Russia

In this paper we consider the unsteady vibration problem for the Euler-Bernoulli beam. For the mathematical formulation of problem we use the model of coupled elastic diffusion processes in a medium with rectangular Cartesian coordinate system (dot denotes the time derivative): \begin{equation}\label{Zem1} \ddot{u}_i=\frac{\partial\sigma_{ij}}{\partial x_j}+F_i,\;\dot{\eta}^{(q)}=-\frac{\partial J_i^{(q)}}{\partial x_i}+Y^{(q)}\; \left(q=\overline{1,N}\right), \end{equation} where $\sigma_{ij}$ and $J_i^{(q)}$ are the components of the stress tensor and the diffusion flux vector respectively, which are defined as follows: \begin{equation}\label{Zem2} \sigma_{ij}=C_{ijkl}\frac{\partial u_k}{\partial x_l}-\sum_{q=1}^N\alpha_{ij}^{(q)}\eta^{(q)},\; J_i^{(q)}=-\sum_{p=1}^ND_{ij}^{(q)}g^{(qp)}\frac{\partial\eta^{(p)}}{\partial x_j}+\Lambda_{ijkl}^{(q)}\frac{\partial^{2}u_k}{\partial x_j\partial x_l}. \end{equation} Here all quantities are dimensionless. For them the following notation is used: $$ x_i=\frac{x^*_i}{l},\;u_i=\frac{u^*_i}{l},\;\tau=\frac{ct}{l},\;C_{ijkl}=\frac{C_{ijkl}^*}{C_{1111}},\; c^2=\frac{C_{1111}^*}{\rho},\;\alpha_{ij}^{(q)}=\frac{\alpha_{ij}^{*(q)}}{C_{1111}}, $$ $$ D_{ij}^{(q)}=\frac{D_{ij}^{*(q)}}{cl},\;\Lambda_{ijkl}^{(q)}=\frac{m^{(q)}D_{ij}^{*(q)}\alpha_{kl}^{*(q)}n_0^{(q)}}{\rho RT_0cl},\; F_i=\frac{F_i^*}{C_{1111}},\;Y^{(q)}=\frac{LY^{*(q)}}{c}, $$ where $t$ is time; $x_i^*$ are rectangular Cartesian coordinates; $u_{i}^*$ are components of the displacement vector; $l$ is length of the beam; $\eta^{(q)}=n^{(q)}-n_{0}^{(q)}$ is the concentration increment of $q$-th component in the $N$-component medium; $n^{(q)}$ and $n_{0}^{(q)}$ are the actual and initial concentrations; $C_{ijkl}^*$ are components of the elastic constant tensor; $\rho$ is density of the medium; $\alpha_{ij}^{*(q)}$ are coefficients characterizing the volumetric changes of the medium due to diffusion; $D_{ij}^{*(q)}$ are coefficients of self-diffusion; $R$ is the universal gas constant; $T_{0}$ is temperature of the medium; $m^{(q)}$ is the molar mass; $F_{i}^*$, $Y^{*(q)}$ is the body perturbations. To construct the beam bending equations, a transition to the variational formulation of the problem \eqref{Zem1}, \eqref{Zem2} is used. According to the variational principle of Hamilton, the relations \eqref{Zem1} can be regarded as a condition for the stationarity of a certain functional $H\left(u_{i},\eta^{(q)}\right)$, the variation of which is written thus: \begin{equation}\label{Zem3} \begin{split} \delta H=\int_{t_1}^{t_2}d\tau\int_G\left(\ddot{u}_i-\frac{\partial\sigma_{ij}}{\partial x_j}-F_i\right)\delta u_idG+\\ +\sum_{q=1}^N\int_{t_1}^{t_2}d\tau\int_G\left(\dot{\eta}^{(q)}+\frac{\partial J_i^{(q)}}{\partial x_i}-Y^{(q)}\right)\delta\eta^{(q)}dG+\\ +\int_{t_1}^{t_2}\iint\nolimits_{\Pi_{\sigma}}\left(\sigma_{ij}n_j-P_i\right)\delta u_idSd\tau+ \sum_{q=1}^N\int_{t_1}^{t_2}\iint\nolimits_{\Pi_J}\left(J_i^{(q)}-\tilde{I}_i^{(q)}\right)n_i\delta\eta^{(q)}dSd\tau. \end{split} \end{equation} where $P_i$, $\tilde{I}_i^{(q)}$ are the surface perturbations. It is assumed that: 1) The side surface is free from mechanical loads and mass transfer through this is absent. 2) The beam material is a homogeneous isotropic ideal solid compound. 3) The bending of beam is considered in plane $x_1Ox_2$. Mass transfer occurs also in the plane $x_1Ox_2$. 4) The hypothesis of Bernoulli-Euler flat sections is used (here the prime denotes the derivative with respect to the spatial variable $x_1$) $$ u_1=u-x_2v',\;u_2 =v,\;u=u\left(x_1,\tau\right),\;v=v\left(x_1,\tau\right), $$ $$ \eta^{(q)}=N_q+x_2H_q,\;N_q=N_q\left(x_1,\tau\right),\;H_q=H_q\left(x_1,\tau\right). $$ Calculating the variation of the functional \eqref{Zem3}, we obtain the formulation of the problem of the bending of an Euler-Bernoulli beam with taking into account diffusion. It is written as follows with respect to deflection $v$: $$ \frac{I_3}{F}\ddot{v}''-\ddot{v}=\frac{I_3}{F}\left(v^{IV}+\sum_{j=1}^N\alpha_{j}H''_j\right)+\frac{q}{F}+\frac{m'}{F},\;\dot{H}_q =D_qH''_q-\Lambda_q v^{IV}+\frac{Z^{(q)}}{I_3}; $$ $$ \left.\left(v''+\sum_{j=1}^N\alpha_jH_j\right)\right|_{x_1=0,1} =\frac{\left.M\right|_{x_1=0,1}}{I_3}; $$ $$\left.\left(v'''+\sum_{j=1}^N\alpha_jH'_j-\ddot{v'}\right)\right|_{x_1=0,1}=\frac{\left.\left(Q-m\right)\right|_{x_1=0,1}}{I_3}; $$ $$\left.\left(D_qH'_q+\Lambda_qv'''\right)\right|_{x_1=0,1}=-\frac{\left.\Omega^{(q)}\right|_{x_1=0,1}}{I_3}. $$ where $F$ is the cross-sectional area of the beam; $I_3$ is the moment of inertia of beam relative to the axis $Ox_3$; $M$ and $Q$ are bending moment and shearing force; $m$ and $q$ are linearly distributed bending moment and shearing force; $\Omega^{(q)}$ and $Z^{(q)}$ are surface and body sources of mass transfer. The solution of the constructed problem is sought by the Laplace transformation and expansion into incomplete trigonometric Fourier series.

The reported study was funded by RFBR according to the research project № 17-08-00663 A.

Vaclav Kus^{1}
, J Kucera^{2}
, D Morales^{3}

^{1}FNSPE, Czech Technical University in Prague, Depatment of Mathematics, Czech Republic

^{2}Czech Technical University in Prague, , Czech Republic

^{3}Universidad Miguel Hernández de Elche, , Spain

Different numerical aspects of Rényi pseudo-distance estimators are studied. These estimators are based on the minimization of information-theoretic divergences between empirical and hypothetical probability distributions. They are not classical distances, because the symmetry or triangle inequality does not hold. Robust properties of the minimum Rényi pseudodistance estimators are required by various applications in mathematical modeling, physics, or material science. Therefore we model the distribution of contaminated data as a mixture of the true distributions P and error distribution Q under different contamination level ε. We focus on the estimators for relatively small data samples or very sparse and scattered data with high variance, which appears mostly in high energy physics (signal and sparse background). In this case, the strict minimization leads to delta functions and it is impossible to obtain satisfactory numerical results. A way of adjusting the Rényi minimum distance estimators to these conditions is proposed. This so called ‘blurring’ is created as a convolution of Rényi distance with averaging Gaussian mask. Simultaneously, the effect of the input parameter alpha to the robustness is presented based on Monte-Carlo simulations for Gaussian model. Thus the Rényi distance is ready to be used in divergence decision trees for the signal versus background separations, e.g. in high energy physics NOvA or DUNE experiments at Fermilab.

Subhash Chandra^{1}
, Sanjay Kumar Pandey^{2}

^{1}IIT(BHU), Dept of Mathematical Sciences, India

^{2}Indian Institute of Technology (BHU), Varanasi, Mathematical Sciences, India

In this paper, we presented a mathematical formulation deals with the flow of micropolar fluid in a circular cylindrical tube of non-uniform cross-sectional area induced by peristaltic waves of increasing amplitude. This is an intended act to model the swallowing of various types of foods in the oesophagus which suffers from a hiatus hernia. Due to sliding hiatus hernia, the cross-section of the lower oesophagus does not remain uniform. The impact of bulging, which is formed by various combinations of divergence and convergence, has been examined. The effects of dilating amplitude, the slope of the tube wall, the coupling number, and micro-polar parameter, have been investigated.

Arkady Seryakov^{1}

^{1}RUDETRANSSERVICE, Veliky Novgorod , scientific laboratory, Russia

Abstract. The results of the numerical simulation of vortex formation near the condensation surface in the Laval-liked vapour channel of short low temperature heat pipes (HPs) at high heat loads are presented. For the first time it was found that the vapour vortex of moist compressible vapour flow near the condensation surface inside the vapour channel of short HPs changes its rotational motion direction, depending on the thermal load magnitude on the HPs evaporator. With low thermal loads, the rotational motion direction of the vapour vortex due to the Coanda effect and sticking moving vapour jets to the walls of the channel occurs from the periphery to the center of the vapour channel. While the thermal load increasing, the direction of rotation of the vapour vortex changes to the opposite, from the center to the periphery. The thickness of the liquid condensate film located under the vortex also depends on the magnitude of the thermal load and the direction of the vortex rotation. The numerical calculation of the condensate film thickness of working fluid were compared with experimental values depending on the thermal load on the HPs evaporator and obtained by capacitive sensors.

Natalia Tuchkova^{1}
, Konstantin Belyaev^{2}
, Andrey Kuleshov^{3}

^{1}FRC CSC RAS, Department of Computing systems and scientific information, Russia

^{2}CIO RAS, Department fisical of gidrology, Russia

^{3}Keldysh Institute of Applied Mathematics of RAS , , Russia

An estimation of the impact of the observed characteristic in Central and South Atlantic is considered as the result of model experiments with and without data assimilation. The special data assimilation technique derived by authors and coupled ocean atmosphere MPIESM from Max Planck institute for Meteorology (Hamburg, Germany) is used. Several experiments have been performed on supercomputer Lomonosov 2 in Moscow State University. Several temperature and salinity profiles obtained from special scientific cruises which conducted by Shirshov Institute of Oceanology Russian Academy of Science are used. The estimation of the meridional mass transport are calculated with the special focus of the of Antarctic Bottom Water transport. This variable plays the significant role in climate variability and global water pollution. The difference between assimilated and control fields are discovered and analyzed and the comparison with independent observations are done.

This research has been supported by Russian Science Foundation, grant 14-11-00434

Antonio Gesualdo^{1}

^{1}University of Naples Federico II, Dept. of Structures for Engineering and Architecture, Italy

The safeguard of museum exhibits like vases and statues, laboratory equipments, storage tanks, or even tall buildings and other structural elements subjected to earthquake and in general to time-dependent forces has been in the last years a topic focus. The classical mechanical problem of the motions of rigid objects simply supported on a base plane, developed in the first year of sixties, thanks to the seminal paper by Housner on the inverted pendulum, can be now take advance due to the development of calculation tools. In some cases the quality of motion can be determined varying the material parameters involved. This is a strong tool to reduce vulnerability, especially when rocking motion should be avoided and sliding motion is welcome, as in the case of artistic heritage. This paper focuses the attention on this last problem, common to a large class of both non-structural and structural elements that can lose their functionality because of earthquake motions. The results of a numerical modelling of sliding and rocking motion in presence of both different excitations and mechanical parameters are presented and compared with experimental data performed by the authors. The results developed are in good agreement with the laboratory tests and this assures the reliability of both the analytical procedure and the determination of the parameters involved. A model for a safeguard proposal is provided.

Hugo Aya Baquero^{1}

^{1}Universidad Distrital Francisco José de Caldas, Facultad de Ingeniería, Colombia

The electromagnetic waves applied on surfaces of materials with high density of free charges induce the generation of plasmonic waves both in volume and surface. Plasmonic waves in metals have been widely studied; also, recently plasmons have been investigated in intrinsic and doped semiconductor materials. In this work, plasmonic waves are simulated in a two-dimensional electron gas. Two-dimensional electron gas may be created at the interfaces between GaAs and AlxGa1xAs. The plasmons propagation speed depends on Fermi velocity and Fermi vector of the two-dimensional electron gas. The 2D plasmons dispersion relations deviate from light dispersion at all frequencies and does not have an asymptotic character as in three-dimensional electron gas; this allows operating plasmonic devices with speeds much lower than the speed of light. At low frequencies the plasmon wave vector is much smaller than the Fermi vector and the geometric capacitance dominates the quantum capacitance. The propagation of plasmons in channels without gate is studied.

H. A. acknowledges partial financial support of Facultad de Ingeniería de la Universidad Distrital through Proyecto Curricular de Ingeniería Electrónica

Gustavo Fatigate^{1}
, Ruy Reis^{2}
, Rafael Neves^{3}
, Eliezer Mafra^{4}

^{1}IFSULDEMINAS, Department of Physics, Brazil

^{2}UFJF, , Brazil

^{3}IFSULDEMINAS, , Brazil

^{4}IFSULDEMINAS, Computer Engineer, Brazil

Cancer is a worldwide health issue responsible by 8.8 millions of deaths according to World Health Organization. The most common treatments are chemotherapy and radiotherapy. A new approach has been developed aiming to improve the efficiency of the most commons therapies, the hyperthermia. This technique consists in a magnetic nanoparticles injection submitted to an alternating magnetic field of low frequency. This procedure increases the tissue temperature up to $43^oC$ inducing cellular necrosis. This research aims to maximize the tumor tissue death and to minimize healthy tissue death, searching the optimized positions to inject the magnetic nanoparticles. The optimization is carried out using the Pennes model, responsible by modeling heat transfer in live tissue, in conjunction with a genetic algorithm which searches injection proper position. All the tissue parameters were obtained from the literature and the numeric method of finite differences was utilized. The space convergence was quadratic, $O(\Delta^2_x)$, and the time convergence linear, $O(\Delta_t)$. Spacial discretization used a centered difference while the temporal used a progressive difference.The simulation presented the algorithm convergence to the death of the whole tumor tissue and about $6\%$ of the health tissue.

Sultan Alpar^{1}

^{1}al-Farabi Kazakh National university, Department of Mechanics and Mathematics, Kazakhstan

Alpar Sultan^{1}, Yekaterina Moisseyeva^{1}, Dauren Zhakebayev^{1}

^{1}al-Farabi Kazakh National university, Department of Mechanics and Mathematics, Kazakhstan

Abstract

Emergencies of natural and anthropogenic origin are often accompanied by the thermal energy release in which a large-scale thermal is formed. Thermal’s high temperature, pressure and complex chemical composition lead to the formation of environmentally hazardous components. The evolution of the thermal is a complex process involving turbulence mixing, combustion of hydrocarbon fuels, and thermal radiation. Soot formation processes are also important, since the presence of soot particles in combustion products increases the emissivity of the flame. The impossibility of experimental measurements of the thermal’s evolution and its interaction with the environment due to the formation and propagation transience necessitates the use of mathematical modelling methods for the short-term prediction of harmful substances’ spread and environmental hazard assessment. A series of numerical studies of two-phase thermals was currently undertaken in [1-3]. The CFD model in [1] was based on the RANS approach, the structure of burning clouds for different Froude numbers was revealed. In [2-3] the large eddy simulation (LES) results were obtained with the use of ANSYS FLUENT and OpenFOAM. Soot formation and oxidation, as well as flame radiation, were evaluated in two buoyant turbulent diffusion flames fuelled by methane and heptane in [3]. For this, three soot models were compared including Moss-Brookes model which demonstrated best agreement with the measurement data.
In this paper, the LES approach is applied for compressible turbulence with the dynamic subgrid Smagorinsky model. Dynamics of the three-dimensional isolated thermal is validated based on the results of [4]. Soot formation and oxidation are described by the Moss-Brookes model. The optical properties of the combustion products of hydrocarbon fuels are described by the weighted-sum-of-gray-gases (WSGGM). The numerical algorithm is based on the Lattice Boltzmann method (LBM) of the second order for space. The model reproduces the main features of the evolution of the buoyant thermal, namely rising and expansion, turbulent mixing and air entrainment, and soot particle distribution. Results demonstrate that the developed mathematical model coupled with the LBM is suited to simulate the three-dimensional dynamics of buoyant two-phase thermals, being able to correctly reproduce the non-equilibrium behavior of gas-particle mixtures with a reduced computational cost with respect to that expected from other computational methods and software.

References

1. G. M. Makhviladze and S. E. Yakush. Modelling of fires following bursts of pressurized fuel tanks. Fire Safety Science – Proc. of the 7th Int. Symp. D. Evans (Ed.), Int Association for Fire Safety Science, 2002, pp. 643–654.

2. A. Snegirev, E. Kokovina, A. Tsoy, J. Harris, T. Wu, The effect of soot modeling on thermal radiation in buoyant turbulent diffusion flames 2016.

3. M. Cerminara, M. Cerminara, T. Esposti Ongaro, and L. C. Berselli, ASHEE-1.0: a compressible, equilibrium–Eulerian model for volcanic ash plumes 2016.

4. A. Brandenburg, Evolution of highly buoyant thermals in a stratified layer 2000

Eren Tosyalı^{1}
, Fatma Aydogmus^{2}

^{1}Istanbul Bilgi University, Vocational School of Health Services, Turkey

^{2}Istanbul University, Physics, Turkey

It is known that Bose-Einstein Condansate is well described by the non-linear Schrodinger equation known as the Gross Pitaevskii Equation (GPE) with the macroscopic wave function which evaluates with time and space. Many studies have been performed on nonlinear properties in Bose-Einstein Condansate. In this study, we present some numerical results of the Gross-Pitaevskii Equation with the external potential under noise. The phase portraits and Poincare sections of the system are simulated numerically both with and without noise.

Koblandy Yerzhanov^{1}

^{1}L.N.Gumilyov Eurasian National University, General and Theoretical Physics, Kazakhstan

Early was proposed the Myrzakulov gravity model was as $F(R,T)$ as generalization of the F(R) and F(T) gravity theories, where T is the torsion scalar. A generalization of this model with scalar fields will allow us to take into account the physics of elementary particles in this model. Was obtained particular cases of some exact solutions of this model with inflation phases in this model.

Sergey Davydov^{1}
, Andrei Zemskov^{2}

^{1}Moscow Aviation Institute (National Research University), Department 311 "Applied software and mathematical methods", Russia

^{2}Moscow Aviation Institute (National Research University), Department 311 "Applied software and mathematical methods", Russia

In this paper, we consider the unsteady problem of the propagation with finite velocity of one-dimensional thermoelastodiffusion perturbations in a homogeneous multicomponent layer. Physical-mechanical processes are described using a locally-equilibrium model of coupled thermoelastic diffusion (the prime denotes the derivative with respect to the dimensionless spatial variable $x$ and the point - to the dimensionless time $\tau$) that includes the equation of motion of the elastic medium, the heat transfer equation and $N$ the mass transfer equations: $${\ddot{u}=u''-b_{u} \vartheta '-\sum _{q=1}^{N}\alpha _{q} \eta '_{q} ,}$$ $$ {\dot{\vartheta }+\tau _{T} \ddot{\vartheta }=k\vartheta ''-b_{T} \left(\dot{u}'+\tau _{T} \ddot{u}'\right)-\sum _{q=1}^{N}\beta _{q} \left(\dot{\eta }_{q} +\tau _{T} \ddot{\eta }_{q} \right) ,}$$ $$ {\dot{\eta }_{q} +\tau _{\eta q} \ddot{\eta }_{q} =D_{q} \eta ''_{q} -\Lambda _{q} u'''-M_{q} \vartheta ''\quad \left(q=\overline{1,N}\right),}$$ At the boundaries of the layer ($x=0$ and $x=1$), surface mechanical, thermal and diffusion effects are specified: $${\left. u\right|_{x=0} =f_{11} \left(\tau \right),\quad \left. u\right|_{x=1} =f_{12} \left(\tau \right),\quad \left. \vartheta '\right|_{x=0} =f_{21} \left(\tau \right),\quad \left. \vartheta '\right|_{x=1} =f_{22} \left(\tau \right),}$$ $$ {\left. \left(\Lambda _{q} u''-D_{q} \eta '_{q} +M_{q} \vartheta '\right)\right|_{x=0} =f_{q+2,1} \left(\tau \right),\quad \left. \left(\Lambda _{q} u''-D_{q} \eta '_{q} +M_{q} \vartheta '\right)\right|_{x=1} =f_{q+2,2}.}$$ The initial conditions are assumed to be zero: $$\left. u\right|_{\tau =0} =\left. \dot{u}\right|_{\tau =0} =\left. \vartheta \right|_{\tau =0} =\left. \dot{\vartheta }\right|_{\tau =0} =\left. \eta _{q} \right|_{\tau =0} =\left. \dot{\eta }_{q} \right|_{\tau =0} \equiv 0.$$ Dimensionless quantities are used above (with the same print, the dimensional values are indicated by asterisk): $$ x=\frac{x_{1} }{L} ,\quad u=\frac{u_{1} }{L} ,\quad \tau =\frac{Ct}{L} ,\quad C^{2} =\frac{C_{1111} }{\rho } ,\quad \alpha _{q} =\frac{\alpha _{11}^{\left(q\right)} }{C_{1111} } ,\quad D_{q} =\frac{D_{11}^{\left(q\right)} }{CL} ,$$ $$\tau _{T} =\frac{Ct_{T} }{L} ,\quad \tau _{\eta q} =\frac{Ct_{\eta }^{\left(q\right)} }{L} ,\quad \Lambda _{q} =\frac{m^{\left(q\right)} n_{0}^{\left(q\right)} D_{11}^{\left(q\right)} \alpha _{11}^{\left(q\right)} }{\rho RT_{0} CL} ,\quad M_{q} =\frac{n_{0}^{\left(q\right)} D_{11}^{\left(q\right)} \ln \left[n_{0}^{\left(q\right)} \gamma ^{\left(q\right)} \right]}{CL} ,$$ $$\kappa =\frac{\kappa _{11} }{\rho c_{0} LC} ,\quad \beta _{q} =\frac{n_{0}^{\left(q\right)} R\ln \left[n_{0}^{\left(q\right)} \gamma ^{\left(q\right)} \right]}{m^{(q)} c_{0} } ,\quad \vartheta =\frac{\vartheta ^{*} }{T_{0} } ,\quad b_{u} =\frac{b_{11} T_{0} }{C_{1111} } ,\quad b_{T} =\frac{b_{11} }{\rho c_{0} };$$ $$f_{1l} \left(\tau \right)=\frac{f_{1l}^{*} \left(t\right)}{L} ,\quad f_{2l} \left(\tau \right)=\frac{Lf_{2l}^{*} \left(t\right)}{T_{0} } ,\quad f_{q+2,l} \left(\tau \right)=\frac{f_{q+2,l}^{*} \left(t\right)}{n_{0}^{\left(q\right)} C} ,\quad l=\overline{1,2}.$$ Here $t$ is time; $u_{1}$ is component of the displacement vector; $L$ is thickness of the layer; $q$ is the component number of the $N$-component medium; $\eta ^{\left(q\right)} =n^{\left(q\right)} -n_{0}^{\left(q\right)}$ is increment of concentration; $n_{0}^{\left(q\right)}$ and $n^{\left(q\right)}$ are initial and actual concentrations; $t_{T} $ is the time of thermal relaxation; $t_{\eta }^{\left(q\right)}$ is the time of diffusion relaxation; $C_{1111}$ is the elastic constant; $\rho$ is the medium density; $b_{11}$ is the temperature constant characterizing thermal deformations; $\alpha _{11}^{(q)}$ is the coefficient characterizing the volumetric change of the medium due to diffusion; $D_{11}^{(q)}$ is the coefficient of self-diffusion; $m^{(q)}$ is the molar mass; $R$ is the universal gas constant; $\vartheta ^{*} =T-T_{0}$ is temperature increment; $T_{0}$ and $T$ are initial and actual temperatures; $\kappa _{11}$ is the coefficient of thermal conductivity; $\gamma ^{\left(q\right)}$ is the activity coefficient; $c_{0}$ is the specific heat capacity at constant concentration and deformation. The solution is sought in the form of convolutions by time of Green's functions and boundary conditions. To find the Green's functions, we use the Laplace transformation in time domain and the expansion into Fourier series in spatial coordinate domain. As a result of the corresponding transformations, the transformants of the unknown functions are expressed as rational fractions with respect to the Laplace transformation parameter. This means that their originals are found using known theorems and tables of operational calculus. This approach allows us to minimize the using of numerical algorithms and analyze the obtained solution.

The reported study was funded by RFBR according to the research projects № 18-31-00437 and № 17-08-00663.

Sergey Gladkov^{1}

^{1}Moscow aviation institute (national research university) (MAI), therd Faculty, Russia

The stationary speed of the aerosol particle entrained by the positive viscous flow was calculated. It was shown that in the first approximation in Knudsen number, that speed was proportional to the temperature gradient and inversely proportional to the viscosity of the medium where the particle moved. It was proved that Maxwell slip effect appeared in the second approximation in Knudsen number.

Alexandr Baidganov^{1}
, Tatiana Petrova^{2}
, Olga Baruzdina^{3}
, Olga Maksimova^{4}
, Vladislav Egorov^{5}
, Andrei Maksimov^{6}

^{1}Cherepovets State University, Laboratory of mathematical and computer modelling of nanostructures, Russia

^{2}Southern Federal University , Institute of Mathematics, Mechanics and Computer Sciences them. I.I. Vorovich, Russia

^{3}Cherepovets State University, , Russia

^{4}Cherepovets State University, Laboratory of mathematical and computer modelling of nanostructures, Russia

^{5}Cherepovets State University, Laboratory of mathematical and computer modelling of nanostructures, Russia

^{6}Cherepovets State University, , Russia

Rough surfaces play an essential role in many physical phenomena including wave scattering, friction, adhesion, electrical conductivity, capacitance and heat transport, and in applications ranging from thin films to sensors [1]. Light scatter, which is faster and less invasive than SPM, can also measure surface roughness via the inverse wave scattering method [2]. The study of the surface of samples of sheet metal with polymeric coatings of microscopy showed that it consists of grooves that collect in star-like condensations scattered over the entire surface of the sheet. When comparing images obtained by an optical microscope and the method of scanning probe microscopy, we observed scale invariance.Therefore, the distribution of defects on the surfaces of polymer coatings of a metal sheet is fractal in nature.To construct a fractal model of surface, a three-dimensional anisotropic model based on the Julia set. Obtained fractal dimension determined. The analysis of influence of parameters various fractal functions on a profile of rough surfaces are carried out. The method of determination of parameters of model is given. Values of these parameters for various samples of the coverings of metal rolling received under various conditions of their formation are received. The research of the scattering of electromagnetic waves from various surfaces of sheet metal rolling with polymeric coverings within the received fractal models is conducted. In the frame of the Kirchhoff method, the average coefficient of light scattering, scattering diagrams for the coating surface are numerically calculated for various values of surface roughness and falling angles. The results of the research made it possible to develop a convergence technique for such a metal roll allowing identifying a correspondence between the color and the texture of the sample surfaces. References 1.M. Zamani, F. Shafiei, S. M. Fazeli, M. C. Downer, and G. R. Jafari. Phys. Rev. E 94, 042809 2. G. R. Jafari, S. M. Mahdavi, A. Iraji Zad, P. Kaghazchi, Surf. Interface Anal. 37 (7) (2005) 641-645.

T.O.P. acknowledge thefinancial support by the Russian Science Foundation under grant“Methods of microstructural nonlinear analysis, wave dynamics and mechanics of composites forresearch and design of modern metamaterials and elements of structures made on its base”(No.15-19-10008-P).

Mikhail Nikabadze^{1}
, Armine Ulukhanyan^{2}
, Tamar Moseshvili^{3}
, Ketevan Tskhakaia^{4}
, Nodari Mardaleishvili^{5}
, Zurabi Arkania^{6}

^{1}Lomonosov Moscow State University , Department of Mechanics of Composites of Faculty of Mechanics and Mathematics , Russian Federation

^{2}Bauman Moscow State Technical University, Mathematical Physics, Russia

^{3}Akaki Tsereteli State University, Designe and Technology, Georgia

^{4}Akaki Tsereteli State University, Energy and Telecommunications, Georgia

^{5}Akaki Tsereteli State University , Department of mechanic end Machine Industry, Georgia

^{6}Akaki Tsereteli State University , Department of Mechanic end Machine Industry, Georgia

Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained, as well as for rigid in the transverse direction of thin bodies under a new parameterization of the domain of thin body. The initial boundary value problems for thin bodies were also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the fifth approximation of the micropolar theory of five-layer thin prismatic bodies are considered in more detail.

this research were supported by the Shota Rustaveli National Science Foundaiton (project no. DI-2016-41).

Seiki Saito^{1}
, Hiroaki Nakamura^{2}
, Keiji Sawada^{3}
, Masahiro Kobayashi^{4}
, Gakushi Kawamura^{5}
, Masahiro Hasuo^{6}

^{1}National Institute of Technology, Kushiro College, Creative Engineering, Japan

^{2}National Institute for Fusion Science, , Japan

^{3}Shinshu University, , Japan

^{4}National Institute for Fusion Science, , Japan

^{5}National Institute for Fusion Science, , Japan

^{6}Kyoto University, , Japan

Because plasma particles are neutralized by the recycling process on the divertor and the first wall of magnetic confinement fusion devices, many neutral particles of fuel exist in the edge plasma. These neutral particles affect the plasma parameters such as the electron and ion temperature and the plasma density by the interaction with plasma such as ionization, charge exchange, and recombination processes. Moreover, the temperature of the neutral particles affect the electron temperature of detached plasma near the divertor. Therefore, the understanding of the transport of the neutral particles in the plasma is one of the critical issue for the achievement of effective fusion devices. The transport of the neutral particles can be numerically simulated by neutral-transport code. However, when neutral-transport code is performed, the parameters of emitted neutral particles such as the energy distribution and the form (atom or molecule) of emitted neutral particles are usually decided artificially without logical reason because they are not clearly obtained from experimental results. The effects of neutral particles on plasma parameter may depend on the parameters emitted neutral particles from the plasma facing material. In this paper, therefore, we perform molecular dynamics simulation to investigate the energy distribution and ratio between number of emitted atoms and molecules especially for hydrogen. The energy distribution is separately investigated for translational, vibration, and rotation energy of emitted hydrogen.

This work is performed with the support and under the auspices of the NIFS Collaboration Research program (NIFS16KOAP031).

Andrey Petrov^{1}

^{1}Don State Technical University, Research and Education Center "Materials", Russia

Studying wave propagation in poroelastic bodies subjected to dynamic loading is important in many engineering applications. The results of such studies are used to develop methods of non-destructive check, in soil structure analysis and earthquake seismology. Very often, an analytical solution to wave propagation problems is only possible in some special cases and for particular kinds of boundary conditions, so that numerical approximation methods, such as the boundary element method (BEM), have to be used. The BEM is especially suitable for wave propagation problems since it requires the formulation of the problem only along the boundary and produces highly accurate solutions. In BEM-modeling of dynamic processes, three main approaches can be conventionally discerned: solving in time domain [1], solving in Laplace or Fourier transforms with the subsequent inversion of the transforms [2], and the dual reciprocity approach [3]. The numerical accuracy of the time-domain BEM with time-stepping discretization is strongly influenced by the time step size. In particular, the time-marching process becomes numerically unstable when using collocation methods where the time step size is smaller than the element size. To overcome this difficulty, a number of stabilization techniques were developed [4, 5]. However, the time-domain approach cannot be used to solve wave propagation problems for poroelastic media due to the absence of fundamental solutions in time. Methods working in the frequency domain also have their limitations, as they require efficient numerical techniques for inverse transforms and are applicable only to problems for which the correspondence principle is valid. In 1988, Lubich [6, 7] introduced the convolution quadrature method (CQM) to discretize the convolution integral. It gained significant interest as a technique (CQ-BEM) for applying the BEM to time-dependent problems where classical time-step schemes suffer from instability and numerical damping [8-11]. Later, it was shown in several studies that the CQ-BEM based on implicit Runge–Kutta provides better accuracy than that based on the linear multistep method [12-16]. Also, Banjai and Sauter reformulated CQ-BEM approach to solve decoupled problems in Laplace domain that works as a transformation method [17]. Retaining the numerical stability properties of the original method, this approach has the time step size as the only parameter determined by the physical parameters of the problem. The applicability of the reformulated CQ-BEM to poroelastic problems was demonstrated by Schanz, but only using the linear multistep method [18]. The present study presents a modification of such a time-step scheme on the nodes of Runge-Kutta methods, exemplified by 2- and 3-stage Lobatto method. The time-step BEM scheme is based on the stepped method for numerically inverting Laplace transforms. This method is similar in its formulation to the CQM but, in contrast to it, is based on the theorem of operational calculus for integrating in time-domain. The application of the scheme to problems of wave propagation in partially saturated media is considered. [1] W.J. Mansur, A Time-Stepping Technique to Solve Wave Propagation Problems Using the Boundary Element Method (UK, University of Southampton, 1983) [2] T.A. Cruse, F.J. Rizzo, J. Math. Anal. Appl. 22, 244 (1968) [3] Aliabadi MH (2002) The boundary element method: applications in solids and structures, vol 2. Wiley, London [4] Y. Guoyou, W.J. Mansur, J.A.M. Carrer, L. Gong, Stability of Galerkin and collocation time domain boundary element methods as applied to the scalar wave equation, Comput. Struct. 74 (2000) 495–506. [5] T. Ha-Duong, B. Ludwig, I. Terrasse, A Galerkin BEM for transient acoustic scattering by an absorbing obstacle, Internat. J. Numer. Methods Engrg. 57 (2003) 1845–1882. [6] C. Lubich, Convolution quadrature and discretized operational calculus I, Numer. Math. 52 (1988) 129–145 [7] C. Lubich, Convolution quadrature and discretized operational calculus II, Numer. Math. 52 (1988) 413–425. [8] M. Schanz, H. Antes, Application of Operational quadrature methods in time domain boundary element method, Meccanica 32 (1997) 179–186. [9] A.I. Abreu, J.A.M. Carrer, W.J. Mansur, Scalar wave propagation in 2D: a BEM formulation based on the operational quadrature method, Eng. Anal. Bound. Elem. 27 (2003) 101–105. [10] Ch. Zhang, Transient elastodynamic antiplane crack analysis of anisotropic solids, Int. J. Solids Struct. 37 (2000) 6107–6130. [11] T. Saitoh, S. Hirose, T. Fukui, Development of an operational quadrature boundary element method for 3-D wave analysis and its application to ultrasonic simulation, in: Proceedings of the RSID5, 2006 (CD-ROM). [12] C. Lubich, A. Ostermann, Runge–Kutta methods for parabolic equations and convolution quadrature, Math. Comp. 60 (1993) 105–131 [13] L. Banjai, M. Messner, M. Schanz, Runge–Kutta convolution quadrature for the boundary element method, Comput. Methods Appl. Mech. Engrg. 245–246 (2012) 90–101. [14] M.P. Calvo, E. Cuesta, C. Palencia, Runge–Kutta convolution quadrature methods for well-posed equations with memory, Numer. Math. 107 (2007) 589–614. [15] L. Banjai, C. Lubich, An error analysis of Runge–Kutta convolution quadrature, BIT 51 (2011) 483–496. [16] L. Banjai, C. Lubich, J.M. Melenk, Runge–Kutta convolution quadrature for nonsectorial operators arising in wave propagation, Numer. Math. 119 (2011) 1–20. [17] Banjai L, Sauter S (2008) Rapid solution of the wave equation in unbounded domains. SIAM J Numer Anal 47(1):227–249 [18] Schanz M (2010) On a reformulated convolution quadrature based boundary element method. CMES Comput Model Eng Sci 58(2):109–128. doi:10.3970/cmes.2010.058.109

The research was carried out under the financial support of the Russian Scientific Foundation (project no. 15-19-10056).

René Verreault^{1}

^{1}Université du Québec à Chicoutimi, Fundamental Sciences, Canada

It is customary to describe the behaviour and stability of oscillators with the help of phase space representation. However, two-dimensional (2D) oscillators like the Foucault pendulum call for a 4D phase space that is not simple to visualize. Applying celestial body perturbation theory to the Foucault pendulum in his doctor dissertation, Nobel laureate Kamerlingh Onnes showed that the essential features of a Foucault pendulum are its inherent circular and linear anisotropies. A spherical differential 2D sub-space can be defined, where the group of the points of a spherical surface with respect to the operation rotation about a diametral axis is isomorphic with the group of sequential states of oscillation of a 2D pendulum with respect to the operation translation in time. Any Foucault pendulum is then characterized by two elliptical eigenstates which are represented by the poles of that rotation axis on the so-called anisosphere. Such poles play the role of attractor/repellor when “dichroic” damping is present. Moreover, they move drastically within a meridian plane when nonlinear restoring torque giving rise to Airy precession occurs. The concept of anisosphere constitutes a very powerful tool for analysing and optimizing actual Foucault pendulum implementations. That feature is illustrated by a numerical model.

René Verreault^{1}

^{1}Université du Québec à Chicoutimi, Fundamental Sciences, Canada

The residual anisotropy of a real Foucault pendulum is responsible for an oscillatory behaviour of the precession angle according to the original description given by Kamerlingh Onnes in his dissertation. Using the anisosphere model, it is shown that the oscillation in precession of a Foucault pendulum and the oscillation of many types of torsional balances are both driven by an extremely shallow gravitational potential wells. Such systems are therefore rendered sensitive to minute horizontal gravitational perturbations. In particular, the periodic apparent motion of the Moon and other celestial bodies can modify the anisotropy characteristic of the precession potential well. This may provide an explanation for the similar responses to Moon position of the paraconical pendulum of Allais during the 1950’s and of the torsion pendulum of Saxl and Allen in 1971.

Sergey Gladkov^{1}
, Sophie Bogdanova^{2}

^{1}Moscow aviation institute (national research university) (MAI), therd Faculty, Russia

^{2}Moscow Aviation Institution (National Research University), Applied Software and Mathematical Methods, Russia

- By the example of two coplanar metal pendulums whose suspension centers are at the same horizontal level at a fixed distance from each other, it is shown the principal possibility of its synchronization due to the effect of electromagnetic interaction between them. It is obtained that a system of nonlinear dynamical equations of motion where the effective of maximum interaction is taken into account when of chaotic approach of pendulums. It is given analytical estimate of synchronization time , provided for numerical solution of obtained equations. It is shown their agreement with the experimental results. It is noted that similar phenomenon’s have clear-cut diffusive character and they are written in a language of diffusion equation.

Sergey Gladkov^{1}

^{1}Moscow aviation institute (national research university) (MAI), therd Faculty, Russia

The paper suggests an alternative approach devoted to the derivation of the equations of the theory of gravitation, the electromagnetic field and other fields, taking into account their interaction with each other. The theory is based on the calculation of the potential and kinetic energy of the surface of a five-dimensional hypermembrane, where the vector of its displacement , associated with fields of purely physical nature and it plays the role of the fifth dimension

Halil Ozer^{1}

^{1}Yildiz Technical University, Mechanical Engineering, Turkey

Adhesive bonding has been widely used in many areas because of having advantages over the mechanical fasteners. Functionally modified adhesive joints provide greater opportunities in reducing the stress concentration and increasing the joint strength. In this study, the variational iteration method is employed to solve the nonlinear boundary value problems arising in functionally graded adhesive joints. The variational formulations were obtained by following the rules of the calculus of variations. Illustrative example has showed that the method is a powerful mathematical tool for modelling and solving the problem studied.

Valery Serov^{1}

^{1}University of Oulu, Mathematics, Finland

This work studies the inverse fixed energy scattering problem for the generalised nonlinear Schrodinger operators in two- and three-dimensional cases. We prove that in a three-dimensional case the unknown compactly supported generalized nonlinear potential (with some restriction for this potential) from $L^2$ space can be uniquely determined by the scattering data with fixed positive energy (meaning that we have the knowledge of the scattering amplitude with fixed non-zero spectral parameter). While in a two-dimensional case we may prove that all singularities of the unknown compactly supported generalized potential belonging to $L^2$ space can be uniquely determined by the scattering amplitude with fixed positive energy. The results are based on the new estimates for the Faddeev Green's function in $L^{\infty}(R^n), n=2, 3$. These results may have applications in nonlinear optics for the saturation model. In particular, the constant coefficients of this model can be uniquely reconstructed by the scattering data with fixed energy. The main achievement in two-dimensional case is the computation of the Born approximation which is carried out using the total variation regularization method (with numerical examples).

Sergei Sidorov^{1}
, Alexey Faizliev^{2}
, Fyodor Smolov^{3}
, Elena Korotkovskaya^{4}
, Andrew Vlasov^{5}

^{1}Saratov State University, Mechanics and Mathematics Department, Russia

^{2}Saratov State University, Risks institute, Russia

^{3}Saratov State University, , Russia

^{4}Saratov State University, , Russia

^{5}Saratov State University, , Russia

This paper examines the problem of choosing the optimal portfolio for an investor with asymmetric attitude to gains and losses described in the cumulative prospect theory (CPT) of A.~Tversky and D.~Kahneman. We consider the problem of finding the portfolio that maximizes CPT-utility over the set of all feasible portfolios along the mean--variance efficient frontier under some conditions on the stochastic behavior both of the portfolio price and the discount factor. It should be noted that since the value function of the CPT-utility is S-shaped and not-differentiable in the origin, numerical approach to the optimization of CPT-utility is not trivial. We introduce a model with stochastic behavior of both the portfolio price and a discount factor. Following routing procedure for Black-Scholes environment we find the closed-form solution for CPT-utility in a simple particular case. Nevertheless, it turned out that in the general case the closed-form solution can not be obtained. Therefore, we use computational methods for studying the properties of CPT-utility. The paper shows that this approach to the optimal portfolio choice for CPT-investor might be promising.

This work is supported by RFBR (grants 18-37-00060)

Vladislav Kozhukhov^{1}

^{1}Moscow State University, Theory of Elasticity, Russia

This paper focuses on modeling the bending collapse behavior of thin-walled circular tubes subjected to three-point bending load. To study this problem, we used the experimental approach and found a non-monotonic dependence of the breaking load on the tube diameter. This allows us to consider and experimentally solve the following optimization problem: for a pipe of constant mass and length to find the diameter (and hence the wall thickness) at which the breaking load will be maximum. Non-monotonicity is the result of the interaction of two processes of destruction: local buckling of thin-wall tube and a bending failure of thick-wall tube (for small tube diameters). The experimental results allow us to draw useful conclusions concerning the construction of thin-walled structures.

Vladislav Egorov^{1}
, Olga Maksimova^{2}
, Alexandr Baidganov^{3}

^{1}Cherepovets State University, Laboratory of mathematical and computer modelling of nanostructures, Russia

^{2}Cherepovets State University, Laboratory of mathematical and computer modelling of nanostructures, Russia

^{3}Cherepovets State University, Laboratory of mathematical and computer modelling of nanostructures, Russia

The Wang-Landau (WL) algorithm has been successfully applied to study the different types of phenomena, including statistical physics models, protein molecules (2017 J. Phys.: Conf. Ser. 905 012016), complex fluids and polymers (2013, Polym. Sci. Ser. C, 55, 23-38). The main advantage of this algorithm is that thermodynamic state functions (free energy, entropy. etc.) can be calculated by means of one simulation in a wide range of temperatures. In addition, the algorithm allows visiting all the accessible phase states much faster than the Metropolis algorithm. The WL algorithm performs a random walk to obtain the density of states. Often we need probability distribution not only of energy but also of other parameters such as magnetization. In that case, a two-dimensional or joint density of states (JDOS) should be estimated by random walk both in the energy and in this parameter. However, the calculation of JDOS by the WL algorithm requires huge computational time. There are some modifications of the WL algorithm to build JDOS: global updates (2006, Phys. Rev. Lett., 96, 120201), the two level method (2016, J Stat Phys, 163, 197–209), the transition matrix algorithm (2014, J. Phys.: Conf. Ser., 487, 012002). In this work, we employ the modified parallel WL algorithm for JDOS, which combines global updates and two level method. This algorithm is represented by the three-dimensional Ising and Potts models.

The work is performed within the framework of the project “Methods of microstructural nonlinear analysis, wave dynamics and mechanics of composites for research and design of modern metamaterials and elements of structures made on its base” (grant №15-19-10008-П of by the Russian Science Foundation).

CESAR FERNANDES^{1}

^{1}FEDERAL INSTITUTE OF SCIENCE AND TECHNOLOGY OF SÃO PAULO, MECHANICS, Brazil

Albert Einstein Theory of General Relativity in 1916 described a universe through its equations, where space and time are dynamical. Einstein suggested that matter and energy change the geometry of space-time through the effect of gravity. The velocity of gravity is one of the great mysteries of nature, and one of the most difficult to be measured, because when trying to obtain this value through observation of astrophysical objects, it is almost impossible to isolate the speed of gravitation and the transport of information. The LIGO (Laser Interferometer Gravitational-Wave Observatory) project made it possible to prove the existence of gravitational waves. To simulate the effects of gravitational waves, solutions for the generation and detection of electromagnetic signals will be integrated. The development of the detection and transduction system is the focus of this research project where 3 sapphire bars will be magneticly suspended in vacuum at a distance of 10 meters between them at a temperature of 4 K. The central Sapphire will be the receiver of the gravitational signals, while the other two sapphire bars will excited throught a PZT (piezoelectricity) system. Sapphire was chosen as a component of this solution because it has a monocrystalline form of aluminum oxide (Al2O3) with ideal characteristics, especially for the propagation of sound, and consequently suitable for the detection of gravitational signals generated by the system.

My advisor Professor Carlos Frajuca, thank you very much for the support, intelligence, conversations, encouragement, help, attention, guidance, contributions and above all your confidence, which were fundamental in the elaboration of this work, being always available throughout the research. My co-advisor Professor Dr. Garabed Kenchian, for the orientations and contributions of this work.

Mikhail Nikabadze^{1}
, Armine Ulukhanyan^{2}

^{1}Lomonosov Moscow State University , Department of Mechanics of Composites of Faculty of Mechanics and Mathematics , Russian Federation

^{2}Bauman Moscow State Technical University, Mathematical Physics, Russia

The representations of some differential operators, the system of equations of motion, the heat flow equations, the constitutive relations of the micropolar elasticity theory and the Fourier thermal conductivity law under a new parametrization of the thin body domain are given. The corresponding relations in the moments of the unknown quantities with respect to the Legendre polynomials are obtained. A system of equations in displacements and rotations in moments for any anisotropic material is derived. Interlayer contact conditions are written out under certain connections of adjacent layers of a multilayer body. The initial-boundary value problems are formulated. The tensor-block operator of cofactors to the tensor-block operator of the equations of motion in displacements and rotations for isotropic homogeneous materials as with a center of symmetry as well without a center of symmetry is constructed. It allows us to decompose the equations and, separately, to obtain the equations with respect to the displacement and rotation vectors. Static boundary conditions for bodies with a piecewise-plane boundary are also decomposed. From the decomposed equations of the micropolar theories of elasticity, the corresponding decomposed equations of the static (quasistatic) problem of the theories of constant thickness prismatic bodies in displacements and rotations are obtained. From the latter systems of equations it is derived the equations in the moments of unknown vector functions with respect to any systems of orthogonal polynomials. As a special case, the system of equations of the ninth approximation in moments with respect to the system of Legendre polynomials is obtained, which splits into two systems. One of them is a system with respect to the moments of even orders of an unknown vector function, and the other is a system with respect to moments of odd orders of the same functions. Based on the obtained operator of cofactors to the operator of any of these systems, an elliptic equation of high order (the system order depends on the order of approximation) is obtained for each moment of the unknown vector function. The characteristic roots of these equations can be easily found. Using the Vekua method for solving such equations, we can find their analytical solution.

this research was supported by the Shota Rustaveli National Science Foundaiton (project no. DI-2016-41).

Gerontiy Sakhvadze^{1}
, Mikhail Nikabadze^{2}

^{1}Blagonravov Institute of Machines Science, Russian Academy of Sciences, Moscow, Russia , laboratory for research of biomechanical systems , Russia

^{2}Lomonosov Moscow State University , Department of Mechanics of Composites of Faculty of Mechanics and Mathematics , Russian Federation

Two-sided laser shock processing (TSLSP) is often employed to improve the surface quality of thin section components, which can reduce excessive plastic deformation induced by one-sided laser shock processing. Residual stresses (RS) field induced in a thin Ti–6Al–4V alloy plate by TSLSP was investigated through numerical simulation. The multiple TSLSP impacts and the increasing laser shock pressure were found to have significant effects on the RS field. Microhardness distribution on the section of specimen, calculated from RS by Carlsson-Larsson model, is analyzed.

the work is financially supported by the Ministry of Education and Sciences of the Russian Federation within the federal target program “Studies and Developments of Promising Trends in Russian Science and Technology Sector for 2014–2020”. Subsidy agreement no. 14.607.21.0191 of September 26, 2017, project RFMEFI60714X0191.

Vitaly Silchev^{1}

^{1}National Research University Higher School of Economics, Faculty of Business and Management, Russia

This paper proposes a simple econophysical model of stock market as an open nonequilibrium system in form of Lorenz–Haken equations with random intensity of external information pumping as one of the parameters. We present solution of the model in adiabatic approximation. This model explains the impossibility of existence of an equilibrium state of the market and shows the presence of deterministic chaos, fractality, long memory and 1/f noise in a stock market.

Mikhail Nikabadze^{1}
, Andrey Khizhenkov^{2}

^{1}Lomonosov Moscow State University , Department of Mechanics of Composites of Faculty of Mechanics and Mathematics , Russian Federation

^{2}Bauman Moscow State Technical University , Basic Sciences, Russia

We consider several questions about the new parameterization of three-dimensional thin body with one small size. The vector parametric equations of the multilayered thin body are given. The geometric characteristics are considered. The representations of several differential operators, the equations of motion, the boundary conditions and the constitutive relations of classic and micropolar theories of elasticity are given. Several interlayer contact conditions are obtained. The definition of the k-order moment of tensor quantity with respect to the system of Legendre polynomials is given. We have obtained the formulations of the problems of the classic and micropolar theories of elasticity for one-, two- and three-layered prismatic thin bodies with respect to the moments of displacement and rotation vectors. The problems of one-, two- and three-layered two-dimensional rectangular cantilever plate and plate with two pinched edges under the distributed load are solved.

This work was supported by the Shota Rustaveli National Science Foundation (project no. DI-2016-41).

Mikhail Nikabadze^{1}
, Maxim Bogatyrev^{2}

^{1}Lomonosov Moscow State University , Department of Mechanics of Composites of Faculty of Mechanics and Mathematics , Russian Federation

^{2}Lomonosov Moscow State University , Department of Mechanics and Mathematics , Russia

The new parametrization of the three-dimensional thin domain of an arbitrary body of revolution is considered, consisting in using several base surfaces in contrast to the classical approaches. The vector parametric equation is given. The geometric characteristics inherent to the new parameterization are determined. Expressions for the translation components of the unit tensor of the second rank and also relations connecting different classes of bases and geometric characteristic generated by them are written. Using the basic recurrence formulas for Legendre polynomials, several additional relations, which play an important role in the construction of various versions of the theory of thin bodies, have been obtained. The determination of the moment of the tensor quantity, as well as their derivatives and repeated derivatives, is given. The moments of the k-th order of several expressions with respect to the Legendre polynomials are written out. Several representations of differential operators, systems of motion equations, heat inflow equations, the constitutive relations of the micropolar theory of elasticity, the Fourier’s law of thermal conductivity and boundary conditions of the physical and thermal contents under the new parameterization of a thin body of rotation are obtained. In addition, from the presented relations in turn, the corresponding relations are obtained in the moments of the unknown quantities with respect to the Legendre polynomials. The statements of initial-boundary value problems in moments are given. As special cases cylindrical and spherical bodies are considered.

this research was supported by the Shota Rustaveli National Science Foundaiton (project no. DI-2016-41).

JIANGANG QI^{1}

^{1} Shandong University at Weihai, School of Mathematics and Statistics, China

This talk is mainly concerned with Lyapunov-type inequalities for nonlocal Sturm-Liouville problems with nonlocal potentials: $$ -y''(x)+v(x)y(1)=w(x)y(x),\ y(0)=0=y'(1)+\int^1_0 v(t)y(t) dt. $$ Such kind of nonlocal operators appears in quantum mechanics with point interaction, nonlocal continuum mechanics, diffusion processes, voltage-driven electrical systems, and population dynamics. Applying the spectral theory of this king of operators, Lyapunov–type inequalities are set up. With applications we study the infimum problem of the $L^1$-norm for the local potentials in the nonlocal problem $$ -y''(x)+q(x)y(x)+v(x)y(1)=\lambda y(x), $$ with boundary conditions $$ y(0)=0=y'(1)+\int^1_0 v(t)y(t) dt $$ if the first eigenvalue is known. We give the quantity of the infimum explicitly in terms of the first eigenvalue and specify where the infimum can be attained.

This research was partially supported by the NNSF of China (Grant 11771253)

Canan Bozkaya^{1}

^{1}Middle East Technical University, Department of Mathematics, Turkey

A numerical investigation of free convection flow in a corrugated cavity filled with a fluid-saturated porous medium in the presence of uniform inclined magnetic field is performed. The steady, viscous, incompressible flow inside the porous medium is assumed to obey Darcy law. The fluid physical properties are constant except the density in the body force term which is treated according to Boussinesq approximation. The fluid and porous medium are in thermal equilibrium, and the radiation heat flux in $y$-direction is considered negligible compared to that in $x$-direction. The governing equations subject to appropriate boundary conditions are solved by using the dual reciprocity boundary element method (DRBEM) which transforms the differential equations into equivalent boundary integral equations by treating the non-homogeneity through a radial basis function approximation. The stream function equation is solved by DRBEM with the fundamental solution of Laplace equation in the usual way, while the DRBEM with fundamental solution to the modified diffusion term by deriving the corresponding radial basis functions is employed for the solution of the energy equation. A parametric study illustrating the influence of the Hartmann number, the Rayleigh number, the magnetic inclination angle and the radiation parameter on the flow and heat transfer characteristics such as the streamlines, isotherms and the average Nusselt number is carried out. It is observed that the heat transfer rate is enhanced with an increase in the Rayleigh number, radiation and wavelength parameters whereas the Hartmann number and inclination angle of the magnetic field do not have significant effects on the temperature distribution.

Manan Dholakia^{1}

^{1}Indira Gandhi Centre for Atomic Research, Materials Science Group, India

In this work, we study the displacment cascades in Y2O3 single crystal using molecular dynamics (MD) and analyse the disorder in the systems using topological methods. Y2O3 is an important material for the nuclear industry, as the nanoparticles of Y2O3, dispersed in the steel matrix provide the radiation resistance to the oxide dispersion strengthened (ODS) steels. However, the constant irradiation can lead to amorphization of the nanoparticles, thus deteriorating their desirable properties. A proper understanding of the underlying processes, such as displacement cascades is the only way to make better materials. Since displacement cascades occur over atomic length and time scales, MD emerges as the most efficient computational technique to study the underlying processes. To simulate the displacement cascades, a Y atom with an energy 8 keV is taken as the PKA. 100 different runs for primary kncok-on atom (PKA) directed along different directions as well as using different random seeds are generated so as to acquire good statistics. Studies of disorder by melting the system at high temperatures are also carried out. The disorder in the system is analysed using radial distribution function (RDF), mean square displacement (MSD), point defects analysis and by using rings statistics which is a manifestation of Graph theory and Topology. The studies show that a radiation damaged and a melted system can have very different topologies, and this difference can be effectively discerned by a topology based analysis. For the given energy, Y2O3 is shown to be amorphization resistant as it quickly reorganizes itself as soon as the cascade subsides.

Victoria Titova^{1}
, Stanislav Boronovskiy^{2}
, Jean-Pierre Mazat^{3}
, Stephane Ransac^{4}
, Yaroslav Nartsissov^{5}

^{1}Institute of cytochemistry and molecular pharmacology, Mathematical modeling and statistical analysis of results, Russia

^{2}Institute of cytochemistry and molecular pharmacology, Department of mathematical modeling and statistical analysis of results , Russia

^{3}Laboratoire de métabolisme énergétique cellulaire IBGC – CNRS, , France

^{4}Laboratoire de métabolisme énergétique cellulaire IBGC – CNRS, , France

^{5}Institute of cytochemistry and molecular pharmacology, Department of mathematical modeling and statistical analysis of results , Russia

Cytochrome c oxidase (CcO) is a terminal enzyme complex of the mitochondrial respiratory chain. It catalyzes oxygen reduction to water alongside with cytochrome c oxidation. These reactions are coupled with proton pumping through the membrane from the N-side to the P-side and therefore the membrane potential is generated. In the present work CcO activity was estimated via the computer algorithm based on stochastic approach where the catalytic cycle of enzyme is reported as a set of consecutive transitions between distinct states with the various numbers of metabolites. In the present study the proton channels activity was explored under different pH. The functioning of CcO was investigated with four different schemes, providing various requirements on proton pumping process initiation. In general the pumping protons could be uptaken not only via the D-channel, but also via the K-channel, and the possibility of the such pumping was implemented in the distinct schemes. The increase of uptaken via the K-channel protons flow was observed under pH raising. Furthermore the D- and the K-proton flows were equal under pH 8.5, and significant predominance of K-proton flow was observed under pH from 8.5 to 10. This fact shows that one can observe a change of channel modes under the alkalization of the enzyme environment: the D-channel activity decreases significantly in favor of the K-channel. K-proton flow increased under raising pH, then the peak under pH from 8.4 to 8.6 and further decrease were observed in the model scheme, assuming K-protons transmembrane transfer. The peak of the pumping efficiency was also obtained when pumping of K-protons occurs under the binuclear center full of protons. However the pumping efficiency monotonically increased when three or four protons were present in the binuclear center.

Maria Fedorova^{1}

^{1}Omsk State Technical University, Foreign languages, Russian Federation

The main problem faced by the specialists of education management in determining the goals of regulation and development of students’ research activities is the imperfection of the evaluation mechanisms of scientific training at the university. The model includes the development of motivational, activity and evaluation components. This article focuses on the design of assessment component of students’ research activities. Assessment component of pedagogical system under consideration, i.e. in a technical university, includes the proposed results and the criterion indicators; mechanisms for diagnosis and improvement methods of students’ research activities. As projected results there may be: a model of high school graduate, the place of research competence in it, a model of research competence, a model of the scientific information environment of high school (resource potential). Depending on this, criteria indicators can serve as an increment of competencies in the composition and performance as well as resource potential of students research work. The purpose of the proposed research is to develop a mathematical model of research activities effectiveness of the technical students. This article discusses the possibility of using the conceptual and methodological apparatus of pedagogical qualimetry in studying the process of scientific training of students. In the article the possibilities of using pedagogical qualimetry terms and methodology in the study of students’ research process are observed. Diagnostics technique is offered to evaluate the efficiency of technical university students’ research efficiency. The result and index of students’ research system development are specified. As a result of the experiment, main components of students’ research competence and their weighting values are revealed. Linear mathematical model which allows diagnosing student’s research efficiency is constructed. Minimum (L min = 33.75), maximum (L max = 101.25) and threshold values of diagnostic assessment and their indicators are presented. This model is a basis for the test software which is licensed and can be used further in practice.

Sergey Astafurov^{1}
, Aleksandr Grigoriev^{2}
, Evgeny Shilko^{3}

^{1}Institute of Strength Physics and Materials Science SB RAS (ISPMS SB RAS), , Russia

^{2}Institute of strength physics and materials science SB RAS (ISPMS SB RAS), Laboratory of computer-aided design of materials, Russia

^{3}Institute of Strength Physics and Materials Science SB RAS (ISPMS SB RAS), , Russia

It is well known that the mechanical response of brittle rocks is largely determined by the features of stress state, including the ratio between volumetric and deviator stresses. In particular, uniaxially compressed (unconfined) rock samples are often broken in a brittle manner by dynamically forming and propagating macrocracks. At the same time, even brittle rocks show strongly pronounced inelastic behavior under confined loading, starting with a certain value of the side pressure. In this case, fracture has a "quasi-viscous" character and is associated with the formation of a localized shear band, or (at still greater side pressures) by a volumetric cataclastic flow. Interpretation and generalization of the results of experimental studies of the behavior of confined brittle porous materials are key problems of both rock mechanics and fracture mechanics. Traditionally, the factors determining the inelastic behavior of such materials are subdivided into structural (dispersion and spatial distribution of pores), material (strength characteristics of pore wall material) and mechanical (characteristic values of external pressure). An experimental analysis of the individual contributions of these factors is rather difficult. Therefore, in the present work such an analysis was carried out using the numerical modeling of the axial compression of model porous samples confined at different side pressures. The considered range of side pressure values was limited to a “brittle fracture” region below the brittle-to-ductile transition threshold. The simulation results showed that at enough large confining pressures ensuring inelastic behavior of porous samples under axial compression, the maximum resistance force to compression is not related to the actual fracture of the sample (which is defined as fragmentation, i.e., separation into a set of contacting blocks). Transition from strain hardening to softening of the confined sample occurs well after the actual fracture and is associated with the formation of a localized shear band and its compaction in the process of relative shear displacement of the parts of the sample separated by this band. The results give grounds to assume that experimentally measured values of strength and ultimate strain of confined macroscopic porous samples (these characteristics are determined at the point of maximum resistance force) can be significantly overestimated in comparison with the true values. We studied the possibility of building a unified strain hardening curve describing the inelastic deformation of porous rock samples at various magnitudes of the side pressure. We showed that the use of plastic flow laws with a linear dependence on the magnitude of the first and second invariants of the stress tensor is strongly limited. In particular, the widely used Mises-Schleicher yield criterion can be correctly applied only for describing the initial stage of inelastic deformation of brittle porous materials (up to the stage of formation of internal cracks with a length of up to 10-15 characteristic distances between pores). To determine the boundary between these stages, it is necessary to carry out several experiments at different magnitudes of side pressure. A more complete description of the inelastic behavior of porous brittle materials up to the stage of fragmentation or the onset of strain softening should be performed using nonlinear (for example, elliptic) yield functions. Simulation results also indicate that the values of the specimen strength obtained in standard uniaxial compression, tensile and/or shear tests should be applied with caution as input parameters for a macroscopic description of porous brittle materials under confined conditions. In particular, the example of the Drucker-Prager failure criterion shows that its constants determined from standard engineering tests differ up to 2 times from those actually corresponding to the behavior of the material under mechanical confinement. In order to obtain adequate values of the constants, one should carry out special tests under conditions close to those in which the behavior of the porous material will be analyzed. The obtained results are important for deepening the understanding of the general laws of inelastic deformation and fracture of brittle porous solids (including rocks) under large compressive stresses. The work was carried out within the project of the Comprehensive Program of Fundamental Researches II.1 of SB RAS.

The work was carried out within the project of the Comprehensive Program of Fundamental Researches II.1 of SB RAS.

Assylzhan Kizbayev^{1}

^{1}Al-Farabi Kazakh National University, Mathematical and Computer Modeling, Kazakhstan

Zhansaya Shakhmugambetova^{1}, Assylzhan Kizbayev^{2}, Kairzhan Karzhaubayev^{3}

^{1}Al-Farabi Kazakh National University, Department of Mathematical and Computer Modeling, Kazakhstan

^{2}Al-Farabi Kazakh National University, Department of Mathematical and Computer Modeling, Kazakhstan

^{3}Al-Farabi Kazakh National University, Department of Mathematical and Computer Modeling, Kazakhstan

Abstract

Thermal plume evolution is important task in the field of atmospheric flow research. Depending on the initial parameters of the plume and various meteorological parameters, thermal plume can rise from a few meters up to many kilometers above the ground. Thermal plumes may contain dangerous contaminants like ashes and species. However, even relatively small thermals may have regional impacts, as fine ash is transported far from the base ground by wind and buoyancy forces and thus, could present a major hazard for the environment. Despite a large amount of work and research the dynamics of thermal plumes are still poorly understood. Motivation of this work is development of computational tool for simulation of the dynamics of plume evolution at different initial and meteorological conditions. To this purpose, we developed LES-LBM solver accelerated by the Graphics Processing Unit (GPU) on the CUDA computational platform, integrating LBM with Large Eddy Simulation (LES). Simplicity of coding is usually an appealing feature of the LBM. Conventional implementations of LBM suffer from high memory consumption and poor computational performance. The main advantage of the solvers based on GPU is their ability to perform significantly more floating point operations per unit time (FLOPS) than a Central Processing Unit (CPU) and a good scalability of explicit parallel algorithms. LES-LBM code was tested on the NVIDIA GeForce GTX 1050 ti and NVIDIA TESLA K80 GPUs. The performance of parallel LES-LBM code was compared with similar OpenMP based LES-LBM code. It is found that GPU can significantly accelerate the computational performance as compared to thread-parallel CPU code. Computed plume parameters, such as height, width and radial spreading of the umbrella cloud were compared with experimental studies and demonstrate the great potential of the LBM for simulation of plume dynamics.

References

1. Christian Obrecht, Frédéric Kuznik, Bernard Tourancheau, Jean-Jacques Roux, Scalable lattice Boltzmann solvers for CUDA GPU clusters, Parallel Computing, Vol.139, issue 6-7, 2013, https://doi.org/10.1016/j.parco.2013.04.001

2. Parmigiani, A., Huber, C., Chopard, B. et al. Eur. Phys. J. Spec. Top. (2009) 171: 37. https://doi.org/10.1140/epjst/e2009-01009-7.

3. Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., & Viggen, E. M. (2017). The Lattice Boltzmann Method. Springer.

Vadim Matveev^{1}

^{1}Midas marketing UAB, Vilnius, Research and Development Group, Lithuania

A mental model that on the example of watercrafts traveling at usual speeds in water describes all known kinematic relativistic effects is constructed. The modeled objects conduct themselves in accordance with the formal laws of the special theory of relativity. Simulation of the symmetry in the proposed model makes it possible to simulate kinematic relativistic effects in the water medium in symmetrical relativistic form and to obtain Lorentz transforms. In our mental model individual watercrafts and groups of watercrafts located on the surface of a flat-bottomed water body with a depth of h, filled with slack water, serve as the objects of our conceptual observation. The watercrafts are equipped with hardware components that perform metrological operations. The hardware components have at their disposal sources of sonar signals. The signals travel between the watercrafts and the watercrafts and the bottom. The velocity of the sonar signals equals V and is unapproachable for the other watercraft, i.e., the velocity of the watercrafts, v, correspond to the inequality v < V. Each watercraft is equipped with a clock, during which a sonar signal that continuously travels along a plumb line (relative to this watercraft) between the watercraft and the bottom performs the function of the pendulum. Each sonar signal travel to the bottom and back requires a time of Δt(1) = 2h/Vz , where Vz – the speed of submersion and surfacing of the signal. The sonar clock “mechanism” controls not only the clock’s hands, but also all the watercraft hardware components, thereby ensuring the proportionality of their working speed to the clock’s working speed. As the watercrafts move at a velocity of v, the Vz submersion and surfacing speed of a signal traveling between a watercraft and the bottom along the hypotenuses of right-angled triangles is equal to V/γ, where γ is Lorentz factor. Time on the moving watercrafts, which we named simulated time t', flows more slowly than time t on watercrafts at rest by γ times. The procedure for maintaining distance between group’s watercrafts consists of the following. A sonar signal is dispatched from each watercraft to the neighboring watercraft, and upon reaching it, the signal goes back again. The watercraft’s hardware components, using their clocks, measure the signal’s time of movement to the neighboring watercraft and back, then approach or move away from the neighboring watercraft as necessary in order to maintain this time and the immutability of the sonar “distance”. It is easy to show that the lateral dimensions of the group are retained in this instance; however, the longitudinal dimensions (in the direction of movement) of the watercraft group in motion are contracted by γ times. In order to derive the symmetry of the processes, we simulated relativistic time synchronizing clocks in such a manner that the speeds of movement of a sonar signal in water flow from point, O, to point, x, and back to the point, O, became identical. Thanks to the artificial clock synchronization on the group of moving watercrafts the results of the measurements made by the hardware components on the watercrafts of the groups in motion and at rest become symmetrical. The same thing is true of the distances.

Zhen Zhu^{1}
, Junhao Chu^{2}

^{1}Key Laboratory of Polar Materials and Devices, Ministry of Education, East China Normal University; School of Electronics and Information Engineering, Suzhou Vocational University, , China

^{2}Key Laboratory of Polar Materials and Devices, Ministry of Education, East China Normal University, , China

Polycrystalline silicon thin film transistors with thin body and thin gate oxide can realize high performance and relative low power consumption simultaneously. Considering their wide applications, a surface-potential-based drain current model suitable for devices with the above structure is derived on charge sheet approximation considering the double exponential trap state distribution, the interface charge and the effect of the back surface potential. According to the calculated areal density of the charges under the gate oxide, the ionized acceptors and the trapped charges, the areal density of the inversion charge is obtained. Under several mathematical treatments, a surface-potential-based drain current model suitable for devices with thin body and thin gate oxide is developed accompanying the quantitative model-validity conditions for high and low trap states respectively. Under high and low state densities respectively, this proposed surface-potential-based drain current model is verified by 2D-device simulation in devices’ transfer characteristics under various drain biases in the situations without or with interface charge.

Peter Golubtsov^{1}

^{1}Lomonosov Moscow State University, Faculty of Physics, Russia

In large-scale research, data is usually collected on many sites, have a huge volume, and new data is constantly generated. Since it is often impossible to collect all the relevant data on a single computer, much attention is paid to the algorithms that provide sequential or parallel accumulation of information and do not need to store all the original data. As an example of information accumulation, a Bayesian update for the linear experiment is analyzed. The corresponding $information$ $spaces$ are defined and the relations between them are studied. It is shown that processing can be unified and simplified by introducing a special $canonical$ $form$ of information representation and transforming all the data and the original prior information into this form. Thanks to the rich $algebraic$ $properties$ of the canonical information space, the sequential Bayesian procedure allows various parallelization options that are ideally suited for distributed data processing platforms, such as Hadoop MapReduce. This opens up the possibility of a flexible and efficient scaling of information accumulation in distributed data processing systems.

Nidhal Mahmud^{1}

^{1}University of Hull, School of Engineering and Computer Science, United Kingdom

State transition models are increasingly being used to describe systems exhibiting sequence-dependent failures. Such representations can be observed in influential modelling languages like the Architecture Analysis and Design Language (AADL) in the aerospace industry and the Electronic and Software Technology Architecture Description Language (EAST-ADL) in the automotive domain. In safety analysis, a set of paths leading to a safety-relevant state can be encoded as a standard sum of product canonical form, and without any loss of the significance of the order in which failure events occur. Such expressions can then be reduced by symbolic calculus into minimal canonical forms out of which all minimal failure scenarios in the source model can be generated. The resulting failure sequences will be assessed for quantitative analysis. In this paper, we formally propose and discuss such minimization algorithm. This approach offers some advantages particularly when choosing to synthesize system failure expressions from the reduced expressions local to the system constituents.

Zoja Tunguskova^{1}
, Nelly Ovchinnikova^{2}

^{1}Lomonosov Moskow State University, Faculty of Mechanics and Mathematics, Russia

^{2}Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Russia

The exact analytical solution of the problem of axial shear of a cylindrical layer fixed on the inner surface is obtained. On the outer surface, either axial tangential stress or axial movement is specified. The model of hypoelasticity with objective stress rates of the corotation type is adopted: the Jaumann stress rate and the Green-McInnis-Naghdi stress rate. It is assumed that at any point of the layer only the axial displacement differs from zero, and it depends on the radius. Such a motion is isochoric. The kinematic characteristics of the motion are determined: the affinor, the distortion rate tensor, the strain-rate and rotation-rate tensor, the rotation tensor and the Green tensor. All components of the Cauchy stress tensor are found. It is shown that, for small deformations, the obtained formulas become known, and a numerical-graphical illustration of the solutions obtained is presented. The analytical solution of the problem is compared with the numerical solution obtained in the ANSYS software package for the hypoelastic material model using the Green-McInnis-Naghdi stress rate.

The work is prepared at support of RFBR (grant No. 16-01-00669)

Lyudmila Vshivkova^{1}
, Vitaly Vshivkov^{2}
, Galina Dudnikova^{3}

^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, , Russia

^{2}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, , Russia

^{3}Institute of Computational Technologies SB RAS, , Russia

Under interaction of plasma flows there are electromagnetic waves are generated in the plasma, which can distribute in different directions, interact with the plasma, change in amplitude and in motion direction. To study the appearing electromagnetic waves in the plasma is impossible due to the nonlinear character of such interaction when modeling. In many problems to solve this problem one place a vacuum domain next to the plasma area, where the electromagnetic waves, generated by the plasma, are studied. Waves in vacuum follow to the linear Maxwell equations and, therefore, it is easy to determine their frequencies and amplitudes using the Fourier analysis. To study this matter, it is sufficient to consider the problem in the 2D case. The problem is in the following: at any time $t$, knowing the values of the functions $E_x$, $E_y$, $E_z$, $B_x$, $B_y$ and $B_z$ (electric and magnetic intensities), to investigate what kind of waves are present in the considering domain, i.e. for each wave number $k_x$ and $k_y$ to find the wave amplitude. As all waves are linearly independent, the only waves having the same $|k_x|$ and $|k_y|$ are considered. The 2D Fourier analysis for the functions $E_z$, $B_x$ and $B_y$ gives amplitudes of all harmonics. However, this expansion mixes four waves which have equal $k_x$ and $k_y$ but have the different motion direction. The aim of the current work is the development of the technique to determine directions and amplitudes of all electromagnetic waves which are present in the solving domain at a definite time. The proposed algorithm will be implemented when solving the 2D/3D problem of the laser impulse and plasma interaction.

The work has been supported by the Russian Science Foundation (RSF) under project number 16-11-10028.

Peter Zagubisalo^{1}
, Andrey Paulish^{2}

^{1}Novosibirsk Branch ISP SB RAS “TDIAM”, Technological Design Institute of Applied Microelectronics, Russia

^{2}Novosibirsk Branch ISP SB RAS “TDIAM”, , Russia

The mathematical model of the piezo-optical transducer of a new design for high sensitive compact strain gauges was created. The strain gauge assembly is the transducer inserted into the adaptive element that is mounted on the controlled object. The transducer was theoretically and experimentally studied. The temperature dependences of the transducer parameters were studied by means of the model. Such dependences are common cause of accuracy limitations of strain gauges. The model performed the accurate numerical simulations of the light passage through transducer's optical elements via Mueller calculus and allowed to determine strain and stresses in the photoelastic element via finite element method. Based on the simulation results, the transducer's sensitivity should't depend on temperature. Nevertheless, the output signal of the transducer depends on temperature. It is explained by two factors: a) asymmetry of the geometry of the controlled object, b) inaccurate installation of optical elements simultaneously with change in the peak wavelength of the LED with temperature. So the installation tolerances of the optical elements are determined.

Vladimir N. Barakov, Mikhail A. Pavlov, Alexander V. Poyarkov, Firma PODIY, Ltd, 107084, Moscow, Russian Federation

Sergei Stradomsky^{1}
, Nikolai Ostapenko^{2}

^{1}Lomonosov Moscow State University, Department of Mechanics and Mathematics, Russia

^{2}Institute of mechanics Lomonosov Moscow State University, , Russia

The problem of the Waverider shape of maximum aerodynamic quality, constructed on a plane shock wave and having a symmetry plane, is set and solved for two isoperimetric conditions: the specific volume of the Waverider and the lift coefficient are given. The upper surface of the Waverider is directed along the incoming stream and does not disturb it. The bottom surface consists of straight lines that make up the same angle α with the undisturbed flow. The leading edge is a curve located in the plane of the shock wave generated by the lower surface and making an angle θ with the direction of the oncoming stream. In addition to the pressure in the interaction model of the flow with the surfaces of the Waverider, there is a local coefficient of friction, varying, as on a plate, independently along each chord from the front to the trailing edge. Extremal - the distribution of the chord length of the Waverider in the plan along the span is found using the method of local variations, adapted to the variational problem with two isoperimetric conditions. It is shown that in the case of a Cx and Cy normalized in a special way, the problem reduces to finding the minimum of the functional Cx, depending on four parameters: α, Mach number, C - is the tangent of the angle determining the sloping cut and m is the parameter characterizing the state of the boundary layer. Thus, within the framework of the model used, the extremal does not depend on the Reynolds number, but depends only on the state of the boundary layer. As an initial contour of the Waverider in the plan, a contour consisting of segments of straight lines was adopted, which, as an analysis of the necessary conditions of the extremum showed, is not an extremal under any combinations of the determining parameters. The shape of the optimal Waverider is determined in the absence and presence of restrictions on the length and scale of the Waverider, and also on the heat flux to the leading edge with various combinations of the determining parameters and the states of the boundary layer. It is established that the optimal Waverider can contain lateral washers, and the curved section of the leading edge both in the absence of a restriction on the heat flux, and in its presence contains an inflection point. It is shown that as the angle α is reduced, the aerodynamic quality of the optimal wavelets increases, and the shape changes insignificantly. The angle of the cutoff of the bottom cutoff of the Waverider has little effect on K. The increase in the Mach number while maintaining the state of the boundary layer leads to a significant change in the shape of the optimal Waverider.

Anastasia Garbuz^{1}
, Stanislav Boronovskiy^{2}
, Yaroslav Nartsissov^{3}

^{1}Institute of cytochemistry and molecular pharmacology, Department of mathematical modeling and statistical analysis of results , Russia

^{2}Institute of cytochemistry and molecular pharmacology, Department of mathematical modeling and statistical analysis of results , Russia

^{3}Institute of cytochemistry and molecular pharmacology, Department of mathematical modeling and statistical analysis of results , Russia

Langevin dynamics and molecular dynamics are actively used for studying of the ions transport in biological transmembrane channels. The above techniques yield to achieve the high level of simulation detail that can reveal atomistic and structural aspects of ion translocation. Nevertheless, these time-consuming approaches require powerful computational tools, especially in the case of large biological systems. Thus, the main purpose of our research is development of a predictive model, which can be used for evaluation of ion localization times in different ion channels based only on their structures. In our model we used a non-parametric method which estimates time values for a research subject. The predictions are based on the structural parameters of various ion channels forming feature space. Their contribution to the result depends on the weights, which are obtained by genetic algorithm. Ion localization times corresponding to the points of feature space are results of Langevin dynamics simulation with respect to hydrodynamic and dielectric friction. The transmembrane potential is simulated by the external uniform electric field, which is directed along the channel axis. Serotonin receptor 5-HT3 was selected as a subject of predictive modeling because it still remains poorly understood. The ion channel was divided into the key areas according to the structure of its transmembrane domain. Corresponding ion localization times were obtained at the external potential values ranging from 0 to 200 mV. Amino acid residues Ile267 and Ile268 were proved to have a strong influence on the time of ion translocation through the channel. Weight distribution analysis showed that the local electrostatic potential is the most significant factor in estimation of times values. However, in the case of 5-HT3 receptor the key area volume is another essential feature for model prediction.

Nikolay Voronin^{1}

^{1}A.A. Blagonravov Institute of Machine Science Russian Academy of Sciences , Department of Tribology, Russia

Topocomposites belong to a class of structured materials providing for specific functional characteristics including tribological ones, which differ essentially from those of compact materials of the same origin. The thickness of coatings and layers received by different types of surface treatments cover five orders of magnitude. To specify the surface layer of the material, differing from the base and with thickness from nanometres to several tens of micrometres, the term "thin coating" is used. Separation of films into thin and thick ones is conventional and relative. The term "thin coatings" is often applied to the surface layers and coatings received by vacuum ion- plasma methods, gas-phase methods, sol-gel technology, as well as others. The tribological properties and characteristics of friction with thin coatings are defined mainly by the coating thickness. If the thickness of a coating (modified surface layer) is comparable to the contact area, or is smaller and the ratio of mechanical characteristics of the materials of the layered system components (base and coating) so that external actions are received and localized, not only in the coating material but also in the base material, then such subsurface volume represents the surface layered composite material and is called a topocomposite. This material, as a subject of scientific and engineering interest of western scientific and technical literature, was called surface engineering. The terms of load, the form of bodies in contact, dimensions and mechanical characteristics of the subsurface volume, which include the base material besides the coating material, is decisive while referring to the class of topocomposites and in providing a certain operating capability and tribotechnical characteristics. A correctly formed composite subsurface layer provides to the friction surface and the article in total a technical efficiency and economic expediency which the elements of the layered system do not give separately. It is known that determination of the stress, the deformation state and deformation—strength parameters of contact areas for homogeneous solid bodies are possible as a result of solving the contact problems. This is an urgent problem for tribotechnical topocomposites and it is in the sphere of the mechanics of contact interaction of solid layered bodies. Some intensity of wear, appearance of microcracks, flaking and plastic deformation in the layered surface of the body is connected with the level, type and localization of stresses in the contact area. Although, at present, the theory of contact interaction has achieved significant advances, due to mathematical difficulties, some gaps between the theory and practical application of these solution for certain problems appears, particularly for the interaction of layered bodies, which is considered a nonclassical problem. The analysis of solutions of certain problems for the layered systems known in the literature shows that exact solutions are obtained with the application of numerical methods requiring use of the PC, software and, in some cases, significant expenditures of computer time. Known asymptotic dependencies are applicable separately for thick and thin layers. But they may have wider use if the specific character of the structure of layered systems is considered, as the physical side of solving the problem is no less important than the mathematical. In this work we propose a novel approach to modelling contact problems of the topocomposites. By this approach we have defined analytical dependencies for calculation of contact parameters, effective elastic compliance; effective yield point and bearing capacity of topocomposites are in terms of thickness of a coating, elastic and mechanical properties of base and coating materials. The local reduction of an effective yield point and bearing capacity is established for specific range of thickness of coatings. Existence of regions of abnormal structural strength for elastic - plastic layered medium is shown allow identifying unequivocally top layered elastic - plastic material (the hardened surface of a solid body), as a topocomposite. The method of study of topocomposites represented in this article enables the scientific design of friction units to choose the method of strengthening treatment, the coating and base material, and to specify the reasonable coating thickness and calculate the life cycle of the strengthened parts of friction units to high precision.

This work is supported by financial support of Ministry of education and science RF, Agreement №14.607.21.0166, 26.09.2017, Project RFMEFI60717X0166

Evgeny Moiseenko^{1}
, Roman Butov^{2}
, Nikolay Drobyshevsky^{3}
, Yuri Tokarev^{4}

^{1}Nuclear Safety Institute of the Russian Academy of Sciences (IBRAE RAN), Laboratory of numerical simulation of thermomechanical processes, Russia

^{2}Nuclear Safety Institute of the Russian Academy of Sciences (IBRAE RAN), Laboratory of numerical simulation of thermomechanical processes, Russia

^{3}Nuclear Safety Institute of the Russian Academy of Sciences (IBRAE RAN), Laboratory of numerical simulation of thermomechanical processes, Russia

^{4}Nuclear Safety Institute of the Russian Academy of Sciences (IBRAE RAN), Laboratory of numerical simulation of thermomechanical processes, Russia

Isolation of high-level nuclear waste (HLW) in deep geological repositories is nowadays considered to be the most feasible way of its disposal. Russia, along with other countries like Finland, Sweden, France, Germany, etc., plans the construction of its own repository. One of crucial aspects of repository functioning is the host rock stability, which is governed by the stress-strain state evolution. For one thing, the HLW is a strong source of heat, for another, the repository is subjected to the pressure of large overlying rock mass, thus its state is influenced by the combination of thermal and mechanical loads. 3D repository structure and host rock inhomogeneity, with dykes and cracks, adds to the complexity of the stress-strain state simulation. 3D code FENIA (Finite-Element Nonlinear Incremental Analysis) is being developed as a tool for realistic modeling of thermal and mechanical evolution of underground repositories. It was applied for modeling of thermal state of a prototypic repository, which implements the concept of placing the vitrified HLW in several dozen meters deep boreholes. For such repository configuration, an uneven temperature distribution with sharp 3D gradients is predicted. This will result in considerable shear stress along with compressive one, which can cause crack appearance and propagation. The estimates for effective (von Mises) stress in case of a prototypic repository with respect to realistic thermal and mechanical properties of HLW, engineered barrier system (EBS) and host rock are obtained with the FENIA code. It is shown that in the repository project the distance between the boreholes should be taken with respect to both thermal (maximum HLW, EBS and rock temperature) and mechanical (von Mises stress) acceptance criteria.

Pragya Singh^{1}

^{1}IIT(BHU) Varanasi, Department of Mathematical Sciences , India

The objective of this article is concerned with a study of a static cruciform crack problem in an infinite orthotropic elastic medium opened by prescribed normal pressure distributions subjected to steady state temperature field. The problem is reduced to Fredholm singular integral equations of the first kind, which are solved using Chebyshev polynomials. Analytical expressions of the stress intensity factors and crack energies are found and the numerical computations for the orthotropic elastic material Boron-Epoxy composite for different particular cases are depicted through figures. The striking feature of the article is the graphical presentation of a possibility of crack arrest and variations of crack energies with the variations of the arms of the cruciform crack.

Nikolai Kazmiruk^{1}
, Stanislav Boronovskiy^{2}
, Yaroslav Nartsissov^{3}

^{1}Institute of Cytochemistry and Molecular Farmacology, Mathematical Modeling And Statistical Analysis Of Results, Russia

^{2}Institute of cytochemistry and molecular pharmacology, Department of mathematical modeling and statistical analysis of results , Russia

^{3}Institute of cytochemistry and molecular pharmacology, Department of mathematical modeling and statistical analysis of results , Russia

Phosphate activated glutaminase (GA, E.C. 3.5.1.2) plays a key role in glutamine/glutamate homeostasis in mammalian brain, catalyzing the hydrolytic deamidation of glutamine to glutamate and ammonium ions. GA is mainly localized in mitochondria, where it has the catalytically active form on the inner mitochondrial membrane (IMM) and the other soluble form, which is supposed to be dormant. At present time, the exact localization of the membrane glutaminase active site remains a controversial and an unresolved issue. In our study, we used a multi-compartment kinetic approach to simulate metabolism of glutamate and glutamine in the astrocytic cytosol and mitochondria. A created system includes four compartments: the mitochondrial matrix, the cytosol of the astrocyte, the space of the synaptic cleft around the astrocyte, and the cytosol of the neuron. Metabolic network in these compartments is carried out using key enzymes and carriers. Due to the fact that the exact location of the mitochondrial membrane GA is the subject of a huge discussion, two system configurations have been proposed. In one type of the scheme, glutaminase is located on the inner mitochondrial membrane and its active site faces the matrix. In another configuration, GA is also located on the inner membrane, but its active center faces the intermembrane space. The influx of glutamate into the system was constant, and concentrations of ATP and NH4+ were fixed. We used physiologically important ratio between the concentrations of glutamine inside the matrix of mitochondria [Glnmit] and glutamine in the cytosol [Glncyt] as a marker for precise functioning of the system. Since this ratio directly depends on the mitochondrial glutamine transporter (MGC) flow parameters, key observation was to investigate the dependence of the [Glnmit]/[Glncyt] ratio on the maximal velocity of MGC at different initial concentrations of mitochondrial glutamate. Another important task was to observe the similar dependence at different inhibition constants of the soluble GA, the exact value of which is unknown. The simulation results confirmed the experimental c-side localization hypothesis, in which the glutaminase active site faces the outer surface of the IMM. Moreover, in the case of such localization of the enzyme, a 3-fold decrease in ammonium production was predicted.

Meruert Zhassybayeva^{1}
, Kuralay Yesmakhanova^{2}
, Ratbay Myrzakulov^{3}

^{1}L.N. Gumilyov Eurasian National University, General and Theoretical Physics, Kazakhstan

^{2}Eurasian National University, General and Theoretical physics, Kazakhstan

^{3}L.N. Gumilyov Eurasian National University, General and Theoretical Physics, Kazakhstan

Integrable nonlinear differential equations are an important class of nonlinear wave equations that admit exact soliton solutions. All these equations have an amazing property which is that their soliton waves collide elastically. One of such equations is the (1+1)-dimensional Fokas-Lenells equation. In this paper, we have constructed an integrable (2 + 1)-dimensional Fokas-Lenells equation. The integrability of this equation is ensured by the existence of a Lax representation for it. We obtained its bilinear form by the Hirota method. Using the Hirota method, exact one-soliton and two-soliton solutions of the (2 + 1)-dimensional Fokas-Lenells equation were found.

Integrable nonlinear differential equations are an important class of nonlinear wave equations that admit exact soliton solutions. All these equations have an amazing property which is that their soliton waves collide elastically. One of such equations is the (1+1)-dimensional Fokas-Lenells equation. In this paper, we have constructed an integrable (2 + 1)-dimensional Fokas-Lenells equation. The integrability of this equation is ensured by the existence of a Lax representation for it. We obtained its bilinear form by the Hirota method. Using the Hirota method, exact one-soliton and two-soliton solutions of the (2 + 1)-dimensional Fokas-Lenells equation were found.

Eugen Anitas ^{1}

^{1}Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Russian Federation

An idealized deterministic multifractal-based model is suggested for describing the structure of complex hierarchical materials. The corresponding small-angle scattering form factor is calculated from a system containing a macroscopic number of randomly oriented multifractals with uncorrelated positions. The scattering curve is characterized by the presence of the three main structural regions: Guinier, fractal, and Porod. We show how to extract various structural information of the fractal from the scattering data, such as: overall and fractal dimensions, and the size of the basic objects composing the fractal. The obtained results give additional insights into the structure of multifractal systems at nano- and micro-scales.

Alexey Setukha^{1}
, Anastasia Semenova^{2}

^{1}Lomonosov Moscow State University, Research Computing Center, Russia

^{2}Lomonosov Moscow State University, Computational Mathematics and Cybernetics, Russia

We propose the numerical method to solve the following linear hypersingular integral equation for a function $g(y)$: \begin{equation*} (I\, g) (x) = \int_{\Sigma}g(y)\dfrac{\partial^2 F(x-y)}{\partial n_x\partial n_y}\,d\sigma_y = f(x),\, x\in \Sigma,\, F(x-y) = \frac{1}{|x-y|}, \end{equation*} where $\Sigma$ is a simple smooth surface that can be closed or opened with the boundary $\partial \Sigma$ (in this case the boundary is the closed piecewise smooth curve), $\vec n(x)$ is the unit outward normal vector to the surface $ \Sigma$ at the point $x$, $\partial/\partial n_x$ is the derivative along the vector $\vec n(x)$ and is calculated as a partial derivative of a function depending on $x$ while the variable $y$ is fixed. The integral is considered in the sense of Hadamard finite value. This integral equation arises if one solves the Neumann boundary value problem for the Laplace equation using the boundary integral equation method and representing the solution in the form of a double layer potential [1]. We construct the numerical scheme for the equation based on the piecewise linear approximation of the unknown function and the collocation method with vertices of the triangle cells as the collocation points. The problem is reduced to the system of linear equations which coefficients can be expressed in terms of the integrals over cells of the partition while the integrand can have a strong singularity. These integrals are calculated analytically and therefore we obtain the precise expressions for the coefficients of the system of linear equations. In case of the closed surface we prove the uniform convergence of the numerical solution to the accurate one on the grid as the triangulation size tends to zero. The testing of the method using the model examples shows its effectiveness both for finding the unknown function $g$ and its surface gradient. The latter is important from the point of view of applications to the solution of boundary value problems. // [1] I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko, Hypersingular Integral Equations and Their Applications, Chapman and Hall CRC, 2004.

Calin Atanasiu^{1}

^{1}National Institute for Laser, Plasma and Radiation Physics, Magurele-Bucharest, Plasma Physics and Nuclear Fusion, Romania

C.V. Atanasiu1, L.E. Zakharov2, K. Lackner3, M. Hoelzl3 1 Institute of Atomic Physics, Bucharest, Romania, 2 LiWFusion, Princeton, US, 3 Max Planck Institute for Plasma Physics, Garching, Germany The key basis for tokamak plasma disruption modeling is to understand how currents ﬂow to the plasma facing surfaces during plasma disruption events. In ITER, the occurrence of a limited number of major disruptions wills deﬁnitely damage the chamber with no possibility to restore the device. In the current exchange plasma-wall-plasma, according to the Helmholtz decomposition theorem, our surface current density in the conducting shell – the unknown of our problem - being a vector ﬁeld twice continuously diﬀerentiable in 3D, has been splited into two components: an irrotational (curl-free) vector ﬁeld and a solenoidal (divergence-free) vector ﬁeld. Developing a week formulation form and minimizing the correspondent energy functionals in a Finite Element approach, we have obtained the space and time distribution of the surface currents. We verified successfully our numerical simulation with an analytical solution with pure homogeneous Neumann B.C. and satisfying the necessary existence condition. By considering the iron core presence in JET tokamak, we have split the magnetization currents — the unknowns in some integral equations — into two components, the first producing a magnetic field in the iron region only and the second producing a magnetic field in the vacuum, obtaining thus a better evaluation of the influence of the iron core on the plasma equilibrium. To reduce the inﬂuence of the singularities appearing during the surface currents determination in multiply connected domains (L-shaped domains) we have used a conformal transformation method.

Yuriy Dimitrienko^{1}
, Igor Krasnov^{2}
, Kirill Zubarev^{3}
, Tatyana Ivanova^{4}

^{1}Bauman Moscow State University, Computational Mathematics and Mathematical Physics, Russia

^{2}Bauman Moscow State University, , Russia

^{3}Bauman Moscow State University, FN-11, Russia

^{4}Bauman Moscow State University, , Russia

Problems of electroinvestigation are solved for determination of parameters of geophysical structures due to comparison of results of the taken measurements of particular type of physical fields with results of their model operation. If application of electromagnetic fields takes place, then mathematical problem definition consists in searching of coefficients of Maxwell's equations and is the inverse task. In this work are considered a problem of restitution of electromagnetic parameters of the nonuniform mediums, by results of intubation by a direct current. The modelled fields are described by the system of difkferentsialny Maxwell's equations. Tasks are considered in a 3-dimensional approximation. The purpose of work is the solution of a direct task, creation of a program complex for electroinvestigation problem solving and also development of a unique algorithm for the solution of the inverse task, but a basis of a program complex for the solution of a direct task. In work the original algorithm of the solution of the inverse problem of electroinvestigation is constructed by a direct current. For the solution of a direct task the finite element method was used. On the basis of the solution of a direct task the algorithm of searching of electrophysical properties of the nonuniform mediums is developed. For the solution of this task the heuristic algorithm allowing to reduce first of all time of searching of the decision was used. When testing of the developed algorithm it was noted that accuracy of results also is acceptable.

Yuriy Dimitrienko^{1}

^{1}Bauman Moscow State University, Computational Mathematics and Mathematical Physics, Russia

The results of the development of the theory of constructing effective constitutive relations for laminated composites with finite deformations are presented. The theory is based on the application of the method of asymptotic averaging for periodic structures and the use of universal models of constitutive relations for composite layers, proposed by Yu.I. Dimitrienko. Universal models allow us to formulate the constitutive relations simultaneously for all the main conjugated stress – strain tensors pairs. The method of asymptotic averaging makes it possible to obtain a relationship between the effective constitutive relations of the composite and its constituent layers. Cases of incompressible layers and layers with compressibility are considered. A numerical algorithm for constructing effective deformation diagrams of layered composites and calculating the material constants in the effective constitutive relations of an anisotropic composite is proposed. Numerical examples of method implementation and numerical algorithms are given.

The reported study was funded by Ministry of Education and Science of the RF according to the research project (Goszadanie) No. 9.3602.2017/ПЧ

Stanislav Khristoforov^{1}
, Vladimir Bochkarev^{2}

^{1}Kazan Federal University, Institute of Physics, Russia

^{2}Kazan Federal University, Institute of Physics, Russia

Nowadays the prediction of ionospheric parameters is an important and acute problem in the field of ensuring the stable operation of radio communication and radio navigation facilities. The network of two-frequency GPS receivers data is used for monitoring the ionospheric condition. Based on these data, a number of laboratories are building global maps of total electron content (TEC). There are strong spatial and temporal correlations in the TEC maps. As a result, in order to successfully solve the problem of TEC prediction, it is advisable to perform preliminary processing of maps data with dimensionality reduction. In this paper, the problem of constructing a low-dimentional model of global distribution of the TEC is solved. In addition, the model of global distribution of the TEC can be useful for the ionosphere dynamics investigation. In this paper, it is proposed to use dense convolutional autoencoders as a base element of the model. This architecture allows us to speed up the neural network learning process and avoid the gradient-vanishing problem in error backpropagation algorithm. To train the network and test the results of the autoencoder, we used a set of two-hour maps from 1998 to 2013, presented by the JPL laboratory. Also, the optimum auto-encoder's latent space dimensionality was selected (in the range from 32 to 128). The application of this technique allowed us to obtain a smaller decompression error for all the maps from the training and test sets in comparison with the principal component analysis (the best result is 2.1 times smaller error) with the same dimensionality of the latent space of the maps.

Pavel Mossakovsky^{1}
, Liliia Kostyreva^{2}

^{1}Institute of Mechanics, Lomonosov Moscow State University, , Russia

^{2}Institute of Mechanics, Lomonosov Moscow State University, , Russia

The impenetrability of the GTE cases after blade breakage is one of the most important requirements in modern aircraft engine building. The need for a reliable numerical method for estimate the impenetrability is particularly significant at the stage of designing a new product. The problem of estimating of the impenetrability, in a more general set, can be considered as a study of the supercritical behavior of the engine construction due to the destruction of the blades or other elements of the rotor. The existence of an adequate numerical methodology for this type of research allows one to set and solve the practically important inverse task of genesis – the determination of the sequence of events by the character of the case and rotor damages, which led to a supercritical stage of the machine operation, as well as other problems closely related to the ones under consideration. One of them is the analysis of rotor dynamics due to the resulting misbalance. Over the past several decades, the computational methods for the estimate of the impenetrability have undergone a significant evolution: from relatively simple semi-empirical techniques based on conservative energy criteria up to modern approaches to the analysis of the supercritical behavior of constructions using direct computer modeling. The problem of the most computational methods based on computer modeling is the insufficient level of the obtained results reliability. The article considers the basic settings of an effective and reliable numerical methodology for estimate of the impenetrability of GTE cases after blade breakage. The method is based on an experimental and computational procedure for solving the corresponding boundary value problem by the finite element method in combination with a system of full-scale and virtual verification experiments used for correction of the numerical scheme. The results of numerical analysis of the impenetrability performed using the presented technique are compared with the data of bench tests, and their good agreement is shown.

George Alexandru Nemnes^{1}
, Tudor Mitran^{2}

^{1}Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), The Department of Computational Physics and Information Technologies (DFCTI), Romania

^{2}Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), , Romania

The graphene bilayer (GBL) has many of the electronic, mechanical, thermal properties of the graphene monolayer counterpart, however with a notable difference: one can induce an electronic gap using a perpendicular electric field, which induces an inversion asymmetry between the two layers [1]. This asymmetry can be provided also by extrinsic doping [2] and, for high doping levels, the system becomes a degenerate semiconductor. In this case a gap opens either above of below the Fermi level, depending on the doping type. We investigate here GLB systems with extrinsic impurities in the presence of external electric field, using density functional theory (DFT) calculations implemented by SIESTA package. We show that the position of the gap can be further tuned by the external field, which may switch the behavior from degenerate to intrinsic semiconductor. Furthermore, we compare the formation energies for several group-III and group-V dopants and the binding energies between the two graphene layers. References: [1] Yuanbo Zhang et al., Nature 459, 820 (2009); [2] Y. Fujimoto et al., Surf. Sci. 634, 57 (2015).

This work was supported by PN 18090205.

Vladimir Bochkarev^{1}
, Vladimir Tyurin^{2}
, Andrey Savinkov^{3}
, Bulat Gizatullin^{4}

^{1}Kazan Federal University, Institute of Physics, Russia

^{2}Kazan Federal University, Radiophysics, Russia

^{3}Kazan Federal University, Institute of Physics, Russia

^{4}Ilmenau Technical University, Polymer Physics, Germany

The problem of fitting time series by the sum of several exponentials with different decay parameters often arises analyzing the data of a physical experiment. In particular, such a problem arises determining the nuclear transverse relaxation times from the spin-echo decay NMR data. Mathematically, the problem can be formulated as a solution of the Fredholm integral equation of the first kind with an exponential kernel, or an equivalent system of linear equations. To solve this equation, various regularization methods are used, based on the L2 norm in most cases. We report the application of the L1-regularization for the problem, using the LASSO algorithm. A comparison was made between the results of L1- and L2-regularization on model time series, which were the sum of several exponential decays with noise. Also the L1- and L2-regularization were applied to analyze the $^1$H NMR experimental data of the spin-echo decays of n-hexane adsorbed in the porous media of aluminum free MFI-type zeolites known as silicalite-1. It was found out that the L1-regularization is effective method for determination of the transverse relaxation times from NMR data.

Alexander Rodnikov^{1}

^{1}Bauman Moscow State Technical University, Fundamental sciences, Russia

We model dynamics of a cabin equipped with a plane solar sail near a heavy artificial space object moving along a heliocentric orbit. We establish that the cabin motion toward the Sun by help of the solar light pressure is possible if the cabin is restricted by some constraint that performs the same function as the sailboat keel. Such constraint could be unilateral and realized by handrail or gantline. Also, the cabin could move along some rigid guides. Analizing the cabin motion equations, we suggest the optimal forms of constraints and the optimal control for the sail orientation to minimize the motion time

This problem formulation is result of discussions with professor Elena N. Polyakhova

Yana Ivanova^{1}

^{1}Saint-Petersburg Electrotechnical University "LETI", Saint-Petersburg, Russia

This work presents advances in complex analysis of nanoheterostructures, which contain quantum wells and delta-doped layers. Experimental investigations by means of admittance spectroscopy with further mathematical processing and modeling of major electrophysical parameters and energy characteristics gives full information about emission properties of the system. Basing on this method we studied impact of delta-layer position on quantum level positions and carrier concentrations in a quantum well

This work was supported by Act 220 of the Russian Government (Agreement No. 14.B25.31.0021 with the host organization IAP RAS).

Yana Ivanova^{1}

^{1}Saint-Petersburg Electrotechnical University "LETI", Saint-Petersburg, Russia

This work presents advances in complex analysis of nanoheterostructures, which contain quantum wells and delta-doped layers. Experimental investigations by means of admittance spectroscopy with further mathematical processing and modeling of major electrophysical parameters and energy characteristics gives full information about emission properties of the system. Basing on this method we studied impact of delta-layer position on quantum level positions and carrier concentrations in a quantum well

This work was supported by Act 220 of the Russian Government (Agreement No. 14.B25.31.0021 with the host organization IAP RAS)

Vasily Shmyrov^{1}

^{1}SPbSU, Department of Control Systems Theory for Electrophysical Facilities, Russia

In this paper we study the problem of flight from a low near-Earth orbit to a neighborhood of collinear libration point of Sun-Earth system in the space of positions. We build impulse controls to achieve invariant manifolds. The spacecraft is in the neighborhood of libration point for a long time on this invariant manifold. We use the apparatus of equations in variations for construction of impulse controls.

Ilia Sheinman^{1}

^{1}Saint-Petersburg Electrotechnical University, Physics, Russia

Self-coordinated transverse dynamics of the high current relativistic electronic bunches used for generation of wake fields in wakefield accelerating structures with dielectric filling is investigated. The Laplase transform method is used for analytical approach to solution of self-coordinated beam dynamics.

Arkady Seryakov^{1}

^{1}RUDETRANSSERVICE, Veliky Novgorod , scientific laboratory, Russia

Abstract. The results of numerical simulation of vortex formation near the condensation surface in the Laval-liked vapour channel of short low temperature heat pipes (HP’s) at high heat loads are presented. For the first time it was found that the calculated value of moist compressible vapour vortex flow near the condensation surface inside the channel of short HP’s changes its rotational motion direction, depending on the thermal load magnitude on the HP’s evaporator. With low thermal loads, the toroidal vapour vortex ring rotational motion direction due to the Coanda effect and moving vapour jets sticking to the channel walls occurs from the periphery to the vapour channel center. While the thermal load increasing, the direction of the vapour vortex rotation changes to the opposite, from the center to the periphery. The thickness of the liquid condensate film located under the toroidal vortex ring also depends on the magnitude of the thermal load and the vortex rotation direction. Using the original capacitive sensors and special electronic equipment we measured the local thickness of the working fluid films inside the HP’s on its condensing surface. The measurement error does not exceed 1•10-3mm. Experimentally detected that the condensate film thickness lessens sharply with the increase of the heat load on the evaporator of the Laval-like low-temperature HP’s due to the change in the radial rotation direction of the vapour toroidal vortex ring on its condensation surface. The numerical calculations results of the condensate film thickness are close to the experimental values.

Anton Oskirko^{1}
, Sergey Ul'yanov^{2}
, Alexey Val'kov^{3}

^{1}Saint Petersburg State University, Physical Department, Russia

^{2}Saint Petersburg State University, Statistical Physics, Russia

^{3}Peter the Great Saint Petersburg Polytechnic University, , Russia

A theory of orientational transitions in plane-parallel chiral nematics liquid crystal (cholesteric – ChLC) cell with account for flexoelectricity, soft anchoring and negative dielectric anisotropy ($\varepsilon_a$) is developed. The main interest will be paid to the electric field inhomogeneity inside the cell. Besides the contribution arising with the voltage U applied to the bounding plates (~ $U^2$), we consider also two terms, which arise in inhomogeneous systems due to the flexoelectricity (first one ~ $\bar{e}U$, and second one ~ $\bar{e}^2U^0$, induced by Maxwell’s equations). Here $\bar{e}$ denotes the mean flexoelectric coefficient. We consider chiral nematics both for $\varepsilon_a > 0$ and for $\varepsilon_a < 0$ (the effect of flexoelectricity on the Fréedericksz transition in non-chiral nematics (NLCs) with $\varepsilon_a > 0$ has been studied in [1,2]). Our analysis showed that in the ChLC with $\varepsilon_a > 0$ the flexoelectric effect promotes the Fredericks transition. It is assumed commonly that there is no Fréedericksz transition in ChLC cells with geometry under consideration and the $\varepsilon_a < 0$ condition in stationary field (nanosecond electro-optics of NLCs with $\varepsilon_a < 0$ had been studied in [3]). Our analysis showed surprisingly that the Freedericksz transition can be driven by the stationary electric field in systems with considerable flexoelectric properties even in the case $\varepsilon_a < 0$. Based on a variation method [4] we investigated the ChLC cell orientational structure above the threshold. To determine type of the transition, continuous or discontinuous, we developed a two-parametrical Landau model with deviations from easy directions at the boundaries taken as the order parameter. It turned out that the Fréedericksz transition in the ChLC cell with the geometry considered can be either continuous or discontinuous depending on the material parameters of the ChLC and on the cell thickness in case εa > 0, while it will be continuous for ChLCs with $\varepsilon_a < 0$. References [1] C. V. Brown and N. J. Mottram, Phys. Rev. E, 68, 031702 (2003). [2] E. Mema, L. Kondic, and L. J. Cummings. Phys. Rev. E, 95, 0123701 (2017). [3] V. Borshch, et al., Phys. Rev. E, 90, 062504 (2014). [4] A. Y. Val’kov, E. V. Aksenova, and V. P. Romanov, Phys. Rev. E, 87, 022508 (2013).

Anis Khasanov^{1}

^{1}Plekhanov Russian University of Economics, Higher Mathematics, Russia

Abstract. The solution of the evaporation problem in diffusion mode in the case of two interacting large stationary aerosol drops is obtained by methods of linear operators. Concentration and temperature fields are presented as series of spherical functions. Search for coefficients in these expansions is reduced to search for infinite-dimensional vectors in a Banach space by means of linear operators. The main goal of this work was to investigate the influence of the factor of interacting of drops on their complete evaporation time. The formulas describing the influence of one drop on the complete evaporation time of another drop are obtained. Corresponding graphs and results of calculations of correction factors for different radii of drops and distances between them are presented. The presented approach is simpler and applicable for solving more problems concerning two interacting aerosol particles than the method using bipolar coordinate system.

Elina Biberdorf^{1}

^{1}Novosibirsk State University, Mechanics and Mathematics, Russia

Using the original numerical method, the characteristics of the van der Waals gas flow between the shock wave and the surface of the cone in the axisymmetric case are calculated. The application of the Lopatinsky condition allows us to establish the type of shock wave: absolutely stable, unstable or neutrally stable. The phase state of the substance is also determined liquid, gaseous, or it is in a transitional two-phase state. For this purpose, the van der Waals isotherms are constructed using the Maxwell's rule of equal areas.

This work was supported by the RFBR Foundation No. 17-01-00791 - "a"

Svetlana Riabova^{1}

^{1}Institute of Geosphere Dynamics of Russian Academy of Science, Near-surface Geophysics, Afghanistan

Wavelet analysis is becoming more popular in geophysics. It is used for numerous researches, including tropical convection, the El Nino-Southern Oscillation, atmospheric cold fronts, temperature variations, the dispersion of ocean waves, and coherent structures in turbulent flows, number of sunspots etc. In this paper we research how informative is the application of wavelet analysis to the analysis of geomagnetic field variations at the mid-latitude observatory "Mikhnevo" of Institute of Geosphere Dynamics of Russian Academy of Science. We review continuous wavelet transform by focusing attention on such aspects as choice of mother wavelet, choice of scales, cone of influence, visualization of results, reconstruction of time series from wavelet transform and its application to estimate the Holder exponents and singularity spectra. In our work we use Morlet wavelet with frequency parameter of 6. In so doing, the reconstruction of the time series from the wavelet transform has a mean square error of 3.4%. The application of wavelet analysis made it possible to distinguish pronounced periodicities of the geomagnetic field with periods of 27, 13.5, 9, 6 days. In solar quiet-day variations is dominated by the 24- 12-, 8-, and 6-hour period components. An analysis of the modulus of the wavelet transform coefficients qualitatively indicates a scaling (close to the monofractal) character of the variations of the geomagnetic field in the diurnal range. Moreover, the intensity of periodic variations of geomagnetic variation isn’t constant in time. The application of the method of wavelet transform modulus maxima confirmed the monofractal character of the diurnal variation for any solar activity. In contrast to the 1-day variation, the 27-day variation and its harmonics show a higher degree of multifractality during a maximum of solar activity in comparison with the minimum of solar activity.

Ruben Danilo BOURDON GARCIA^{1}
, Javier BURGOS-SALCEDO^{2}

^{1}CIINAS Corporation, Bogotá, Distrito Capital, Colombia

^{2}CIINAS Corporation, Scientific Direction, Colombia

Mirror symmetry breaking (MSB) is of crucial significance in the proposal of scenarios in abiotic chemistry. MSB is commonly studied by classical methods using differential ordinary equations but Stoichiometric Network Analysis (SNA), which uses matrix systems, emerges as an alternative to the study of very complex reaction networks. This, because SNA reduces the number of variables and the complexity of the solutions but retaining all the information of the chemical system. We study the application of SNA to MSB, specifically for the production of amino acids by the Strecker reaction. Results on the dynamics for the generation of MSB in six autocatalytic systems are discussed, including limited enantioselectivity network models, cross inhibition and two factors for MSB (stochastic fluctuactions and electron-nucleus electroweak interactions).

Andrey Dmitriev^{1}

^{1}National Research University Higher School of Economics, School of Business Informatics, Russia

By 2018, there are more than 70 million people suffering from various forms of one of the most common neurological diseases – epilepsy. In fact epipelsy is a central nervous system (neurological) disorder , manifesting itself in anomalous brain activity. Nowadays, most of the patients do not have the ability to foresee the onset of an attack in advance, which is due to complex symptoms that are difficult to predict. One of the best way to analyze such kind of behaviour is electroencephalography (EEG). This research paper considers the problem of EEG epileptic seizures prediction from the point of the theory of nonlinear dynamical systems. According to the study, signals of cerebral cortex neural networks corresponds to multifractal nature which gives an opportunity to analyze changes between states in phase space during the abnormal electrical activity. Therefore, monitoring the preictal stage and early indication of an attack may help patients to avoid problems related to sudden seizures. This research will provide valuable information regarding the mechanism of epilepsy onset and introduce the forecasting model utilizing machine learning algorithm which is based on recurrence quantification analysis.

The work was supported by the Russian Foundation for Basic Research (grant 16-07-01027)

Vasily Shmyrov^{1}
, Alexander Shmyrov^{2}

^{1}SPbSU, Department of Control Systems Theory for Electrophysical Facilities, Russia

^{2}SPbSU, Mechanics of Controlled Motion, Russia

Usually, the accuracy is the main criterion in comparing of numerical methods of integration. However, often the qualitative properties of the trajectory is more important than the accuracy. For example, this is the numerical study of a planetary systems motion on large time intervals. In this case that conservative methods show their effectiveness. In this paper we propose the results of a comparison of conservative and nonconservative first-order methods in a numerical study for a well-studied two body problem. It is shown that the nonconservative method leads to exponential departure. The symplectic method of integration describes a chaotic motion close to regular. The results of numerical integration are illustrated graphically.

Ivan Goriachuk^{1}
, Viktor Sokolov^{2}

^{1}Federal Scientific Research Center "Crystallography and Photonics", RAS, Institute on Photonic Technologies, RAS, Russia

^{2}Federal Scientific Research Center "Crystallography and Photonics", RAS, Institute on Photonic Technologies, RAS, Russia

Direct and inverse problems are distinguished in the analysis of electromagnetic radiation interaction with thin film structures. The direct problem is a calculation of the characteristics of reflected electromagnetic field given the known parameters of the structure. The inverse problem is the determination of the parameters of the structure from the characteristics of reflected electromagnetic field. In the present work we analyze the inverse problem of determining the parameters of multilayer metal-dielectric structures using prism-coupling technique. For a relatively long time prism waveguide mode excitation in the geometry of frustrated total internal reflection has been used for measuring refractive index and thickness of thin dielectric films (from 1 to 10 mkm thick). Ordinary algorithms employed for the measurements results processing in the experiments with multilayer samples commonly face serious problems. They do not account for the presence of finite angular divergence of the probe laser beam and they fail to work with non-uniform distribution of the electric characteristics over the thickness for semitransparent metal layers. Many models for the light reflection from multilayer media are not applicable for anisotropic films investigation. Experimental results processing method is proposed for these measurements. It permits to solve all the above mentioned problems simultaneously. The computer algorithm based on gradual variation of the theoretical model parameters and the comparison of the refractive index dependence on the incidence angle, obtained from this model, with the experimental one lies in the foundation of this method. The suggested data processing algorithm may find applications, for example, in the development of integrated optical circuits with anisotropic light-guiding layers and semitransparent metal claddings. It suits for the analysis of any layered systems, when the layers are several microns in thickness and the number of them reaches four and more.

This work was supported by RFBR research project № 18-32-00948 in part of multilayer structures fabrication and by the Federal Agency of Scientific Organizations (Agreement No 007-ГЗ/Ч3363/26) in the part of development of mathematical models and algorithms for calculating optical parameters of thin films.

Vladimir Bochkarev^{1}
, Anna Shevlyakova^{2}
, Eduard Lerner^{3}

^{1}Kazan Federal University, Institute of Physics, Russia

^{2}Kazan Federal University, Institute of Oriental Studies and International Relations, Russia

^{3}Kazan Federal University, Institute of Computational Mathematics and Information Technologies, Russia

Creation of the Google Books Ngram corpus opened up new opportunities for studying language evolution. This corpus consists of a large amount of digitized books written in 8 languages and contains information on frequencies of words, word combinations and syntactic dependencies of the last 500 years. In this paper, we present data on changes in the key characteristics of syntactic relations in English and Russian and propose a model which allows us to explain the observed changes. The clustering coefficient for the networks of syntactic relations of both languages is quite large. It is 0.70-0.77 for English (in 1800-2008) and 0.42-0.44 for Russian (in 1925-2008).The assortativity coefficient is negative in both cases, and varies from -0.27 to -0.24 for English and from -0.129 to -0.124 for Russian. It should be noted that these parameters are stable enough. The obtained values of the clustering and assortativity coefficients doesn’t let us use most of the common models of complex networks for describing dynamics of the network of syntactic relations. We proposed a new model of network growth and compared its predictions with empirical data on the networks of syntactic relations of English and Russian. Theory of prototypes provided the basis for construction of the model. The backbone of the theory is existence of a prototype, that is, an abstract image embodying a multitude of similar forms of the same object or pattern, the most representative example of a concept that posesses its typical properties. When a new word appears, it inherits a number of semes of its prototype. In accordance with this statement, the distribution of the new word will be similar to the distribution of the word-prototype. Formalizing these considerations mathematically, we get a model in which a new word inherits some of the relations of the word-prototype, and also receives at random a number of new relations. We used Google Books Ngram data (1800-2008) and performed modelling of network growth. Then, we compared the characteristics of the obtained model networks with the characteristics of the network of syntactic relations of the English and Russian languages. It was shown that selection of two parameters of the model allows us to obtain a very good correspondence between the changes in the clustering coefficient, the assortativity coefficient, and word distribution by the number of relations in the model network and in real networks of syntactic relations.

This research was financially supported by the Russian Government Program of Competitive Growth of Kazan Federal University, state assignment of Ministry of Education and Science, grant agreement № 34.5517.2017/6.7 and by RFBR, grant № 17-29-09163

Ruben Danilo BOURDON GARCIA^{1}
, Javier BURGOS-SALCEDO^{2}

^{1}CIINAS Corporation, Bogotá, Distrito Capital, Colombia

^{2}CIINAS Corporation, Scientific Direction, Colombia

Nowadays, the research structure of an institution or an enterprise is the core to do the investigation, development and innovation (ID+I). For this reason, if this structure fails, the institutional ID+I also fails. Based on this, we studied the research structure of Saint Matthew Foundation for the High Education by means of Formal Concept Analysis (FCA). This is the baseline to check the correspondence between the Foundation research aims and the Foundation research structure. From the structure found the concept lattice, formal concepts, subconcept-superconcept relations, partially ordered sets, supremum and infimum of the lattice and implications between attributes (Duquenne-Guigues base), were determined.

Alper Selver^{1}
, Mustafa Secmen^{2}
, Yesim Zoral^{3}

^{1}Dokuz Eylül University, Electrical and Electronics Engineering, Turkey

^{2}Yasar University, Electrical and Electronics Engineering, Turkey

^{3}Dokuz Eylul University, Electrical and Electronics Engineering, Turkey

xtraction of distinguishing features and decision of classifiers are highly influenced by the low signal-to-noise ratio (SNR) rates, when target identification from scattered electromagnetic waves is considered. In order to increase the correct identification rates, smoothing operations, which should increase the SNR without greatly distorting the signal, are occasionally employed. However, this operation is mostly performed to the complete scattered signal via an over-complete basis. On the other hand, Savitzky-Golay filters can de-noise the signal by fitting successive sub-sets of adjacent scattered data points with a low-degree polynomial through the use of linear least squares. Thus in this study, both computational burden and accuracy of Savitzky-Golay filters are compared to three well-established smoothing techniques in time domain, frequency domain and time-frequency analysis. The analyses are performed with both simulated and measured data from various conductor and dielectric targets having different size, geometry and material type.

Jorge Linares^{1}
, Imad Zemouri^{2}
, Ilies Benhabbour^{3}
, Sylvain Chevallier^{4}
, Luc Chassagne^{5}
, Kamel Boukheddaden^{6}

^{1}Université de Versailles St. Quetin en Yvelines, GEMAC, France

^{2}Université de Versailles St. Quetin en Yvelines, LISV, France

^{3}Université de Versailles St. Quetin en Yvelines, GEMAC, France

^{4}Université de Versailles St. Quetin en Yvelines, LISV, France

^{5}Université de Versailles St. Quetin en Yvelines, LISV, France

^{6}Université de Versailles St. Quetin en Yvelines, GEMAC, France

In this contribution, we propose a new approach to calculate the heat capacity in triangular Ising nanoparticules subject to a nonzero external magnetic field H ≠ 0. The well known Ising system contains a variety of spin situated on the vertices of a crystalline lattice. The spin at the k node can be either "up" (σk=1) or "down" (σk=-1). A macrostate state of the system is fully characterized by the knowledge of the two macro-variables "m" (the average system magnetization) and "s" (the correlation: the sum of the nearest-neighbors interactions between the spins) and their distribution function, D(m,s). In this new approach, we first we determine the density of macrostates, D(m,s), by scanning the full spin configurations of the triangular lattice for systems containing 7, 19, 37, 61, 101, 179, 227 magnetic atoms. The knowledge of D(m,s) at each temperature allows to calculate the thermal average of the energy

We thank the CHAIR Materials Simulation and Engineering, the UVSQ and Université Paris Saclay, the "Agence National de la Recherche" (ANR Project BISTA-MAT: No. ANR-12-BS07-0030-01) and LIA IM-MED

Jorge Linares^{1}
, Ikbel Sahbani^{2}
, Pierre Dahoo^{3}
, Kamel Boukheddaden^{4}

^{1}Université de Versailles St. Quetin en Yvelines, GEMAC, France

^{2}Université de Versailles St. Quetin en Yvelines, LATMOS, France

^{3}Université de Versailles St. Quetin en Yvelines, LATMOS, France

^{4}Université de Versailles St. Quetin en Yvelines, GEMAC, France

J.Linares., I.Sahbani., P.R.Dahoo, K.Boukheddaden We consider a 2D spin crossover (SCO) nanoparticles under the influence of an external environment. The nanoparticle is made of a triangular lattice leading to a hexagonal-shaped systems. Each molecule located in the bulk has then six nearest-neighbours (nn) while those situated at the surface have three or four nn. Adapting the entropic sampling Monte Carlo method, using three parameters: nHS (the total high spin fraction), s (the correlation: the sum of nn interactions the molecules) and nsHS (the high-spin fraction at the surface), we determined the density of macrostates D(nHS, nsHS ,s) by scanning the full spin configurations. This information is then used to analyse the thermal behaviour of the system taking into account for the short- (J) and long-range (G) interactions as well for the interactions (L) between the molecules at the surface and the matrix (environment). The present contribution is devoted to the investigations of the required physical conditions leading this triangular 2D lattice to exhibit a re-entrant phase transition, in particular when the size of the nanoparticles is reduced. References [1] J. Linares, C.M. Jureschi, K. Boukhedaden, magnetochemistry 2016, 2, 24 [2] C. Harlé, S.E.Allal., D. Sohier, T. Dufaud, R. Caballero, F. de Zela , P. R. Dahoo, K. Boukheddaden, J.Linares , Journal of Physics SC, 936 (2017) 012061

We thank the CHAIR Materials Simulation and Engineering, the UVSQ and Université Paris Saclay, the "Agence National de la Recherche" (ANR Project BISTA-MAT: No. ANR-12-BS07-0030-01) and LIA IM-MED

Jorge Linares^{1}
, Imad Zemouri^{2}
, Ikbel Sahbani^{3}
, Sylvain Chevallier^{4}
, Luc Chassagne^{5}
, Pierre Dahoo^{6}
, Kamel Boukheddaden^{7}

^{1}Université de Versailles St. Quetin en Yvelines, GEMAC, France

^{2}Université de Versailles St. Quetin en Yvelines, LISV, France

^{3}Université de Versailles St. Quetin en Yvelines, LATMOS, France

^{4}Université de Versailles St. Quetin en Yvelines, LISV, France

^{5}Université de Versailles St. Quetin en Yvelines, LISV, France

^{6}Université de Versailles St. Quetin en Yvelines, LATMOS, France

^{7}Université de Versailles St. Quetin en Yvelines, GEMAC, France

To the well know Ising Hamiltonian, we include a new term in order to taking account for the interaction between the atoms at the surface with their immediate environment. So, as function of macro-variables, the total Hamiltonian writes as: H= -hm - Js - L ms, where "m" is the system magnetization, "s" the sum of the nearest-neighbours spin-spin correlation, and "ms" is the total magnetization of the atoms located at the surface. Here σ (resp. σs) is the bulk (resp. surface) Ising spin state operator. A macrostate of the system is then fully characterized by the knowledge of m, s and ms. The three-variable density of states, D(m,s,ms), is the number of micro-states having the same system energy and then the same Boltzmann factor. The Monte Carlo entropic sampling, previously presented in [1, 2] is adapted in this contribution to calculate this three-variable density of states for the specific problem under study. The main idea is to sample the density directly from the entropy of a system at infinite temperature. Importance sampling using the well-known Metropolis Algorithm, is shown to be a particular case of this new algorithm. A comparison with the Metropolis algorithm shows that this Entropic Sampling algorithm is well adapted to study magnetic and spin-crossover nanoparticles: it is more general and more efficient to study the properties of metastable states. Reference [1] J. Lee, Physical Review Letters, 71 (1993) 211-214 [2] I. Shteto, J Linares, F. Varret, Physical Review E, 56 (1997) 5128

We thank the CHAIR Materials Simulation and Engineering, the UVSQ and Université Paris Saclay, the "Agence National de la Recherche" (ANR Project BISTA-MAT: No. ANR-12-BS07-0030-01) and LIA IM-MED

Vladimir D. Stepanov^{1}

^{1}Russian Academy of Sciences, Steklov Mathematical Institute , Russia

.

V. Burenkov^{1}

^{1}RUDN University, , United Kingdom

.

Mikhail Goldman^{1}

^{1}Peoples' Friendship University of Russia, Mathematics, Russia

.

Victor Vlasov^{1}

^{1}Lomonosov Moscow State University, Mechanics and Mathematics, Russia

We study integro-differential equations with unbounded operator coefficients in Hilbert space. The principal part of these equations is an abstract hyperbolic operator perturbed by summands of Volterra integral operators. Operator models of such type have many applications in the linear viscoelasticity theory, homogenezation theory, heat conduction theory in media with memory, etc. In particular these integro-differential equations can be realized as the system of integro-partial differential equations: \begin{multline} \rho \ddot u(x,t) - Lu(x,t) + \int\limits_0^t {{\Gamma _1}(t - s)L_1u(x,s)ds} +\\+ \int\limits_0^t {{\Gamma _2}(t - s)L_2u(x,s) ds} = f(x,t), \end{multline} were $u = \vec u(x,t) \in {\mathbb{R}^3}$ is displacement vector of viscoelastic anisotropic media, $t>0$, $x \in \Omega \subset {\mathbb{R}^3}$ is bounded domain, $u$ satisfy Dirichlet conditions in a domain with smooth boundary $\Omega $, $L_1=\mu \cdot (\Delta u + 1/3\cdot{\text{grad div}}u)$, $L_2=\lambda \cdot {\text{grad div}}u$, $Lu = (L_1+L_2)u$ is Lame operator of elasticity theory, ${\Gamma _1}$, ${\Gamma _2}$ are memory relaxation functions that are the series of decreasing exponents with positive coefficients. A spectral analysis of the operator-valued functions, which are the symbols of considered integro-differential equations is provided. The structure and localization of spectra for these operator-valued functions are analyzed.

This work was supported by Russian Science Foundation project no. 17-11-01215.

Zhi-gang Xu^{1}
, Kai-zhong Qin^{2}
, Wei-min Liu^{3}

^{1}shandong university, mechanical, China

^{2}shandong university, mechanical engineering, China

^{3}Shenzhen research institute of shandong university, mechanical engineering, China

In order to decrease the material handling costs and cope with the closeness of the non-material relationships etc, genetic algorithm was introduced in the typical System Layout Planning (SLP) problems. The above-mentioned two design objectives are to be realized in a hybrid algorithm, i.e. the traditional mathematical method and GA system. A software system is developed on the bases of the hybrid algorithm, and a design example of a transducer plant planning is introduced to verify the efficiency of the algorithm and the software system. Effects of the genetic parameters, genetic paradigms and the punishing factors etc are also talked about in this paper, the results are encouraging.

The above-mentioned research work is supported by the Science, Technology and Innovation Commission of Shenzhen Municipality, JCYJ20160510165328965 and the Chinese NSFC, 61272017, P.R. China, and the National Demonstration Center for Experimental Mechanical Engineering Education （Shandong University）.

Diana Rakhmatullina^{1}
, Natalia Larionova^{2}
, Julia Varlamova^{3}
, Lilia Zulfakarova^{4}

^{1}Kazan Federal University, the Institute of Management, Economics and Finance, Department of Economic Methodology and History, Russia

^{2}Kazan Federal University, the Institute of Management, Economics and Finance, Department of Economic Methodology and History, Russia

^{3}Kazan Federal University, the Institute of Management, Economics and Finance, Department of Economic Methodology and History, Russia

^{4}Kazan Federal University, the Institute of Management, Economics and Finance, Department of Economic Methodology and History, Russia

The BRICS countries are the largest economic bloc, and in many forecasts it is believed that the prospects for the growth of the world economy are related to their development (World Bank 2015, Randers 2012). On the one hand, the backwardness of the banking systems of the BRICS countries may be a factor in the financial vulnerability of countries in the future; on the other hand, the creation of institutional conditions for the effective functioning of the banking system is a potential source of economic growth in the BRICS countries. In this article the institutional environment of the banking system of the BRICS countries are analyzed. There is exploration of the key statistical characteristics of the banking sector, the main trends in its development. The institutional structure of the banking system is represented by external and internal institutions. The main aspect of the study is to reveal the characteristics of the external institutional environment. The purpose of our study is to test the following hypothesis: factors of the external institutional environment affect the banking system. The subject of the study is the BRICS countries banking system. The period of study: 2004 - 2015. We selected three characteristics: the degree of development of the banking system, the stability and efficiency of the banking system. We used the construction of econometric models to test the hypothesis. The findings of the investigation have distinguished the main characteristics of the external institutional environment that have an impact on the banking sector and determine its effectiveness: the size of the public sector and the degree of state intervention in the economy; tax freedom; freedom of doing business; historical and political characteristics; investment and monetary freedom.

Nailya Yakupova^{1}
, Ekaterina Kadochnikova^{2}
, Liliya Zulfakarova^{3}
, Zulfira Zapparova^{4}

^{1}Kazan Federal University, Institute of Management, Economics and Finance, Russia

^{2}Kazan Federal University, Institute of Management, Economics and Finance, Russia

^{3}Kazan Federal University, Institute of Management, Economics and Finance, Russia

^{4}Kazan Federal University, Institute of Management, Economics and Finance, Russia

The article emphasizes the need to forecast financial indicators in order to assess the value of a company in the interests of its owners and investors. This proves the need for using econometric models to measure quantitative economic interrelations of net profit as the main indicator of cash flow and internal factors of growth. The result of the study is an interval forecast estimate of the net profit of the trading company. The authors propose the construction of the forecast of net profit based on regression on the main components in the conditions of collinearity of regressors - indicators of the financial state of the company. Empirical results are obtained in Gretl's software environment in order to reveal the interrelationships of net profit and growth factors on the basis of specific economic data. The study confirmed the existence of a causal relationship between the net profit of the trading company and the turnover of inventory. In future research it is possible to apply the methodological approach presented in the article to obtain prognostic estimates of profit based on regression with non-financial indicators of the company and environmental factors, taking into account qualitative factors and territorial features of business.

Tugce Toprak^{1}
, Burak Belenlioglu^{2}
, Sinan Dogan^{3}
, Burak Aydın^{4}
, Alper Selver^{5}

^{1}Dokuz Eylul University, Electrical and Electronics Engineering, Turkey

^{2}Kentkart, , Turkey

^{3}Kentkart, , Turkey

^{4}Kentkart, , Turkey

^{5}Dokuz Eylül University, Electrical and Electronics Engineering, Turkey

Pedestrian Detection (PD) is now almost a mature field of signal processing in which numeruous application areas exists. Emerging one of these applications is driving autonomy that requires advanced driver assistance systems (ADAS) for PD. Although various benchmark data sets are collected and several methods are already developed, the reported results show that current systems lack generalization capacity due to strong dependence of system performance on PD training set. In other words, the systems trained with a pedestrian data set usually perform considerably well on a test set taken from the same benchmark, but their performance drop drastically when tested on a different benchmark data set. In order to overcome this limitation, one of the approaches might be fusing outcomes of different systems. Here, “different systems” can refer either to train the same model with different data sets or using entirely different models. In both cases, the outcomes of different systems should be diverse enough to cover the entire solution space. Up to our knowledge, the performance of existing well-established models for PD is frequently compared to each other but their diversity is not analyzed in detail yet. In this study, the feasibility of PD systems, whose effectiveness on PD is shown by earlier studies, was examined by systematically training and testing them with different benchmark data sets and their combinations. For this purpose, pairwise and non-pairwise measures of diversity have been employed and several conclusions are driven about the complementarity of existing PD models.

This study is conducted as a part of EUREKA-EURIPIDES project “ADORAS [Advanced Onboard data Recording and Analysis System]” and supported by TUBITAK TEYDEB under grant 9150121, which is being conducted by Kentkart).

Vladimir Smirnov^{1}
, Maria Korovina^{2}

^{1}Moscow Aviation Institute (National Research University) , "Applied software and mathematical methods", Russia

^{2}Lomonosov Moscow state University, faculty of computational mathematics and Cybernetics, Russia

Partial differential operator with holomorphic coefficients of order $m$ on manifold with cusp-type singularities can be represented in the form $$H(r, r^{k+1} {d\over dr} ,x,-i{\partial \over \partial x} )u=0,~~~~~~~~~~~~~~~~(1) $$ where $x$ varies on some compact manifold without boundary (e.g. see [1]). $$H\left(r,r^{k+1} {d\over dr} ,x,-i{\partial \over \partial x} \right)=\sum _{j+l=0}^{m}a_{jl} (x,r)\left(-i{\partial \over \partial x} \right)^{j} \left(r^{k+1} {d\over dr} \right)^{l} $$ $a_{ij} \left(x,r\right)$ are holomorphic in some neighborhood of the point $r=0$. These equations can be interpreted as equations with respect to the functions with values in Banach spaces, namely $$\hat{H}\left(r,-{1\over k} r^{k+1} {d\over dr} \right)u=0,\, $$ where $\hat{H}:E_{k} \left(S_{R,\varepsilon } ,B_{1} \right)\to E_{k} \left(S_{R,\varepsilon } ,B_{2} \right)$. Let $S_{R,\varepsilon }$ denote the sector $S_{R,\varepsilon } = \{r| -\varepsilon < \arg r < \varepsilon , \ |r| < R \}$. We say that the function $f$ analytic at $S_{R,\varepsilon } $ has at most k-exponential growth if there exist nonnegative constants $C$ and $\alpha $ such that the inequality $$|f| < Ce^{\alpha {1\over \left|r\right|^{k} } } $$ holds in the sector $S_{R,\varepsilon } $. Let $E_{k} \left(S_{R,\varepsilon } \right)$ denote the space of holomorphic functions of k-exponential growth. Here $B_{i} ,\, \, i=1,2$ denote some Banach spaces (e.g. the space$H^{s} \left(\Omega \right)$). The degree k of the function $u$ grows exponentially for $r\to 0$. For fixed $r$ and $p$ $$\hat{H}\left(r,p\right):B_{1} \to B_{2} $$ is a bounded operator acting in Banach spaces which is polynomial dependent on $p$ and holomorphic in the neighbourhood of zero. In what follows, we will assume that the operator family $\hat{H}_{0} \left(p\right)=\hat{H}\left(0,p\right)$ is a Fredholm family. This implies that the operator $\hat{H}_{0} \left(p\right)$ is a Fredholm operator for each fixed $p$, and there exists $p_{0} \in C$ such that the operator $\hat{H}_{0} \left(p_{0} \right)$ is invertible. Let the above conditions be satisfied, then solutions of equation (1) is resurgent function (see [2]). Example of such task is Laplace's equation on a manifold with a caspidal singularity. We consider the Laplace equation $\Delta u=0$ on the 2D-Riemannian manifold with a second order caspidal singularity. The Riemannian metric induced from $R^{3} $ with the help of embedding which is defined as the surface by the rotational of the parabolic branch $y=r^{2} $ around axis $0r$ in $R^{3} $. We choose local coordinates $r,\varphi $ on the manifold in the neighbourhood of zero. The metrics on this manifold is given by $$ds^{2} =dr^{2} +4r^{2} dr^{2} +r^{4} d\varphi ^{2} =\left(1+4r^{2} \right)dr^{2} +r^{4} d\varphi ^{2} .$$ For the Laplace operator we obtain $$\Delta u=divgrad\, u={1\over r^{2} \sqrt{4r^{2} +1} } {\partial \over \partial r} \left(r^{2} {1\over \sqrt{4r^{2} +1} } {\partial u\over \partial r} \right)+{1\over r^{4} } \left({\partial \over \partial \varphi } \right)^{2} u.$$ In other words $$\Delta :E_{k} \left(S_{R,\varepsilon } ,H^{s} \left(S_{1} \right)\right)\to E_{k} \left(S_{R,\varepsilon } ,H^{s-2} \left(S_{1} \right)\right),$$ We rewrite Laplace equation in the form $$\left(r^{2} {d\over dr} \right)^{2} u+{\partial ^{2} u\over \partial \varphi ^{2} } =-4r^{2} {\partial ^{2} u\over \partial \varphi ^{2} } +{4r^{3} \over \left(4r^{2} +1\right)^{} } \left(r^{2} {\partial u\over \partial r} \right)\, \, ~~~~~~~~~~(2) $$ The principle symbol of this operator is equal $H(o,p)=p^{2} +\left({\partial \over \partial \varphi } \right)^{2} $. Therefore the solutions of equation (2) is resurgent function. ASSERTION. Any solution of equation of exponential growth (2) can be represented as $$u\left(r,\varphi \right)={C_{-1} \over r} +\sum _{j}u_{j} \left(r,\varphi \right) +O\left(e^{-{A\over r} } \right),$$ where A is an arbitrary positive number, and the sum contains a finite number of summands, each of which corresponding to a point $p=k $ located in the half-plane $Rep>-A$, and $u_{j} \left(r\right) $ have asymptotic expansions $$u_{j} (r,\varphi )=e^{{j\over r} } e^{i\varphi j} \sum _{l=0}^{\infty }M_{l}^{k} r^{l} $$ References 1. Sternin, B. and Shatalov, V., Borel–Laplace Transform and Asymptotic Theory. Introduction to Resurgent Analysis, New York–London–Tokio: CRC, 1996. 2. Korovina M. V., Existence of Resurgent Solutions for equations with Higher–Order Degenerations, Differ. Uravn, 2011, vol. 47, no.3, pp. 349-357.

We would like to express deep gratitude to V.E. Shatalov for permanent discussions throughout the work

Vladimir Sukhanov^{1}

^{1}Sankt-Petersburg state university, physical faculty, Russia

The paper is concerned with the inverse scattering problem for the Stark operator on the line with the potential from the Schwartz class. In our study of the inverse problem, we use the Riemann-Hilbert formalism. This allows us to overcome the principal technical difficulties which arise in the more traditional approaches based on the Gel'fand-Levitan-Marchenko equations and indeed solve the problem. We also produce a complete description of the relevant scattering data (which has not been obtained in the previous works on the Stark operator) and establish the bijection between the Schwartz class potentials and the scattering data.

HOCINE MEGLOULI^{1}

^{1}University of Boumerdes, Automatique, Algeria

.CHARACTERIZATION OF COMPLEX MATERIALS BY ULTRASOUND USING PRISM TECHNIQUE H. Meglouli, A.Khebli, M’HamedBougara university, faculty of hydrocarbon and chemical SUMMARY In this article, we are interested in measuring the propagation velocity of ultrasonicwaves to characterizecomplexmaterialssuch as concrete. This work uses the new technique calledprism technique, whichisbased on the calculation of the time of flight in the specimen, during a measurementintervalbetween 24 hours and 28 days. The resultsfound are comparedwiththosefound by classical techniques (cylindrical and cubic technique). Key words: Non destructive testing, ultrasound, prism, characterization, concrete.

Bakhtier Vasiev^{1}

^{1}University of Liverpool, Mathematical Sciences, United Kingdom

One of the most important problems in contemporary science, and especially in biology, is to reveal mechanisms of pattern formation. On the level of biological tissues, patterns form due to interactions between cells. These interactions can be long-range if mediated by diffusive molecules or short-range when associated with cell-to-cell contact sites. Mathematical studies of long-range interactions involve models based on differential equations while short-range interactions are modelled using discrete type models. In this work, we use cellular automata (CA) technique to study formation of patterns due to short-range interactions. Namely, we use von Neumann cellular automata represented by a finite set of lattices whose states evolve according to transition rules. Lattices can be considered as representing biological cells (which, in the simplest case, can only be in one of the two different states) while the transition rules define changes in their states due to the cell-to-cell contact interactions. In this model, we identify rules resulting in the formation of stationary periodic patterns. In our analysis, we distinguish rules which do not destroy preset patterns and those which cause pattern formation from random initial conditions. Also, we check whether the forming patterns are resistant to noise and analyse the time frame for their formation. Transition rules which allow formation of stationary periodic patterns are then discussed in biological terms and applied to the segmentation process in fly embryos.

This work has been supported by the EPSRC scholarship to M.R. and BBSRC grant BB/K002430/1 to B.V. We are grateful to Dr Igor Potapov for fruitful discussions of the ECA and obtained results.

Constantin MEIS^{1}

^{1}CEA , National Institute for Nuclear Science and Technology, France

We show that the zero-energy electromagnetic field ground state (EFGS), a quantum vacuum component issued from the vector potential quantization enhanced at a single free of cavity photon, permits to remedy to the vacuum energy singularity in quantum electrodynamics. The EFGS is composed of a real universal electric potential field whose oscillations generate photons. The elementary charge is expressed precisely by the EFGS quantized amplitude entailing that photons as well as leptons/antileptons are issued from the same vacuum field. We also show that the gravitational constant is equally expressed precisely through the EFGS quantized amplitude putting in evidence the electromagnetic nature of gravity and linking the cosmological equations to the quantum vacuum. Fluctuations of the EFGS contribute to the cosmic electromagnetic background and may be at the origin of the dark energy and matter in the universe

Kairat Myrzakulov^{1}
, Sabit Bekov^{2}
, Ratbay Myrzakulov^{3}

^{1}L.N.Gumilyov Eurasian National University, Departament General and Theoretical Physics, Kazakhstan

^{2}L.N.Gumilyov Eurasian National University, FACULTY OF PHYSICS AND TECHNICAL SCIENCES, Kazakhstan

^{3}L.N. Gumilyov Eurasian National University, General and Theoretical Physics, Kazakhstan

In this paper, we consider cosmological model of fermion fields that is non-minimally coupled to F(T,B) gravity in Friedman-Robertson-Walker space-time. In order to determine forms F function, coupling and potential function of fermionic field, we use Noether symmetry approach. Exact cosmological solution is obtained. This solution shown that the fermion field can describe late-time acceleration of Universe.

Naceri Mostepha^{1}

^{1}oran school of economic, Mathematics, Algeria

In this paper, we present some new Lyapunov and Hartman type in equalities for nth- order equations with mixed nonlinearities: x^((n) ) (t)+p(t)|x(t)| ^(β-1) x(t)+q(t)|x(t)|^(γ-1) x(t)=0, where p(t), q(t) are real-valued functions and 0<γ<1<β<2. No sign restrictions are imposed on the potential functions p(t)and q(t). The inequalities obtained generalize and compliment the existing results for the special cases of this equation in the literature.

Alexey Tsukanov^{1}

^{1}Institute of strength physics and materials science SB RAS, Physicochemistry of Fine Dispersed Materials, Russia

At present, molecular level computer simulations such as molecular dynamics methods family have become the powerful and widely used tool for study of the nanoscale systems and processes. In the paper we applied nonequilibrium molecular dynamics (NEMD) approach with all-atom models to investigate the salt water permeation through the naturally occurring mineral (hydroxyapatite) with nanosized pores. We showed that the sub-nanometer slit-shaped nanopores in hydroxyapatite demonstrate the good salt rejection and comparatively high water permeability at the same time. This makes the nanoporous materials with connected system of slit-shaped nanopores the prospective base for the development of equipment for water treatment including water purification, desalination, decontamination and other technical and environmental applications.

A.A. Tsukanov and E.V. Shilko gratefully acknowledge the financial support from the Russian Science Foundation Grant No. 17-11-01232. S.G. Psakhie carried out the research within the Fundamental Research Program of the State Academies of Sciences for 2013–2020 years (Priority direction III.23). The research is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University.

Slimane Zaiem^{1}

^{1}University Batna1, Physique, Algeria

We obtain exact solutions to the two-dimensional Klein-Gordon oscillator in a non-commutative complex phase space to Örst order in the non-commutativity parameter. We derive the exact noncommutative energy levels and show that the energy levels split to 2m levels. We Önd that the non-commutativity plays the role of a magnetic Öeld interacting automatically with the spin of a particle induced by the non-commutativity of complex phase space. The effect of the non-commutativity parameter on the thermal properties is discussed. It is found that the dependence of the heat capacity CV on the non-commutative parameter gives rise to a negative quantity. Phenomenologically, this e§ectively conÖrms the presence of the e§ects of self-gravitation induced by the non-commutativity of complex phase space.

Sergey Lurie^{1}
, Dmitriy Volkov-Bogorodsky ^{2}

^{1}Institute of Applied Mechanics of Russian Academy of Science, Moscow, Russia and Lomonosov State University, Moscow, Russia and Computing Center FRC CSC RAS, Moscow, Russia, Methods of Mathematical Physics of Computing Center FRC CSC RAS, Russia

^{2}Institute of Applied Mechanics of Russian Academy of Science, Moscow, Russia, , Russia

A generalized theory of elasticity for nonlocal generalized displacements constructed by way of averaging over finite regions with accuracy up to high order derivatives is considered. This generalized theory, written in terms of generalized displacements, coincides with the classical elasticity theory based on the Lamé operator and on the classical defining relationships formulated with respect to the generalized deformations and stresses. At the same time, the true displacements are the solutions of the generalized gradient elasticity with an operator of higher order. This operator is represented in the form of the product of the Lamé and Helmholtz operators (or in general case, of generalized Helmholtz operator). As a result, the generalized theory contains a scale scalar parameter, which allows one to take into account the scale effects in applied problems. The solution of a boundary value problems for the formulated gradient theory of elasticity is constructed through potentials that satisfy the Laplace and Helmholtz equations on the base of the generalized Papkovich-Neuber representation. This representation allows us to introduce fundamental systems of functions for solving the problems of the generalized theory by means of introducing radial multipliers with scale parameter, which is transforming to the classical systems of functions by a passage to the limit with respect to the scale parameter. On the basis of the radial multipliers method the solution of the generalized non-local elasticity is constructed explicitly for a number of important applied problems used in the modeling and prediction of the effective properties of dispersed-reinforced composites (with allowance for scale effects) and in problems of nonsingular fracture mechanics. Theorems about the structure of fundamental solutions in the radial multipliers method and about their connection with fundamental solutions of the Helmholtz and Laplace equations are presented.

the work was supported by the Russian Foundation of Basic Research grants 16-01-16-01-00623a, 18-01-000553a and Government Foundation of Institute of Applied Mechanics of RAS

Pantea Davoudifar^{1}

^{1}Research Institute for Astronomy and Astrophysics of Maragha, Astroparticle Physics, Iran (Islamic Republic of)

A simple model were developed to give fast results about the Flux of secondary neutrons above the atmosphere. This model produces the neutron Flux in the vicinity of Earth due to events on the sun. The cases for a simple spherical ejects and a cone model with a moving axis were considered and the result are discussed for few solar neutron events.

Mikhail Surnachev^{1}
, Yuriy Alkhutov^{2}

^{1}Keldysh Institute of Applied Mathematics RAS, Computational AeroAcoustics Laboratory, Russia

^{2}Vladimir State University, , Russia

We shall discuss regularity of solutions to elliptic equations with $p(x)$-Laplacian, the model case being the $p(x)$-Laplace equation $-\mathrm{div}\, (|\nabla u|^{p(x)-2}\nabla u)=0$, $x\in D\subset \mathbb{R}^n$. The exponent $p(\cdot)$ is assumed to be a measurable bounded function, satisfying $1<\alpha\leq p(x)\leq\beta<\infty$. Thorough mathematical investigation of equations of this type and their solutions was initiated by V.V. Zhikov in the beginning of 1980s. We shall give an account of results concerning regularity of solutions to elliptic equations with $p(x)$-Laplacian. The majority of well-known results in this direction were obtained under the celebrated Zhikov's log-condition. The direction of our study is relaxing the log-condition and obtaining results on continuity, boundary regularity, Harnack's inequality and related problems under minimal assumptions on the exponent $p(\cdot)$.

This work was supported by the Ministry of Education and Science of the Russian Federation, project 1.3270.2017/4.6.

Larisa Muravleva^{1}

^{1}Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Russia

Viscoplastic or yield-stress fluids are materials which behave like a solid below a critical yield stress and flow like a viscous fluid for stresses higher than this threshold. Such materials are now commonly found in civil engineering, petroleum or food industries. Computation of viscoplastic fluid flows has always been a challenging task. Viscoplastic models are intrinsically discontinuous at the yielded-unyielded interface, which leads to numerical difficulties. For this reason, several modeling or numerical approaches have been proposed, the most popular being the Papanastasiou regularization (PR) and the Augmented Lagrangian (AL) methods, respectively. The augmented Lagrangian algorithm became the most popular way to solve viscoplastic flow problems because of its accuracy, despite the fact that the regularization approach runs in general faster. Recently, Goldsteinhis at al (2014) have suggested the accelerated variant of the alternating direction method of multipliers (ADMM) to solve the minimization problem of nondifferentiable objectives. The proposed acceleration is of the form first proposed by Nesterov for gradient descent method. We suggest to use this accelerated ADMM for the numerical solution of viscoplastic flow problem. To test the efficiency of this algorithm the benchmark lid-driven cavity problem is examined and the results are compared to relevant results using AL method. The accelerated algorithm captures the yield surfaces correctly, while maintaining a low computational cost. This approach is much faster than the AL algorithm. Furthermore, we solve numerically the squeeze flow of a viscoplastic Bingham medium with no-slip and slip yield boundary conditions at the wall. We compute the shape of the yield surface, the velocity, stress fields and squeeze force. We confirm numerically the recently obtained asymptotic solution of Muravleva (2017). Depending on the ratio of two dimensionless parameters partial slip (stick-slip) or full slip at the wall (slip) are possible. The analytical solution is in very a good agreement with the numerical results.

Anatoly Muravlev^{1}
, Ekaterina Muravleva^{2}

^{1}Lomonosov Moscow State University, Mechanics and Mathematics, Russia

^{2}Skolkovo Institute of Science and Technology, Hydrocarbon Recovery, Russia

We consider an approach to Ilyushin elastoplastic processes theory generalization on finite deformations of viscoplastic material. This approach is based on usage of two different finite strain tensors for construction (on the base of polar frame) of loading process image, divided on scalar and vector parts. Within the framework of this approach, the generalization for postulate of isotropy from Ilyushin elastoplastic processes theory on finite deformations is analyzed. This generalization uses the splitting of loading process image on scalar and vector parts. We consider the methods of experimental data processing. Data received from torsion experiments on solid cylindrical specimens. Our methods take into account the emerging heterogeneity of the stress-strain state along the radius of the specimen and allow investigate shear viscoplastic material properties at finite deformations. The strain rate influence on the variation module and deviation angle laws for Ilyushin stress vector is analyzed in simple shear process of eutectic tin and lead. This analysis shows that the deviation angle for Ilyushin stress vector has an order of magnitude less dependent on the strain rate as compared with the variation module law for this stress vector in simple shear processes (at the area of optimum strain rate for superplasticity). As a generalization of this conclusion, we formulate the properties of constitutive relations for the materials in a superplasticity state.

this work was partially supported by RFBR projects №16-01-00669, №18-08-01375, №18-501-12011.

Aleksey Kostenko^{1}

^{1}University of Ljubljana, Faculty of Mathematics and Physics, Slovenia

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Kirchhoff Laplacians on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Moreover, we establish a connection with the combinatorial isoperimetric constant, which enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. If time permits, we'll demonstrate our findings by considering trees, antitrees and Cayley graphs. Based on joint work with N. Nicolussi

Nikolaj Shamarov^{1}

^{1}MSU, Mech.-Math., Russia

Such a generalization of the Weyl quantizing of polynomials is supposed that it does not depend on the finiteness of dimension of the quantized Hamiltonian system with real separable Hilbert phase space $E=Q\oplus P$, $Dim(E)\in 2N\cup\{+\infty\}$. The construction uses Fourier transform of good enough measures on the configuration space $Q$ or on the momenta spase $P=Q^* \cong Q$. To avoid the using of the generalized measures (that was suggested by O.G.Smolyanov) for the infinite dimensional $E$, we can use Kolmogorov integrals of simplest ``transition functions’’ which are functions of a point and of a set, $f(x,A)=\phi(x)\mu(A)$ or $f=\phi\otimes \mu$. Even if the quantum observables are operators in a space of functions defined on infinite dimensional $Q$, for polynomial classical observables it is necessary to use differentiation of Borel measures on $P$.

The work was partially supported by the state program ``5—100’’ for the increasing of the competitiveness of the leading universities of Russian Federation among the leading scientific and educational centers of the world.

Viktor V. Siksin^{1}

^{1}Russian Academy of Sciences, Lebedev Physical Institute, Russia

A mathematical model of the TV-type detector calculator operation with analytical approximation of the Bragg curve in the simple form for fast calculations is proposed. An analytical formula is obtained, based on the combination of parabolic cylinder functions, well consistent with numerical methods and experimental data, which is valid for proton energies from 60 to 200 MeV. The basic problem in planning beam therapy is the determination of depth-dose dependence curves (Bragg curves). To improve the proton dosimetry efficiency, a new detector type, i.e., the TV-type detector , was proposed. The TV-type detector employs a new technique for diagnosing dose fields, recording and calculating accurate distributions and values of main beam characteristics in designing proton therapy. The mathematical model with analytical approximation of the Bragg curve in the simple and convenient form was developed, which allows fast calculations under on-line operating conditions.

Mark Dorodnyi^{1}

^{1}St. Petersburg State University, Department of Mathematics and Mathematical Physics, Russia

In $L_2 (\mathbb{R}^3 ; \mathbb{C}^3 )$, we consider a self-adjoint operator $\mathcal{L}_\varepsilon$ , $\varepsilon > 0$, generated by the differential expression $\operatorname{curl} \eta(\mathbf{x}/\varepsilon)^{-1} \operatorname{curl} - \nabla \nu(\mathbf{x}/\varepsilon) \operatorname{div}$. Here the matrix function $\eta(\mathbf{x})$ with real entries and the real function $\nu(\mathbf{x})$ are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators $\cos(\tau \mathcal{L}_{\varepsilon}^{1/2})$ and $\mathcal{L}_{\varepsilon}^{-1/2} \sin(\tau \mathcal{L}_{\varepsilon}^{1/2})$ for $\tau \in \mathbb{R}$ and small $\varepsilon$. It is shown that these operators converge to $\cos(\tau (\mathcal{L}^0)^{1/2})$ and $(\mathcal{L}^0)^{-1/2} \sin(\tau (\mathcal{L}^0)^{1/2})$, respectively, in the norm of the operators acting from the Sobolev space $H^s$ (with a suitable $s$) to $L_2$. Here $\mathcal{L}^0$ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation $\partial_\tau^2 \mathbf{v}_\varepsilon = -\mathcal{L}_{\varepsilon} \mathbf{v}_\varepsilon$, $\operatorname{div} \mathbf{v}_\varepsilon = 0$, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to $\mathbf{1}$ and the dielectric permittivity is given by the matrix $\eta(\mathbf{x}/\varepsilon)$. The talk is based on the joint work with Tatiana Suslina.

mohamed roudane^{1}

^{1}universityof Blida 1, Mechanics, Algeria

The aim of this work is to develop a new numerical calculation program for designing new contours for a supersonic axisymmetric MLN dual bell nozzle, giving a uniform and parallel flow at the exit section, to make the trip with maximum thrust in both schemes of flight altitudes, one at low altitude, usually to ensure the take-off of the aerospace craft (missile and supersonic aircraft), and the other at high altitude, usually in space when the ambient pressure decreases considerably. The design method is based on the use of the method of characteristics in axisymmetric part of a perfect gas, calorically and thermally perfect. Solving the equations is obtained by the predictor corrector algorithm of finite difference method. The nozzle shape is unknown at priori. It will be determined point by point in order to meet the desired condition. The validation of results is ensured when the critical sections ratio, numerically calculated by our computing program for high discretization, tend to theoretically critical sections ratio. In this case all other design parameters such as the thrust delivered by the nozzle so that the mass of the structure and also converge in an automatic manner to the desired solution.

Sergey Simonov^{1}

^{1}St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, ., Russia

Let a metric space $\Omega$ be given. Consider the lattice of open sets $\mathfrak O$ in it and the lattice $\mathfrak F$ of functions from $(0,+\infty)$ to $\mathfrak O$. For every open set the family of metric neighborhoods defnes a function from $\mathfrak F$, and such functions form a subset of $\mathfrak F$ which we denote by $I\mathfrak O$. Consider its closure in the topology of ponitwise convergence on $\mathfrak F$. We show that under certain assumptions the atoms of this closure form a metric space $\widetilde\Omega$ which is isometric to the original metric space $\Omega$.

The research was supported by the grants RFBR 18-31-00185 mol a, RFBR 16-01-00443 and RFBR 16-01-00635.

Tatiana Mozzhorina^{1}
, Elena Gubareva^{2}

^{1}BAUMAN MOSCOW STATE TECHNICAL UNIVERSITY (National research university of technology), Department of computational mathematics and mathematical physics, Russia

^{2}BAUMAN MOSCOW STATE TECHNICAL UNIVERSITY (National research university of technology), Department of computational mathematics and mathematical physics, Russia

The paper presents the results of optimization of the flight program for a passenger aircraft. In mathematical modeling, real operational limitations and possible changes in atmospheric parameters were taken into account. The mathematical model is characterized by the description of the flight of an airplane as a material point and the description of a gas turbine engine by a model of the first level of accuracy. The following operational limitations were considered: the possibility of cruising at certain levels, the restriction on the vertical speed with a decrease, the possibility of transition to the next echelon with an available stock of thrust of 20%. The optimization of the flight program is based on the criterion of the minimum amount of fuel expended for the flight, at a given range. Calculations were conducted for several standards of air temperature change, (depending on the climatic zone). A comparative analysis of the obtained optimization results is carried out, the degree of influence of the change in atmospheric conditions is estimated. When comparing flights in different climatic zones, the fuel costs were compared for a flight program optimized for the given atmospheric conditions and for a flight program optimized for use in the mathematical model of the standard international atmosphere.

Elena Martynova^{1}
, Nina Stetsenko^{2}

^{1}Moscow State University, Mechanics and Mathematics , Russia

^{2}Moscow State University, mechanics and mathematics, Russia

The constitutive equations for viscoelastic finite strains that generalize the differential elementary Maxwell model and use the Cauchy stress tensor, the strain rate tensor and the one-parameter family of objective Gordon – Showalter time derivatives are studied in the paper. Objective derivatives of Oldroyd, Cotter – Rivlin and Jaumann are the particular cases of this family for values of the parameter a equal to 1, -1, 0 respctively. The proposed family of constitutive equations is used in two problems for incompressible viscoelastic material: simple shear and cylinder torsion without elongation. The obtained analytical solutions of these problems showed that the used constitutive equations allow to describe the Poynting effect for any value of the parameter a. For the stepwise process of deformation of the cylinder, the result also qualitatively coincides with the available experimental data. The effect of the parameter a on the stress state in the mentioned above problems is also revealed.

the work is prepared at support of RFBR (grant No. 16-01-00669).

George Brovko^{1}

^{1}Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Russia

In the Newtonian approach to the description of movements and interactions of bodies, the concepts of objective tensors of various ranks and types are introduced, differing in the rules of transformation when replacing the reference frame. The tough classification of objective tensors is given including tensors of material and spatial types, the diagrams are constructed for non-degenerate ("analogous") relations between tensors of one and the same (any) rank and of various types of objectivity. Mappings and equations expressing dependence between objective tensor processes of various ranks and types are considered, and the fundamental concepts of frame-independence of such mappings and equations are introduced as being inherent to constitutive relations of various physical and mechanical properties in Newtonian approach. The criteria for such frame-independence are established. Simultaneously revealed became the mathematical restrictions imposed on the frame-independent mappings and equations by the objectivity types of connected tensors. The absence of such restrictions is established exclusively for mappings and equations linking tensors of material types. Using this, a generalizing concept of objective differentiation of tensor processes in time and a new concept of objective integration are introduced. The axiomatic construction of the generalized theory of stress and strain tensors in continuum mechanics is given, which leads to the emergence of continuum classes and families of new tensor measures. The axioms are proposed and a variant of the general theory of constitutive relations of mechanical properties of continuous media is constructed, generalizing the known approaches by Ilyushin and Noll, taking into account the possible presence of internal kinematic constraints and internal body-forces in the body. The concepts of the process image and the properties of the five-dimensional Ilyushin’s isotropy are generalized on the range of finite strains.

The work is prepared at support of RFBR (grant No. 16-01-00669)

Eleonora Zavoychinskaya^{1}

^{1}Moscow State University, Mechanical and Mathematical Dept., Russia

A phenomenological model of brittle fatigue fracture of metals and alloys at proportional loading is presented. The model is constructed within the framework of the physical-mechanical approach as a system of hypotheses about the probability of defect growing on different scale structural levels (such as brittle micro- and macro cracks). From the whole scale hierarchy here is considered six scale structural levels in accordance with the stages of the metal structure evolution and different physical mechanisms. Here are formulated constitutive relations for the fracture probability at each level. The amplitude of the maximum principal stress is selected as a variable of these relations, and the material functions are determined on the results of fatigue tests at symmetric uniaxial and biaxial loading and shear and take into account the principal stress relations according to known fatigue strength criteria of metals. Here were obtained fatigue curves on each of six defect levels and fatigue macro crack formation curves for proportional tension - torsion loading and biaxial loading with different amplitude ratios. It is shown the correlation between calculated and experimental data.

the work is financially supported by the Ministry of Education and Sciences of the Russian Federation within the federal target program "Development of physical and mechanical foundations and methods for complex system calculation in solid mechanics at thermo-mechanical loadings"(No. AAAA-A16-116071210055-5).

George Alexandru Nemnes^{1}
, Tudor Mitran^{2}

^{1}Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), The Department of Computational Physics and Information Technologies (DFCTI), Romania

^{2}Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), , Romania

The graphene bilayer (GBL) has many of the electronic, mechanical, thermal properties of the graphene monolayer counterpart, however with a notable difference: one can induce an electronic gap using a perpendicular electric field, which induces an inversion asymmetry between the two layers [1]. This asymmetry can be provided also by extrinsic doping [2] and, for high doping levels, the system becomes a degenerate semiconductor. In this case a gap opens either above of below the Fermi level, depending on the doping type. We investigate here GLB systems with extrinsic impurities in the presence of external electric field, using density functional theory (DFT) calculations implemented by SIESTA package. We show that the position of the gap can be further tuned by the external field, which may switch the behavior from degenerate to intrinsic semiconductor. Furthermore, we compare the formation energies for several group-III and group-V dopants and the binding energies between the two graphene layers. References: [1] Yuanbo Zhang et al., Nature 459, 820 (2009); [2] Y. Fujimoto et al., Surf. Sci. 634, 57 (2015).

This work was supported by PN 18090205 and by Romania-JINR cooperation project JINR Order 322/21.05.2018, no. 29.

Valery Timofeev^{1}

^{1}BAUMAN MOSCOW STATE TECHNICAL UNIVERSITY (National research university of technology), Department of computational mathematics and mathematical physics, Russia

The procedure of the mathematical simulation of the subsonic flow around the lengthening bodies for the flow conditions with the flow separation in the environment of the line of ground shear is developed on the base of the concept of binding- inviscid interaction. The subsonic motions of gas with the moderate values of the module of the velocity vector of the incident flow were investigated. Adapted the diagram of flow with the equivalent body, whose flow around of the inviscid flow determined speeds and pressures on the surface being investigated. For the purpose of the reduction of the expenditures of computational power it was proposed not to carry out the complex iterative process of the multilevel refinement of the forms of the nose and tailed sections of the equivalent body. The configuration of the surface of the tail section of the equivalent body it was proposed to assign in the form the totality of sections of sufficiently simple (simplex) form. The semi-infinite section of tail section downstream departed to infinity, preserving in the cross section the constant value of effective diameter. Transition section connected semi-infinite with the nose section of the equivalent body on the line of flow breakaway. For determining the parameters, which characterize the geometry of the tail section of the equivalent body, were used Horner's formula for the aerodynamic coefficient of base drag and relationship the theories of distant track. The numerical simulation of the inviscid flow around equivalent body was carried out by the method of single vortices. The results of the mathematical simulation of the flow around axisymmetrical cylindrical bodies with the head part of the ogival form are represented. When, before the ground shear, the additional tapered conical part is present, calculated distribution the pressures were compared with the experimental data. In the case of the absence of the tapered part are represented the dependences, which connect the geometric parameters of the streamlined body and tail section of the equivalent body, and also the distribution of speed and pressure on the surface being investigated.

Igor Faskheev^{1}

^{1}MSU, Mechanics and mathematics, Russia

The work is aimed at solving a complex problem of the rapid flow of liquid through a solid porous framework, taking into account its finite strains. In this work modern model of a porous saturated media with complex internal interactions was used. This model generalizes classical works in the field of the poromechanics.

The work is prepared at support of RFBR (grant No 16-01-00669).

Olga Zagubnaya^{1}

^{1}Institute of Cytochemistry and Molecular Pharmacology, Mathematical modeling and statistical analysis of results, Russia

Excitatory amino acid transporters (EAATs 1-5) bring into action the precise control of synaptic transmission of major mammalian excitatory amino acid L-glutamate acting as secondary active transporters and ion channels at the same time. EAATs are essential for the most aspects of normal brain functioning including cognition, memory and learning. However, excessive glutamate stimulation can result in excitotoxicity (glutamate-related cell injury or death). This type of damage may contribute to a large number of severe neurodegenerative disorders. Therefore, understanding of the glutamate transporter functional mechanism is of great importance. In order to clarify the aspects of EAAT functioning virtual computer simulator was proposed. So far, the overall structure of EAATs is not fully described. However, mammalian transporters are considered to be trimeric proteins and each protomer is an independently and stochastically functioning subunit. The transport cycle of protomer is described as a co-transport of 1 glutamate with 3 sodium ions and 1 proton into the cell, and counter-transport of 1 potassium ion into the external solution. The sequence of elemental events in glutamate transporter cycle is included in a model as a sequence of stochastic steps. The protein undergoes several conformational changes during its whole transport cycle. Some of them are related with substrate binding and its translocation inside the cell across the membrane, another comes along with substrate dissociation and relocation of the transporter to the outside position. The probabilities and the times of these processes should be also included in the model. In addition, EAATs possess thermodynamically-uncoupled anion conductance, which is a chloride current under physiological conditions. Structure and functional properties of an anion channel are vigorously investigated. It is a common understanding now, that anion channel opening is achieved via additional branching conformational change and not a part of the transport cycle. Existence of such a current has been proposed to preserve excessive cell excitation and prevent additional glutamate release. The probability model of glutamate transport through the membrane can highlight the uncertain functional properties of the enzyme. The possibility to take into account environment conditions, physiological parameters and structure peculiarities makes the computer modeling of the neurotransmitting process of great importance.

Zagubnaya Olga. A., Boronovskiy Stanislav. E., Nartsissov Yaroslav. R.

Vladimir Vlasov^{1}

^{1}Federal Research Center "Computer Science and Control" of the RAS, Applied Mathematical Physics, Russia

Let the domain $\mathcal{B}_\varepsilon$ on the complex $w$--plane be the exterior of a semi--infinite slot of width $2 \varepsilon$ with sides parallel to the real axis and with a piecewise smooth bottom $\sigma_\varepsilon \ni \{0\}$. More precisely, a family of domains $\{\mathcal{B}_\varepsilon\}$, $\varepsilon \in (0, 1]$, is defined by the relation $\mathcal{B}_\varepsilon = \{w:\, w / \varepsilon \in \mathcal{B}_1\}$. Suppose further that a subdomain $g_\varepsilon \subset \mathcal{B}_\varepsilon$ is simply--connected, and its boundary consists of two links, $\partial g_\varepsilon = \gamma_\varepsilon \cup l_\varepsilon$, where the arc $\gamma_\varepsilon$ includes the bottom $\sigma_\varepsilon$, along with the contiguous sections of the contour $\partial \mathcal{B}_\varepsilon$, and the arc $l_\varepsilon$ is located in domain $\mathcal{B}_\varepsilon$ in such a way that $l_{\varepsilon_1} \subset l_{\varepsilon_2}$ for $\varepsilon_1 > \varepsilon_2$. As $\varepsilon$ tends to zero, the bottom $\sigma_\varepsilon$ is homothetically contracted to the point $w = 0$, the slot bounded by $\partial \mathcal{B}_\varepsilon$ becomes the cut $\partial \mathcal{B}_0 = \mathbb{R}_-$ (negative real semiaxis), and the domain $g_\varepsilon$ with a slot becomes the domain $g_0$ with a cut. In the domain $g_\varepsilon$ we consider the Dirichlet problem for the Poisson equation $\Delta u_\varepsilon (w) = f (w)$ with boundary conditions: $u_\varepsilon (w) = 0, w \in \gamma_\varepsilon$; $u_\varepsilon (w) = \varphi (w), w \in l_\varepsilon$. The right-hand side $f$ is H\"older in $\mathcal{B}_0$ and, in addition, $f$ is equal to 1 in some neighborhood of $\partial\mathcal{B}_0$. The boundary function $\varphi$ belongs to $L_2 (l_0)$. The solution is sought in a special class of Hardy type, in which the indicated problem is uniquely solvable. An analytic-numerical method for solving this problem is based on the use of a conformal mapping $w = \mu_\varepsilon (\zeta)$ of the upper half-plane $\mathbb{H}$ onto the domain $\mathcal{B}_\varepsilon$ satisfying the normalization: (1) $\mu_\varepsilon (0) = w_\varepsilon$, where $w_\varepsilon$ is a point of the arc $\sigma_\varepsilon$; (2) $\mu_\varepsilon (\zeta) \sim - \zeta^2$, $\zeta \to \infty$. Numerical implementation of the method is performed for several specific bottom shapes, for which the mapping was constructed in explicit form. Using this mapping, we translate the problem into the one on the plane $\zeta$ for the equation $\Delta U_\varepsilon (\zeta) = F_\varepsilon:= f \circ \mu_\varepsilon (\zeta) |\mu_\varepsilon (\zeta)|^2$ with respect to the function $U_\varepsilon (\zeta) := u_\varepsilon \circ \mu_\varepsilon (\zeta)$ in the domain $G_\varepsilon := \mu^{-1} (g_\varepsilon)$ with boundary conditions $U_\varepsilon|_{\Gamma_\varepsilon} = 0$, $U_\varepsilon|_{\mathcal{L}_\varepsilon} = \Phi_\varepsilon \in L_2 (\mathcal{L}_\varepsilon)$, where $\Gamma_\varepsilon := \mu^{-1} (\gamma_\varepsilon)$ and $\mathcal{L}_\varepsilon} := \mu^{-1} (l_{_\varepsilon})$. Let $\mathbb{K}_R = \{\zeta: |\zeta| < R, \mathrm{Im}\, \zeta > 0\}$ be a semidisk of radius $R$ such that $G_\varepsilon \subset \mathbb{K}_R$. Noting that the function $F_\varepsilon (\zeta)$ admits a smooth extension to the whole semidisk, we consider in it the boundary-value problem: $\Delta V_\varepsilon = F_\varepsilon (\zeta), \zeta \in \mathbb{K}_R$; $V_\varepsilon (\zeta) = 0, \zeta \in \partial \mathbb{K}_R$. Its solution is written out in an ordinary way in an explicit form in terms of the normalized eigenfunctions of the Laplace operator in the semidisk. In the polar coordinate system $(r, \varphi)$ those functions have the form $V_{k,\, m} (r, \varphi) =(\sigma_{k,\, m})^{-1} J_k (j_{k,\, m} r) \sin (k \varphi)$, $k,\, m \in \mathbb{N}$, where $J_k$ are the Bessel functions, the numbers $j_{k,\, m}$ are their zeros, and $\sigma_{k,\, m} = \sqrt{\pi / 2}\, |J'_k (j_{k,\, m})|$. They form, as is well known, an orthonormal basis in $L_2 (\mathbb{K}_R)$. Returning to the plane $w$, we obtain $v_\varepsilon (w) = V_\varepsilon \circ \mu^{-1}_\varepsilon (w)$. Then we arrive at the following problem for the function $\psi_\varepsilon = u_\varepsilon - v_\varepsilon}$ in the domain $g_\varepsilon$ on the plane $w$: $\Delta \psi_\varepsilon (w)= 0, w \in g_\varepsilon$, $\psi_\varepsilon (w) = 0, w \in \gamma_\varepsilon$, $\psi_\varepsilon (w) = \varphi (w) - v_\varepsilon (w), w \in l_\varepsilon$. We construct its solution by means of an approximate system of functions $\Omega_n = \mathrm{Im} (\mu^{-1})^n$, $n \in \mathbb{N}$, that are harmonic in the domain $g_\varepsilon$, vanish on the arc $\gamma_\varepsilon$; in addition, the system $\{\Omega_n\}$ is complete and minimal in $L_2 (l _\varepsilon)$. The asymptotics of the gradient of the solution $u_\varepsilon$ obtained at the point $w_\varepsilon$ at the bottom of the slot when it is narrowed has the form $\mathrm{grad} u_\varepsilon (w_\varepsilon) = i A_1 (0) \overline{\mathcal{F}_1' (w_1)} \varepsilon^{- 1 / 2} + O (1), \varepsilon \to 0$.

The work was supported by the Ministry of Education and Science of RF (the Agreement 02.a03.21.0008), and by RFBR (projects 16-01-00781, 16-07-01195).

Yulia Meshkova^{1}

^{1}St. Petersburg State University, Chebyshev Laboratory, Russia

The talk is devoted to homogenization of periodic differential operators. In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider matrix elliptic second order differential operator $B_\varepsilon$. Coefficients of the operator depend on $\mathbf{x}/\varepsilon$. Operator $B_\varepsilon$ has terms of the first and zero order. For the semigroup $e^{-B_\varepsilon t}$, $t>0$, we obtain approximation in the $L_2$-operator norm with the error estimate of the order $O(\varepsilon)$ for fixed time. In this approximation, the corrector is taken into account. Our result is generalisation of the result by E. S. Vasilevskaya (2009) for the operator without the lower order terms. The method is based on the scaling transformation, Floquet-Bloch theory, and analytic perturbation theory.

Research is supported by the Russian Science Foundation grant №14-21-00035.

Yuriy Dimitrienko^{1}
, Andrey Zakharov^{2}
, Mikhail Koryakov^{3}

^{1}Bauman Moscow State University, Computational Mathematics and Mathematical Physics, Russia

^{2}Bauman Moscow State Technical University, Computational Mathematics and Mathematical Physics, Russian Federation

^{3}Bauman Moscow State Technical University, Fundamental Sciences, Russia

An algorithm for the numerical solution of the conjugate problem of external aerodynamics of high-speed aircraft and thermomechanics of heat-shielding structures from thermally decomposing polymer composite materials is proposed. The algorithm is step-by-step and at each time step includes the steps of solving the Navier-Stokes equations with chemical reactions, calculating the heat fluxes to the structure, calculating the internal heat transfer in the material of the structure, calculating the structural ablation, calculating the temperature field in the structure, calculating the stress-strain state of the composite structure materials taking into account phase transformations and temperature degradation of the materials characteristics. In the calculations, the temporary effects of elastic characteristics changing and shrinkage caused by the processes of thermal destruction of the composite are also taken into account. The results of simulation for a model aircraft design illustrating the application of the proposed algorithm are presented.

SHAGUFTA TABASSUM^{1}

^{1}MAHARASHTRA UDAYAGIRI MAHAVIDYALAYA, UDGIR, PHYSICS AND ELECTRONICS , India

The study of static dielectric properties in a binary mixture of 1,2 dichloroethane (DE) and n,n dimethylformamide (DMF) polar liquids has been carried out in the microwave frequency range of 10 MHz to 30 GHz using time domain reflectometry technique for 11 different concentrations at 10ºC temperature. The complex permittivity spectra were obtained by the time domain reflectrometry (TDR) technique. The relaxation process in dielectric can be consider as passing of a dipole across a potential barrier that separates the minima of energy, the change in energy can be interpreted by using thermodynamic parameters.

The department of Science and Technology, New Delhi is grateful acknowledged (Project no. SR/S2/LOP-25/2007) for laboratory facility. We are thankful to the A. C. Kumbharkhane for providing TDR facility and Dr. S. C. Mehrotra, Ramanujan Chair Professor, Department of Computer Science and IT, Dr. B.A.M. University, Aurangabad for their valuable discussion and suggestions.

Igor Krasnov^{1}
, Anna Salnikova^{2}

^{1}Bauman Moscow State University, , Russia

^{2}BAUMAN MOSCOW STATE TECHNICAL UNIVERSITY (National research university of technology), Department of computational mathematics and mathematical physics, Russia

The paper presents deals with the problem of finding the geometric dimensions of a cavity-type defect by measuring the thermal fields. As a structural element, a plate with a rectangular cavity is considered. The method of solving this problem is based on the optimization approach. Optimization is carried out using methods of finding extremums without calculating the derivative, in this paper this is the Hook-Jeeves method. The results of solving this problem make it possible to improve the accuracy of determining the location of the defect and show the advisability of generalizing the problem under consideration to more complex structural elements. The results obtained in this paper can be used in problems of thermal non-destructive testing.

Tatyana Suslina^{1}

^{1}St. Petersburg State University, Department of Mathematics and Mathematical Physics, Russia

We study homogenization of a stationary Maxwell system with periodic coefficients in ${\mathbb R}^3$ and in a bounded domain $\mathcal{O}\subset {\mathbb R}^3$. Assume that the dielectric permittivity and the magnetic permeability are given by $\eta({\mathbf x}/\varepsilon)$ and $\mu({\mathbf x}/\varepsilon)$, $\varepsilon>0$. Here $\eta$ and $\mu$ are positive definite and bounded $(3\times 3)$-matrix-valued functions, periodic with respect to some lattice. The classical results show that, as $\varepsilon \to 0$, the solutions of the Maxwell system with such coefficients converge weakly in $L_2$ to the solution of the homogenized Maxwell system with the effective coefficients $\eta^0$ and $\mu^0$. We improve the classical results and find approximations for the solutions in the $L_2$-norm. The approximations involve not only the solution of the homogenized Maxwell system, but also some correctors of zero order. For the problem in ${\mathbb R}^3$, the error of approximation is $O(\varepsilon)$, while for the problem in a bounded domain the error is $O(\sqrt{\varepsilon})$. This is explained by the boundary influence.

Nikita Senik^{1}

^{1}St.Petersburg State University, Department of Physics, Russia

In homogenization theory, one is interested in studying asymptotic properties of solutions to differential equations with rapidly oscillating coefficients. We will consider such a problem for a matrix strongly elliptic operator $\mathcal A^\varepsilon = -\operatorname{div}A(x,x/\varepsilon)\nabla$ on $\mathbb R^d$, where $A$ is H\"older continuous of order $s\in[0,1]$ in the first variable and periodic in the second. We do not require that $A^*=A$, so $\mathcal A^\varepsilon$ need not be self-adjoint. It is well known that the resolvent $(\mathcal A^\varepsilon-\mu)^{-1}$ converges, in some sense, as $\varepsilon\to0$. In this talk, we will discuss results regarding convergence in the uniform operator topology on $L_2(\mathbb R^d)^n$, i.e., the strongest type of operator convergence. We present the first two terms of an approximation for $(\mathcal A^\varepsilon-\mu)^{-1}$ and the first term of an approximation for $(-\Delta)^{s/2}(\mathcal A^\varepsilon-\mu)^{-1}$. Particular attention will be paid to the rates of approximation.

Igor Bataronov^{1}
, Viktor Dezhin^{2}

^{1}Voronezh State Technical University, Higher mathematics and physical and mathematical modeling, Russia

^{2}Voronezh State Technical University, Higher mathematics and physical and mathematical modeling, Russia

The vibration spectra of the edge, mixed and screw dislocations, depending on the dislocation segment length, are considered. The value of Poisson's ratio 1/3 is assumed. The normalized values of the eigenfrequencies and damping coefficients for different frequency numbers by equating the inverse matrix of dislocation oscillators generalized susceptibilities [1] to zero were found. It is seen from the dependences obtained that in the case of a screw dislocation the transformation of the dislocation oscillators generalized susceptibility poles occurs. For edge and mixed dislocations the transformation of the poles was not observed. This agrees with the results of the vibration spectrum orientation dependence study. The authors suggest that these differences are due to the presence of a screw dislocation quasilocal vibration branches [2]. 1. Bataronov I.L. and Dezhin V.V. The generalized susceptibility of dislocation segment in nondissipative crystal. J. Phys.: Conf. Ser. V.738. 012108. (2016). 2. Bataronov I.L. and Dezhin V.V. On the natural small vibrations of dislocation in an isotropic medium. J. Phys.: Conf. Ser. V. 936. 012035. (2017).

Igor Bataronov^{1}
, Viktor Dezhin^{2}

^{1}Voronezh State Technical University, Higher mathematics and physical and mathematical modeling, Russia

^{2}Voronezh State Technical University, Higher mathematics and physical and mathematical modeling, Russia

Dependences of the dislocation segment vibration eigenfrequencies and the damping coefficients on the angle between the dislocation line and its Burgers vector are found. For the ratio of the transverse sound waves velocity to the longitudinal sound waves velocity, the value 1/2 was used, the different lengths of the dislocation segment were considered. When calculating and plotting the dependency curves, the normalized values of the eigenfrequencies, damping coefficients, and lengths of the dislocation segment were used. The normalized values of the eigenfrequencies and damping coefficients for different frequency numbers by equating the inverse matrix of dislocation oscillators generalized susceptibilities [1] to zero were found. At large values of the dislocation segment normalized lengths, sharp changes in the frequencies values with numbers 5 and more were observed on the orientation dependence graphs. To explain this, the surface level of the real and the imaginary part of the inverse generalized susceptibility built. [1] Bataronov I.L. and Dezhin V.V. The generalized susceptibility of dislocation segment in nondissipative crystal. J. Phys.: Conf. Ser. V. 738. 012108. (2016).

Xu Zhang^{1}

^{1}Shandong University at Weihai, School of Mathematics and Statistics, China

This work is concerned with the chaotic dynamics of a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the existence of strange attractors, and shows that some maps are chaotic in the sense of Li-Yorke or Devaney.

A K Fazlul Haque^{1}

^{1}University of Rajshahi, Department of Physics, Bangladesh

A new model for the calculation of the yield of the electron-induced secondary electron is proposed. We modify the modified electron stopping power formula derived by Nguyen-Truong by adding a multiplicative effective charge factor for the incident electron and an additive term within the logarithmic function of the formula. We then connect this modified stopping power expression with the secondary electron yield formula given by Batrakin et al., with the inclusion of an adjustable species-dependent parameter, to frame the proposed model. The new model is applied to calculate the secondary electron yield of 42 elemental solids from Be to U over the incident energy 10 eV - 100 keV. The present prediction is compared with the available experimental data and the prediction of our previous model. We find a reasonable agreement of the present model with the experimental data of almost all the species considered herein and notice the superiority in performance of present model over the previous model.

Kwok Yip Szeto^{1}

^{1}Hong Kong University of Science and Technology, Physics, Hong Kong

Chun Yin Yip and Kwok Yip Szeto Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong phszeto@ust.hk The covering time problem of a network by random walker is important for search when there is no knowledge of the target. To address this problem, we introduce a two-colored-random-walker model. Given a network, all unvisited nodes are initially white and become permanently colored by its first visitor. Each random walker preferentially walks to a white neighboring node or follows the strategies predefined in its intelligence sequence and become peer-avoiding or self-avoiding if there are no white nodes near him. We apply this model to square lattices and small-world networks and study the cover time, i.e. the time steps needed to color all nodes. Furthermore, we search for the combination of strategy sequence that will minimize the cover time. For square lattices, the cover time of the combined intelligence sequence is more likely to be shorter if the walkers’ sequences are neutral or have a shorter distance between them. For small-world networks, the cover time is in general shorter due to the effect of long-range rewiring. Genetic algorithm is also applied to search for good sequence combinations to cover a network.

Yip Chun Yin acknowledges the support of UROP-HKUST

Arslan Kusmaev^{1}
, Ekaterina Kots^{2}
, Karina Kienskaya^{3}
, Makar Sardushkin^{4}

^{1}Mendeleev University of Chemical Technology of Russia, Department of Pharmaceutical Technologies and Biomedical Preparations, Russia

^{2}Lomonosov Moscow State University, , Russia

^{3}Mendeleev University of Chemical Technology of Russia, , Russia

^{4}Mendeleev University of Chemical Technology of Russia, , Russia

Steroids play an important role in the life of insects. However, transport of steroids and other hydrophobic molecules through hydrophilic media requires a specialized transport system. In the yellow mealworm Tenebrio molitor that function is implemented by the carrier protein THP12. This work focuses on study of dynamic properties of free protein and its complex with an ergosterol using molecular modelling. Analyses of calculated molecular dynamics trajectories for both free and ligand-bound THP12 revealed a conformational shift arising from a ligand binding. Further free binding energy and dissociation constant of a complex were determined.

This research was supported by the grant of the Mendeleev University of Chemical Technology of Russia (project number 003-2018). Authors are grateful to the M.V. Lomonosov State University Research Supercomputer Center. Authors express special gratitude to prof. Alexander Nemukhin and prof. Vladimir Tsirelson.

Aleksandr Ivanov^{1}

^{1} St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Laboratory of Mathematical Problems of Physics, Russia

In this work a matrix formalism for calculating the trace part of the heat kernel of the covariant Laplace operator is described. It is well known that the heat kernel can be expanded in a series in powers of time. The coefficients of this series are named Seeley-DeWitt coefficients and they play important role in physics, but the calculating of these is very hard problem. In this work the author show that the last one can be presented as algebraic problem without integrating and taking the derivatives. For this purpose the author constructs the space of matrices and operators on the last ones. Also the created formalism has interesting applications in other fields (for example, relation of operators with the generating functions of numbers).

Revaz Kavtaradze^{1}
, Tamaz Natriashvili^{2}
, Yichun Wang^{3}

^{1}Bauman Moscow State Technical University, Reciprocating Engine (E-2), Russia

^{2}Rafael Dvali Institute of Machine Mechanics, Mindeli Str.10, Tbilisi, 0186, Georgia, , Georgia

^{3}School of Mechanical Engineering, Beijing Institute of Technology, 5 South Zhongguancun Str., Haidian District, Beijing, 100081, China, , China

The concept of a hydrogen diesel with direct gaseous hydrogen injection, despite the obvious advantages as compared to a traditional diesel, is poorly explored. A detailed survey of the operating process, the formation of nitrogen oxides and local heat transfer in the combustion chamber of a hydrogenous diesel has been carried out in three-dimensional space using a mathematical model verified by experimental data. The specific features of the thermophysical processes in the combustion chamber of the hydrogen diesel engine have been specified, providing the applicable ecological and efficient indices of a hydrogen diesel engine.

Boris Kryzhanovsky^{1}
, Leonid Litinskii^{2}

^{1}Scientific Research Institute for System Analysis, Russian Academy of Sciencies, Center of Optical Neural Technologies, Russia

^{2}Scientific Research Institute for System Analysis RAS, Center of Neural Optical Technologies, Russia

The partition function of a thermodynamic system of N spins can be calculated with the aid of its spectral density, which characterizes the degeneracy of each energy level. We compare two approximations for the spectral density, which are the approximation by polynomials [1] and the nv-approximation presented earlier by us [2]. The approximation by polynomials provides the results that contradict the fundamental physical principles, namely, the system reaches the ground state at a finite temperature. In addition, this approximation fails to predict correctly the phase transition type. The nv-approximation is free from these defects. Furthermore, it allows us to calculate the critical temperature, which deviates from the commonly accepted value by less than 3%.

The work was supported by the RBRF grants 16-01-00626 and 18-07-00750. The authors are grateful to Dr. Inna Kaganova and Dr. Marina Litinskaya for help during the course of the work.

Ruben Khachaturov^{1}

^{1}Dorodnicyn Computing Centre, FRC CSC RAS, 23, Russia

The advances in the development of new technologies are closely related to the advance in nanotechnologies. Therefore, the development of mathematical models and methods for the analysis of properties of multilayer nanostructures is of great importance. An important parameter of multilayer nanostructures is the quality of internal surfaces, which strongly depends on the methods used to grow such structures and on the dynamics of their growth. The analysis of experimentally obtained angular spectra of the intensity of reflected X-rays is an appropriate method for determining static characteristics of inhomogeneities in internal surfaces that allows one to perform nondestructive control of the quality of nanostructures. This work is devoted to the development of mathematical models and methods for determining the value of the interlayer roughness (the depth of the transient layer) and other parameters of multilayer nanostructures from the angular spectrum of the intensity of reflected X-rays. The proposed mathematical model for solving the direct problem of X-ray propagation and the distribution of their electromagnetic field within a multilayer nanostructure takes into account the refraction effect due to the inclusion of the second derivative with respect to the structure depth. A numerical method for solving the resulting problem is developed, and the numerical results are analyzed. The equivalence set method is described and proposed to solve the inverse problem of determining various parameters of nanostructures from the angular spectrum of the intensity of reflected X-rays. To verify the correctness of the proposed mathematical model and the accuracy of the computational method, we compared the computed angular spectra of the intensity of reflected X-rays with the experimentally measured data for various multilayer nanostructures for which the roughness is known. The average relative error was in the range from 0.10 to 0.25%, which does not exceed the error of the experimental data. Such a high accuracy of solving the direct problem suggests that the inverse problem of determining various parameters of nanostructures from the angular reflection spectrum can be stated and considered. It is of great practical importance to find the values of the parameters (for example, the mean roughness, the number and the order of layers, their composition, and so on) for which the intensity of reflected X-rays has a specified (or desired) form. The proposed approach can be extended for the case of determining several independent parameters of a multilayer nanostructure simultaneously. In this case, instead of the grid in one parameter, a grid in the space of all the parameters under consideration must be defined. It is clear that the dimension of the problem increases, but theoretically the approach is quite realizable.

Tatiana Gubaidullina^{1}
, Natalia Ivanova^{2}
, Svetlana Absalyamova^{3}
, Tatiana Yerina^{4}

^{1}Kazan Federal University, Institute of Management, Economics and Finance, Russia

^{2}Kazan Federal University, , Russia

^{3}Kazan Federal University, Institute of Management, Economics and Finance, Russia

^{4}Kazan Federal University, Institute of Management, Economics and Finance, Russia

The problem of interaction between the economy and the environment gains special significance every year. At present, anthropogenic impact of industrial production on the environment has exceeded the sustainability limits of natural systems. Further conventional development of the economy is confronted with the limitations of the self-restoration potential of ecosystems: the non-renewable character of productive resources and the limited character of the environment to take production waste. The purpose of the paper is to identify the main directions for the implementation of national policies of sustainable development with account of the fundamental conceptual premise. The paper outlines the main principles, policies and scenarios for sustainable development. New components of environmental policy are presented. National strategies for sustainable development of the United States, China, Russia, the EU countries are considered, special attention is paid to the strategy adopted in the Netherlands. Their specific features reflecting the interests of these countries’ population and the role of national economies in world economic relations are determined. The paper explores the possibilities of the SDG Index for the identification, assessment and comparative analysis of countries' actions to achieve the goals of sustainable development. Based on the results of the Sustainable Development Index for 2017, we carried out a cross-country analysis of the sustainability strategies of countries from the perspective of achieving sustainable development goals (LRC). It helped to identify the problems of using the index to track progress towards the LRC. The strengths and weaknesses of national sustainable development strategies of different countries, including Russia, have been determined. It is shown that Russia has achieved high positions in eradicating poverty, with a significant lag behind other goals of sustainable development, especially such as the elimination of inequality, the provision of a healthy lifestyle, access to justice and the creation of effective institutions. Finally, the paper indicates the main directions of sustainable development in Russia.

Marko Turzynsky^{1}
, Olga Rozanova^{2}

^{1}Moscow State University, Mathematics and Mechanics, Russia

^{2}Moscow State University, Mechanics and Mathematics, Russia

We show that with the multidimensional system of gas dynamics with a special forcing one can associate a quadratically nonlinear ODE system which describes a special class of motion. The system can be obtained by two different ways. In particular, we study the influence of Coriolis and frictional terms. We review the result about the non-frictional case and study the influence of constant dry friction.

Alexander Skaliukh^{1}

^{1}Southern Federal University, Institute of Mathematics, Mechanics and Computer Sciences named after II. Vorovich, Russia

Modeling of irreversible processes of polarization and deformation of polycrystalline ferroelectric-ferroelastic materials is an actual problem in the light of new applications of piezoceramic materials in the form of thin films and as elements of composite materials. In the basis of modeling the processes of polarization by an electric field or depolarization by mechanical stresses lies down the constitutive relationships, both for reversible and irreversible parameters. Reversible parameters are related to the parameters of external loads by linear algebraic operators, and irreversible parameters are related either by differential or hereditary operators. However, reversible and irreversible parameters are interrelated; therefore, increments of ones and others are calculated by one computational scheme. The appearance of residual parameters alters the structure of the material, leading to phase transitions of the "solid" - "solid" type. Material, passing from unpolarized to polarized state, changes the anisotropy class. The question of the class of anisotropy of partially polarized or depolarized ceramics is important and little studied in the contemporary literature. On the other hand, the simulation of irreversible phenomena in which the residual parameters vary is also an actual, far from fully studied task. In the present investigation, the constitutive relationships between reversible parameters and loading parameters are obtained in the form of algebraic operator relations. Functional relationships between the elastic, piezoelectric and dielectric constants of the material and residual parameters are established for any polarization-depolarization processes. On the basis of the thermodynamics of irreversible processes, the relations for tensors of elastic constants, piezoelectric and dielectric modules are obtained as linear functions of residual polarization vector and the residual strain tensor. It is shown that the tensors of elastic and dielectric constants depend only on the residual strain, while the tensor of the piezoelectric constants depends only on the residual polarization vector. Constitutive relations for irreversible parameters are obtained in the form of equations in differentials. For this purpose, a generalized model of polarization-depolarization is constructed on the basis of taking into account the material structure, the energy criterion of domain switching, and the energy balance in a representative volume. Comparison with the results of known models is given, large and small loops of dielectric and strain hysteresis are constructed, and cases of anisotropy for reversible parameters are investigated. It is shown that the constructed model successfully extends for many types of ceramics. The present study is the component part of the complex model of irreversible processes of polarization-depolarization of complete ferroelectrics-ferroelastics.

This work was supported by the Russian Foundation for Basic Research under Grant 17-08-00860-a

Viktor Dezhin^{1}

^{1}Voronezh State Technical University, Higher mathematics and physical and mathematical modeling, Russia

Two adjacent segments of an infinite rectilinear dislocation in a crystal that perform small oscillations under the action of an external force are considered. The crystal was simulated by a continuous isotropic medium. Integral equation of two dislocation segments vibrations in view of fixing conditions is written down. It had the form of an integral Fredholm equation of the first kind. Solution for the dislocation displacement is sought in the form of a series. The external force is also expanded into a series of functions in this system. The equation of two adjacent dislocation segments oscillations in the matrix form is written. Expressions for the matrix elements of dislocation oscillators are found. Similar expressions in [1] were obtained earlier. An equation describing the natural oscillations of two adjacent dislocation segments is obtained. [1] Bataronov I.L. and Dezhin V.V. The generalized susceptibility of dislocation segment in nondissipative crystal. J. Phys.: Conf. Ser. V. 738. 012108. (2016).

Armine Ulukhanyan^{1}

^{1}Bauman Moscow State Technical University, Mathematical Physics, Russia

Expressions for the components of the transversely isotropic and orthotropic tensors of the moduli of elasticity for the classical and micropolar medium are presented. The equations of motion for an arbitrary homogeneous anisotropic classical elastic medium, as well as for an arbitrary anisotropic micropolar elastic medium with a center of symmetry are written out. Based on these equations, general dispersion equations for both medium are obtained. Furthermore, the dispersion equations and the wave propagation velocities in three different directions for transversely-isotropic and orthotropic micropolar infinitely elastic medium are given.

JESUS GONZALEZ^{1}
, Liliana Selene Martínez Chima 57439768^{2}
, Henry Sánchez^{3}
, Jesús Beltrán^{4}
, José Escobar^{5}
, Sandra Arciniegas ^{6}

^{1}Universidad del Magdalena, Magdalena, Colombia

^{2}Universidad del Magdalena, Magdalena, Colombia

^{3}Universidad del Magdalena, Facultad de Educación, Colombia

^{4}Universidad del Magdalena, Facultad de Ingeniería, Afghanistan

^{5}Universidad del Magdalena, Facultad de Ciencias Básicas, Colombia

^{6}Universidad del Magdalena, Education, Afghanistan

The aim of this study is to develop a computational method that improve the ability by using novel virtual physical experimentation environment helping their problem solving processes in Newton' s laws of motion. Undergraduate students’ understanding of physics concepts in the domain of mass and force, Newton' s First and Second Laws: Inertial Frames, the Third Law and Conservation of Momentum, was tested in a pre- and posttest design that involved 132 participants assigned to the four experimental groups and 32 participants assigned to the control group. Conceptual tests were administered to assess students’ understanding before, during and after instruction. The results revealed that the experimental and virtual conditions were equally effective in promoting student´s understanding of concepts in the domain of Newtons' s laws of motion; hence, virtual manipulation, at least in a context like the one of the present study, reveals an important method in physics learning.

Florin Spineanu^{1}
, Madalina Vlad^{2}

^{1}Natl Inst Laser Plasma Rad Phys, Plasma and Fusion, Romania

^{2}Natl Inst Laser Plasma Rad Phys, Plasma and Fusion, Romania

The connection between the solution of some nonlinear differential equations and the dynamics of poles in the $x$-space extended to complex plane, can help to understand integrability and/or the study of the formation of singular states. We show that in the case of the Burgers equation, the two branch points on the three-sheeted cover of the complex plane offer an alternative representation of the formation of the pre-shock as a particular structure of a Riemann surface. This surface also results from the interaction of closed strings in a reduced version of a supersymmetric field theoretical model of fermion and scalar fields. In the evolution before the pre-shock the Burgers equation and the string interaction have a dynamics that can be described by deformation of the Riemann surface. When the viscosity is introduced in the Burgers equation a series of singularities (poles) are generated along the imaginary axis. We will extend this connections to other physical systems that exhibit extreme states.

This work has been supported by the Contracts PN 18-01 and by CEA-RO C4.

Madalina Vlad^{1}
, Florin Spineanu^{2}

^{1}Natl Inst Laser Plasma Rad Phys, Plasma and Fusion, Romania

^{2}Natl Inst Laser Plasma Rad Phys, Plasma and Fusion, Romania

Turbulence is of fundamental importance in fluid mechanics, plasma physics, astrophysics, atmosphere and ocean sciences. It is a complex nonlinear process that mixes disorder and order. Quasi-coherence or order appears at the basic level of tracer trajectories in the case of smooth velocity fields that have finite correlation lengths. It lasts for a short time of the order of the time of flight. In two-dimensional incompressible turbulence, long time coherence can appear essentially because the trajectories are trapped in the correlated zone and/or because they reflect the structure of the velocity field. Tracer trajectories are random sequences of trapping or eddying events and long jumps, which yield strongly modified diffusion coefficients. They have a high degree of order, which consists of the generation of quasi-coherent structures of trajectories and average velocities (direct transport). In addition to that, we have found more subtle effects, namely the hidden drifts (HDs). Here we define the HDs and analyze their characteristics. Essentially, the HDs appear in the presence of an average velocity Vd as organized components of the motion oriented perpendicular on Vd. They are two opposite average velocities that compensate one another. As a consequence, the HDs have no direct influence on transport. However, they represent a reservoir for direct transport. We show that any perturbation that determines a weak compressibility of the velocity field perturbs the equilibrium of the HDs leading to average velocities of the test particles. We analyze this nonlinear process based on the HDs in two systems • impurity turbulent transport in confined plasma taking into account the polarization drift; • magnetic field line random walk in space plasmas in the presence of perpendicular and parallel large scale gradients of the main (average) magnetic field. The conclusion is that the direct transport can be rather large even at small perturbations. Moreover, there are conditions for which the convective flux determined by the HDs is opposite to the diffusive flux. The results are obtained using a semi-analytical approach, the decorrelation trajectory method [1]-[3], which is able to describe the complicated mixture of the random and quasi-coherent aspects that charaterizes tracer advection in stochastic fields. 1. M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, Phys. Rev. E 58, 7359 (1998). 2. M. Vlad, F. Spineanu, Phys. Rev. E 70, 056304 (2004). 3. M. Vlad, F. Spineanu, New J. Phys. 19, 025014 (2017).

Aigerim Abdibekova^{1}

^{1}Kazakh National University named al-Farabi, Mathematical and modelling department, Kazakhstan

This paper considers direct numerical simulation (DNS) of the homogeneous incompressible magnetohydrodynamic (MHD) turbulence decay based on the lattice Boltzmann equation (LBE) method in a periodic boundary conditions. The modelling of the turbulent process is based on the solution of a filtered unsteady three-dimensional Navier-Stokes equation and the equation of magnetic field strength. Patterns are defined for micro and macro scale turbulence, average kinetic energy and average dissipation rate, velocity profiles, velocity-derivative skewness and velocity-derivative flatness of turbulent flow, and all the key statistical quantities are compared with those from the pseudospectral method at different magnetic Reynolds number. The present work contributes to the overall assessment of the lattice Boltzmann equation method, and it provides a basis for further applications of LBE in studying the physics of turbulent flows.

Yuriy Dimitrienko^{1}
, Olga Dimitrienko^{2}

^{1}Bauman Moscow State University, Computational Mathematics and Mathematical Physics, Russia

^{2}Bauman Moscow State University, Scientific educational center for supercomputer modeling and software engineering, Russia

A fundamentally new approach is proposed to the development of models of large arrays of dynamically changing data in the economy, based on the construction of a multidimensional feature space and the transition to a continual description of the dynamics of the movement of data clusters. For the motion of clusters of data, the laws of conservation of multidimensional continua are formulated, which are analogs of the laws of conservation of mass, momentum, and angular momentum in classical mechanics. Models of rigid and deformable multidimensional continua are developed. Examples of application of these models for the analysis of economic data on sales in large supermarkets are given.

Andrej Kon'kov^{1}

^{1}Moscow Lomonosov State University, Differential Equations, Russia

Assume that $p > 1$ and $p - 1 \le \alpha \le p$ are real numbers and $\Omega$ is a non-empty open subset of ${\mathbb R}^n$, $n \ge 2$. We consider the inequality $$ {\rm div} \, A (x, D u) + b (x) |D u|^\alpha \ge 0, $$ where $D = (\partial / \partial x_1, \partial / \partial x_2, \ldots, \partial / \partial x_n)$ is the gradient operator and, moreover, $A : \Omega \times {\mathbb R}^n \to {\mathbb R}^n$ and $b : \Omega \to [0, \infty)$ are some functions with $$ C_1 |\xi|^p \le \xi A (x, \xi), \quad |A (x, \xi)| \le C_2 |\xi|^{p-1}, \quad C_1, C_2 = const > 0, $$ for almost all $x \in \Omega$ and for all $\xi \in {\mathbb R}^n$. For solutions of this inequality we obtain estimates depending on the geometry of $\Omega$. In particular, these estimates imply regularity conditions of a boundary point.

Vladimir Sloushch^{1}

^{1}S:Petersburg State University, High mathematics and mathematical physics, Russia

Suppose that $\Gamma$ is a connected locally finite $\mathbf{Z}^{d}$-periodic graph embedded in $\mathbf{R}^{d}$. Let $H$ be a Schr\"o{}dinger operator on $\Gamma$. Let $V$ be a nonnegative function on $\Gamma$ such that \begin{equation}\label{Sloushch0.1} V(x)\sim\vartheta(x/|x|)|x|^{-d/p},\ \ |x|\to\infty,\ \ p>0. \end{equation} Let $(\alpha,\beta)$ be a gap in the spectrum of $H$. Fix a number $\lambda\in[\alpha,\beta]$. The spectra of the perturbed operators $H_{\pm}(t)=H\pm tV$, $t>0$, inside the gap $(\alpha,\beta)$ are discrete. Denote by $N_{\pm}(\lambda,\tau,V)$ the number of eigenvalues of the operator $H_{\pm}(t)$ that have passed through the point $\lambda$ as $t$ was increasing from $0$ to $\tau$. %Counting functions %$N_{+}(\alpha,\tau,V):=\lim\limits_{\lambda\to\alpha+0}N_{+}(\lambda,\tau,V)$, %$N_{-}(\beta,\tau,V):=\lim\limits_{\lambda\to\beta-0}N_{-}(\lambda,\tau,V)$ %can be interpreted as the number of eigenvalues, "born" on the %left (right) edge of the gap. The main result can be represented in the following form: {\it If the condition {\rm(\ref{Sloushch0.1})} holds true, then the counting functions have a power-like asymptotics: \begin{equation}\label{Sloushch0.2} N_{\pm}(\lambda,\tau,V)\sim \Gamma_{p}^{\pm}(\lambda,H,V)\tau^{p},\ \ \tau\to+\infty,\ \ \lambda\in(\alpha,\beta). \end{equation} The coefficients $\Gamma_{p}^{\pm}(\lambda,H,V)$ can be calculated in terms of the operator $H$ and the perturbation $V$. Under certain conditions, the asymptotics {\rm(\ref{Sloushch0.2})} holds also at the edges of the gap.\/} The talk is based on the joint work with E.~Korotyaev.

Nurzhan Saduyev^{1}
, Vladimir Oskomov^{2}
, Alexander Sedov^{3}
, Orazaly Kalikulov^{4}
, Inesh Kenzhina^{5}
, Saken Shinbulatov^{6}
, Shinbolat Utey^{7}
, Askhat Zhumabayev^{8}

^{1}National Nanotechnology Laboratory of Open Type of al-Farabi Kazakh National University , , Kazakhstan

^{2}National Nanotechnology Laboratory of Open Type of al-Farabi Kazakh National University , , Kazakhstan

^{3}Institute of Experimental and Theoretical Physics of al-Farabi Kazakh National University , , Kazakhstan

^{4}Institute of Experimental and Theoretical Physics of al-Farabi Kazakh National University , Laboratory of Cosmic Rays Variations, Kazakhstan

^{5}National Nanotechnology Laboratory of Open Type of al-Farabi Kazakh National University, , Kazakhstan

^{6}National Nanotechnology Laboratory of Open Type of al-Farabi Kazakh National University, , Kazakhstan

^{7}National Nanotechnology Laboratory of Open Type of al-Farabi Kazakh National University, , Kazakhstan

^{8}National Nanotechnology Laboratory of Open Type of al-Farabi Kazakh National University, , Kazakhstan

The proposed method of measuring moisture reserves is based on the absorption of cosmic rays neutrons. It is necessary to take into account the regularities of the spatial variation of the neutron field of cosmic radiation, depending on moisture reserves in snow and soil. Special attention is paid to the influence of the main disturbing factors, such as: changes in the moisture of the underlying soils, when measuring moisture reserves in snow, and the effect of humus content in soils, when measuring moisture reserves in the soil. A methodology for measuring and determining soil moisture has been also developed

This work was supported by program BR05236494 of the Ministry of Education and Science of Kazakhstan.

Vladimir Gorbachev^{1}

^{1}Lomonosov Moscow State University, Department of Mechanics and Mathematics, Russia

We consider linear differential equations of mathematical physics with variables coefficients, depending on the coordinates and time. Along with the initial equations, we also consider the accompanying equations with constant coefficients of the same type and with the same input data. It is shown that the solution of the original equation is expressed through the solution of the concomitant equation with the help of an integral formula, the core of which contains the difference between the initial and accompanying coefficients multiplied by the Green function of the original equation. From the integral formula we obtain equivalent representation of the solution of the original equation in the form of series with respect to all possible derivatives of solutions of the concomitant equation. The coefficients of these series are continuous functions of either coordinates, or time, or both coordinates and time. They are called structural functions. We obtain a system of recurrence equations for these structural functions, and this system in some cases is solved analytically in general form.

The study was carried out with the financial support of the Ministry of Education and Science of the Russian Federation Federation (project 14.577.21.0207, unique identifier of the project RFMEFI57715X0207). FGBOU VPO "TGPU named after L. N. Tolstoy is a recipient of a grant from the Ministry of Education and Science.

Olga Lipko^{1}
, Roman Parovik^{2}

^{1}Vitus Bering Kamchatka State University, Faculty of Phisics and Mathematics, Russia

^{2}Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences., Laboratory of Physical Process Modeling, Russia

In this paper, oscillograms and phase trajectories are constructed using numerical simulation to study the limiting cycles of a nonlinear FitzHugh-Nagumo oscillatory system with power memory. The simulation results showed that in the absence of power memory (α=2, β=1) or the classical dynamic FitzHugh-Nagumo system, there is a single stable limit cycle, i.e. the Lienard theorem is fulfilled. In the case of viscous friction (α=2, 0< β<1), there is a family of stable limit cycles of different shapes. In other cases, the limit cycle destroyed according to two scenarios: Hopf bifurcation (limit cycle-limit point) or (limit cycle-aperiodic process). Further continuation of the research may be related to the construction of the spectrum of maximum Lyapunov exponents for a purpose of identifying chaotic oscillatory regimes for the considered hereditary dynamic system (HDS).

This work was supported by the Russian President's Grant MK-1152.2018.1.

Yuriy Dimitrienko^{1}
, Ilya Bogdanov^{2}

^{1}Bauman Moscow State University, Computational Mathematics and Mathematical Physics, Russia

^{2}Bauman Moscow State Technical University, Computational Mathematics and Mathematical Physics, Russia

The paper proposes a physico-mathematical model of a weakly compressible fluid and a two-scale model of the spatial flow of a weakly compressible fluid that displaces the gas medium in a porous composite structure. The model is based on an asymptotic analysis of the three-dimensional Navier-Stokes equations. Algorithms for the numerical solution of local problems of the spatial flow of liquid and gas on the periodicity cells of composite structures and the algorithm for calculating the permeability tensor are developed. A numerical algorithm for solving the global liquid flow problem that displaces gas from a porous composite structure is developed. Examples of numerical modeling of local spatial flows of liquid and gas on the periodicity cell of typical composite structures are given, and also results of numerical modeling of the macroscopic flow of a liquid binder displacing the gas medium in a typical porous composite structure are obtained.

Dilip Sheth^{1}

^{1}Sir Parashurambhau College, Tilak Road Pune , India, Mathematics, India

For any natural number n and let S_n = 1^1 + 2^2 + 3^3 + 4^4 + ....... + n^n, be the hyper triangular function of n. Using SageMath-Cloud, I have computed S_n for n = 1 to 2486, and checked that S_n is prime or not. It is observed that S_2, S_5, S_6, S_10, S_30 are primes and all other S_n's are composite. Then S_n was computed for n upto 10000. All Sn's except S_2, S_5, S_6, S_10, S_30 are composite. Note that S_2 = 5, S_5 = 3413, S_6 = 50069, S_10 = 10405071317, S_30 = 208492413443704093346554910065262730566475781 all are prime numbers. This is a sequence of numbers n such that nl(5^n+5). Mohammad K. Azarian in 2007 have given upper and lower bounds for S_n. A Fermat number, is a positive integer of the form F_n = 2^(2^n) + 1, where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617. It is interesting to see that there are Five F_n's for n = 0, 1, 2, 3, 4 those are primes and only five S_n 's, for n = 2, 5, 6, 10, 30 are primes, no other S_n or F_n are seems to be prime.

Author is thankful to Prof. S. A. Katre, SPPU for guidance and Shikshana Prasaraka Mandali, Pune for their support and encouragement.

Roman Parovik^{1}

^{1}Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences., Laboratory of Physical Process Modeling, Russia

The paper is devoted to the method of investigating the rest points of systems of differential equations of fractional order [1]. In this paper, we will investigate the stability of the rest points of the hereditary dynamical systems by the example of certain fractal oscillators. We will consider the two types of hereditarity dynamical systems: commensurable and incommensurate, for which the corresponding stability theorems for rest points are valid. Further examples of the application of these theorems will be considered for a fractal linear oscillator [2] and the Duffing fractal oscillator [3]. The results of the stability study of the rest points of the hereditarity dynamical systems were confirmed by phase trajectories for the fractal oscillators under consideration.

The work was supported by the grant of the President of the Russian Federation No. MK-1152.2018.1 and on the topic of the research of Kamchatka named after Vitus Bering "Application of fractional calculus in the theory of oscillatory processes" No.АААА-А17-117031050058-9.

Jan Glasa^{1}
, Lukas Valasek^{2}
, Peter Weisenpacher^{3}

^{1}Institute of Informatics, Slovak Academy of Sciences, Department of parallel computational methods and algorithms, Slovakia (Slovak Republic)

^{2}Institute of Informatics, Slovak Academy of Sciences, , Slovakia

^{3}Institute of informatics, Slovak Academy of Sciences, Department of Parallel Computational Methods and Algorithms, Slovakia (Slovak Republic)

Fires in road tunnels produce a large amount of smoke threatening passengers´ lives and health and tunnel facilities. The spread of fire smoke is influenced by various factors. In this paper, the Fire Dynamics Simulator system is used to study the spread of smoke produced by fires with various heat release rates in two 900 m long tunnels which differ from each other by slope. The impact of slope on smoke backlayering and smoke spread in tunnel tube is illustrated. The simulation results show that buoyant forces induced by fire accelerate the smoke spread in ascending sections and slow the smoke spread in descending sections of the tunnel. The analysis indicates that the tunnel slope can have significant impact on smoke spread in road tunnels.

Alexandre SEMEROK^{1}
, S Fomichev^{2}

^{1}DEN-SEARS, CEA, Université Paris-Saclay, , France

^{2}NRC Kurchatov Institute, , Russian Federation

New technological methods for non-destructive testing (NDT) for in situ and real time control of complex physico-chemical processes are under constant development in the French Alternative Energies and Atomic Energy Commission (CEA) [1-12]. These methods should be developed along with mathematical model and computer code to provide both the adequate analytical physical model and the rapid computation of a process under research. In our NDT-studies with Laser Active Radiometry method, samples representative for nuclear and thermonuclear (ITER) installations were under repetitive heating by a pulsed laser radiation. The development of the analytical models and simplified solutions was aimed to simulate laser heating and related temperature distribution T(x, y, z, t) in relation with numerous physical and chemical features of this complex multi-parametric process. The results of our mathematical modeling were validated by the experiments. The simulation and experimental results were in good correlation. The principal results of our theoretical studies, their applicability for rapid NDT and the prospects for future models development for heating temperature calculation in application to complex surfaces with micrometric layers or undersurface defects will be presented and discussed. References [1] Semerok A, Fomichev S V, Brygo F, Weulersse J-M, Thro P-Y and Grisolia C 2006, Proceedings of “LTL Plovdiv 2005” (IV International Symposium Laser Technologies and Lasers “LTL Plovdiv 2005” (Bulgaria, Plovdiv, 8–10 October 2005) pp 76–83 [2] Grisolia C, Semerok A, Weulersse J-M, Le Guern F, Fomichev S V, Brygo F, Fichet P, Thro P-Y, Coad P, Bekris N et al. 2007 J. Nucl. Mater. 363–365 1138 [3] Semerok A, Fomichev S V, Weulersse J-M, Brygo F, Thro P-T and Grisolia C 2007 J. Appl. Phys. 101 084916 [4] Melyukov D, Sortais C, Semerok A, Thro P-Y, Courtois X and Farcage D 2010 10th International Conference on Quantitative Infra-Red Thermography (July 27-30, 2010, Québec, Canada) http://dx.doi.org/10.21611/qirt.2010.089 [5] Courtois X, Sortais S, Melyukov D, Gardarein J-L, Semerok A and Grisolia C 2011 Fusion Eng. Des. 86 1714 [6] Semerok A, Fomichev S V, Brygo F, Thro P-Y and Grisolia C 2012 J. Nucl. Mater. 420 198 [7] Semerok A, Jaubert F, Fomichev S V, Thro P-Y, Courtois X and Grisolia C 2012 Nucl. Instr. Meth. Phys. Res. A 693 98 [8] Semerok A, Jaubert F, Fomichev S V, Thro P-Y and Grisolia C 2012 Fusion Eng. Des. 87 267 [9] Semerok A, Grisolia C, Fomichev S V and Thro P-Y 2013 Proc. of SPIE vol 9065 (Bellingham, WA) 90650A. [10] Semerok A, Fomichev S V, Jaubert F and Grisolia C 2014 Nucl. Instr. Meth. Phys. Res. A 738 25 [11] Pham Tu Quoc S, Cheymol G and Semerok A 2014 Rev. Sci. Instrum. 85 054903 [12] Semerok A, Pham Tu Quoc S, Cheymol G, Gallou C, Maskrot H and Moutiers G 2016 EPJ Nuclear Sci. Technol. 2 20

Petr Permyakov^{1}
, Fedotovich Zhirkov^{2}
, Nikolaevich Zheleznyak^{3}

^{1}Larionov Institute of the Physical-Technical Problems of the North of the Siberian Branch of the RAS, Department of Physical Chemistry of Materials and Technologies, Russia

^{2}Russian Academy of Sciences, , Russian Federation

^{3}Russian Academy of Sciences, , Russian Federation

The process of air vapor condensation is extremely widespread in nature. It occurs in the atmosphere, on open surfaces and in the upper layers of the lithosphere. The process of condensation is exothermic (with release of heat), which proves the need for it to be taken into account in balance calculations in practically any natural and technogenic conditions. Given this fact, water vapor condensation study becomes especially important in the areas of the southern mountainous cryolithozone, where insignificant deviations from the thermal balance cause drastic changes in permafrost conditions. This in turn leads to the development of dangerous cryogenic geological processes. Problems of condensation and evaporation of ground water were studied by many scientists, who have conducted various experimental and field studies. As a result, various hypotheses appeared. Some researchers have established that condensation and evaporation proceed very intensively, while others suggest that these processes can be ignored in practical calculations. Mass exchange in soils occurs with the absorption of heat during evaporation and vice versa with the release of heat during condensation. This work is devoted to the mathematical modeling of heat and moisture exchange taking into account the condensation and evaporation of subsoil moisture during seasonal thawing of permafrost. Moisture motion can be described using the transfer equation in humid or potential (Richards equation) forms in saturated and unsaturated soils. Based on the experimental and full-scale data, the parameters of condensation and evaporation of pore moisture were verified. The numerical realization of the nonlinear heat and mass transfer problem is carried out by method of finite differences using Newton’s method. Validity of the proposed model is shown in the case of seasonal thawing of permafrost. As a result of numerical experiment in the conditions of Central Yakutia, it has been established that when the soil moisture evaporates and condenses, the total moisture content of the soil increases. In the spring-summer period there we observe intense evaporation of intra soil and a decrease in temperature, and the summer-autumn period shows increasing condensation of water vapor, accompanied by an increase in the heat content of the soil. At the end of the summer season, in late September and early October, there occurs a “blocking” of the wet thawed layer. The suggested mathematical model more adequately reflects the process of freezing of the “blocked” zone. The condition for intensive process of water vapor condensation in soils is large temperature gradient (heat exchange at the atmosphere-soil interface), presence of moist air, and porosity of the rocks. The most favorable conditions for the progress of this process are possible in regions with a sharp continental climate and a large amount of precipitation falling in short time periods.

Dimitri Georgievskii^{1}

^{1}Moscow State University, Mechanical and Mathematical Department, Russia

The formulations and the new exact self-similar solutions of the diffusion-vortex problems in terms of stresses simulating a nonsteady one-dimensional shear in some curvilinear orthogonal coordinate system of two-constant rigid viscoplastic medium (the Bingham solid), are analysed. Both the diffusion of plane and axisymmetric vortex layers as and the diffusion of vortex thread belong to these type of incompressible flows. A shear is realized inside the certain expanding in time subdomains of infinite space with beforehand unknown bounds. Herewith one possible way for formulation of additional condition at infinity is described. We introduce into consideration the generalized diffusion of vortex which contains several material parameters and an order of stress irregularity in zero. The self-similar solutions where an order of irregularity corresponds or does not correspond to the kind of shear in the selected coordinate system are constructed.

Oleg Sutyrin^{1}
, Pavel Georgievskiy^{2}
, Vladimir Levin^{3}

^{1}Lomonosov Moscow State University, Institute of Mechanics, Russia

^{2}Lomonosov Moscow State University, Institute of Mechanics , Russia

^{3}Lomonosov Moscow State University, Institute if Mechanics, Russia

Shock-bubble interaction (SBI) is the classical and widely studied gasdynamic problem that is closely related to Richtmyer-Meshkov instability (RMI). The applications of this flow class cover a wide range of scales, spanning from shock-wave lithotripsy in medicine and supersonic combustion to inertial confinement fusion and shocked interstellar media. In gases, such interaction result in complex flow with shock focusing phenomenon and formation of long-living vortex rings. The flows are divided naturally into two major classes: the light bubble class (Atwood number $At<0$) and the heavy bubble class ($At>0$). Although the shock focusing is present for the both cases, in the case of heavy bubble it is much more expressed. In this scenario, for high Atwood numbers, there is an intense localized shock collapse followed by the formation of high-speed axial jets. This effect may allow an otherwise insufficiently strong incident shock to initiate the combustion of bubbles filled with a reactive gas mixture. $$$$ The shock focusing phenomenon results from the velocity of the transmitted shock inside the bubble being lower than the velocity of the incident shock; the latter shock diffracts around the bubble and collapses at the axis of symmetry while several transversal shocks interact inside the bubble. There are two different interaction modes - external and internal shock refraction patterns - depending on the shock Mach number, bubble geometry and Atwood number. $$$$ The present study is concerned with numerical simulation of interaction of a plane shock with heavy gas bubble ($At>0$). Shock focusing process features depending on bubble density ratio are investigated in detail. Unsteady two-dimensional (plane and axisymmetrical) perfect gas flows are modeled using Euler's equations which are solved with finite-difference implementation of 5th order WENO-Z scheme. $$$$ Three distinct pressure peaks are observed on the symmetry axis (or symmetry plane) during shock focusing process: one of them is outside of the shocked bubble and two are inside. Shock focusing intensity - most notably the third pressure peak - rise with density ratio increase. The highest increase is observed upon transition from external to internal shock focusing pattern.

This study was performed using the computational resources of the Moscow State University cluster with partial financial support of the Russian Foundation for Basic Research (projects no. 16-29-01092 and 18-01-00793)

Anton Belyakov^{1}
, Alexander Seyranian^{2}

^{1}Lomonosov Moscow State University, Moscow, Russia, Moscow School of Economics, Russia

^{2}Lomonosov Moscow State University, Moscow, Russia, Institute of Mechanics, Russia

We develop here the method for obtaining approximate boundaries of parametric resonance in the space of parameters. We have Floquet theory applied to the system with parametric excitation. The monodromy (Floquet) matrix is found by averaging method. We study domains of stability with the use of fourth order approximations of monodromy matrix on example of inverted position of a pendulum with vertically oscillating pivot. Addition of small damping shifts the stability boundaries upwards, thus resulting to both stabilization and destabilization effects.

Olga Azarova^{1}
, Tatiana Lapushkina^{2}
, Alexander Erofeev^{3}
, Oleg Kravchenko^{4}

^{1}Federal Research Center “Computer Science and Control” of RAS, Department of Mathematical Modeling of Computer-Aided Design Systems, Russia

^{2}Ioffe Institute, , Russia

^{3}Ioffe Institute, , Russia

^{4}Scientific and Technological Center of Unique Instrumentation of RAS, , Russia

The process of passing a strong shock wave ($M=5-6$) through the region of pre-formed ionization instability in gas discharge plasma in air with pressure equal to 7 Torr and gas-discharge current $200-300\,mA$ has been studied experimentally and numerically. In the experiments the ionization spherical strata have been obtained arising in the gas discharge region due to the development of the ionization instability at high degree of ionization and non-equilibrium (gas temperature $T\sim300-600\,K$, electron temperature $T_e\sim1000\,K$). As a result of the interaction of an initially plane shock wave with plasma area of ionization instability, the formation of a new shock-wave configuration consisting of the shock wave and contact surface is observed. A form of the both discontinuities has been shown to change from smooth-homogeneous to serrated, acquiring an unstable character. Note, that this phenomenon was observed experimentally for the first time. Numerical modeling was carried out on the basis of the Euler and Navier-Stokes systems of equations with the parameters corresponding to the experimental conditions. The energy source was modeled by a set of thermal layers with varying characteristics. Changes in the physico-chemical properties of the medium were described by varying the adiabatic index. Stratified shock-wave structures consisting of modified wavy shock-wave and contact discontinuity fronts have been obtained as a result of the interaction of the shock wave with the region of ionizing-unstable gas-discharge plasma. Generation of the Richtmyer-Meshkov instabilities has been obtained on the thermal strata in the vicinity of the shock wave front curvatures. Results of the study can be used to control of high speed flows and shock-wave configurations, as well as mixing processes.

Yakov Rudaev^{1}
, Daria Kitaeva^{2}

^{1}Kyrgyz-Russian Slavic University, Department of Mechanics, Kyrgyzstan

^{2}Peter the Great St.Petersburg Polytechnic University, Institute of Civil Engineering, Department of Hydraulics and Strength, Russia

Mathematical modeling of behavior of group of the industrial aluminum alloys with initial varying grain size structure showing superplastic properties in certain temperature and strain rate ranges is presented. It is known that the structural superplasticity is connected with facilitated by fine-grained structure formation (1…10 microns) at the preliminary stage. However, for realization of superplasticity of “dynamic type” there has to be a replacement of an initial varying grain size structure state of material with another, ready for superplasticity. Therefore the used definition “the dynamic superplasticity” reflects consecutive change of states which happens in material with initial varying grain size structure under the changing temperature-rate conditions: initial varying grain size → equiaxed fine-grained structure (4…7 microns) formed in the temperature-rate conditions of superplasticity → coarse-grained at further increase in strain rate. These changes are caused by simultaneous action of deformation rates and structural (phase) transitions of evolutionary type in open nonequilibrium systems. In particular, for the considered commercial aluminum alloys with initial varying grain size structure such irreversible transition is dynamic recrystallization. The association of deformation process with metal flow, that has irreversible structural and phase transition of indistinct type at one of the stages, allows using synergetic approach. The mathematical model formulated from positions of solid mechanics use allow to research nonequilibrium system reaction to behavior of thermodynamic response functions – the specific heat and entropy – and to establish implementations features of the irreversible indistinct phase transitions observed in the conditions of dynamic superplasticity for aluminum alloys.

Khurram Mahmood^{1}

^{1}UET Taxila, Computer Science, Pakistan

Cloud computing is an amalgam of techniques and technologies that provide software, infrastructure, platform and storage services that are packaged to provide a new paradigm of scalability, agility, cost effectiveness, availability and flexibility for the benefit of public at large. Along with the provision of useful services, there is also a huge concentration of risks which cloud users are facing. These risks include security, integrity, authenticity, transparency, trust and authorization issues. Information security companies provide offensive and defensive cyber security solutions to guard different organizations from cyber-attacks. They establish policies and measures for providing secure services but still there are many risks that security companies are facing. These risks are associated with keeping sensitive data on cloud. People have a lack of trust due to these issues. Many studies have been conducted previously in which the trust deficiencies at public side for cloud are addressed due to the sensitivity of data, difficulties faced by information security companies while providing cloud services and recommendations were provided to resolve the issue. In this paper the reasons for the lack of trust in cloud environment are addressed as most of the trust issues are stemmed because of security problems so the solutions to these issues for enhancing trust in cloud computing environment are also presented in this paper. For this purpose qualitative study is conducted and interviews were conducted from different security companies. Data is analyzed using Nvivo and results have provided the measures that can be taken to resolve the issues

Thanks to the support of University of Engineering and Technology, UET Taxila, Pakistan

Khanh Pham^{1}
, Sergey Suslov^{2}

^{1}Swinburne University of Technology, , Australia

^{2}Swinburne University of Technology, Department of Mathematics, Australia

Ferrofluids are modern strongly magneto-polarisable nanofluids. Their flows can be non-intrusively controlled by applying an external magnetic field. One of their prospective applications is as a heat carrier in thermal management systems operating in conditions where natural convection is suppressed due extreme confinement (e.g.~microelectronics). The linear and weakly nonlinear flow stability analyses that will be presented here illustrates an intricate interplay between thermogravitational and thermomagnetic mechanisms of convection in one of the practically important geometrical setups, an inclined fluid layer heated from below. The low-dimensional amplitude evolution equations of Landau type are derived to model the physical phenomena of interest. The solutions of the so-obtained dynamical system show that the application of magnetic field can indeed trigger convection in regimes where natural convection cannot exist, thus enhancing heat transfer. At the same time in regimes where both magnetic and gravitational buoyancy mechanisms are active the competition between the two may suppress the overall heat exchange. The physical reasons for such an intriguing behaviour will be illuminated.

Weidong Liu^{1}
, Liangchi Zhang^{2}

^{1}UNSW Sydney, School of Mechanical and Manufacturing, Australia

^{2}UNSW Sydney, School of Mechanical and Manufacturing, Australia

Most of the thermally-deformed polymer glasses can recover back to their original shapes after reheating, but the underlying mechanisms and the potential applications have not been fully explored. With the aid of finite element simulation, this paper studies the shape memory mechanism of a polymer glass (PMMA) and proposes a new moulding technique: shape memory moulding. The simulation results showed that after thermal compression at high temperature and cooling down, the polymer sample can keep its deformed shape and store significant residual stress due to its large viscosity. In the reheating process, however, the residual stress could be released at around 90 ˚C and promote the shape of sample recover back to its original shape. Based on this effect, we further established a numerical platform for exploring the feasibility of shape memory moulding. Particularly, the shape recovery ability of the polymer was evaluated at different pre-treatment temperatures and strains. It was found that the pre-treatment strain can significantly affect the shape recovery ability of polymer, reflected by the contact pressure. The higher the pre-treatment strain, the higher the contact pressure, and thus the better the shape recovery ability one can get. However, the influence of pre-treatment temperature is trivial, and the major shape recovery occurs near the glass transition temperature during the reheating.

Yong-Ho Kim^{1}

^{1}University of Science, Mathematics, Korea, Democratic People's Repub

The Navier-Stokes equations with damping describe the flow with the resistance to the motion such as porous media flow and drag or friction effects and a mathematical viewpoint, the equations can be viewed as a modification of the Navier-Stokes equations with the regularizing term. This paper is concerned with the incompressible Navier-Stokes equations with damping term and homogeneous Dirichlet boundary conditions in 3D bounded domains. First, we prove a certain regularity of weak solutions of the problem . Next, we prove that the weak solution which satisfies an energy inequality is unique provided one of the conditions is satisfied: (i) , (ii) (iii) , (iv) , where is the kinematic viscosity.

Weiwei Sun^{1}

^{1}City University of Hong Kong , Mathematics, China

In this talk we present our recent work on which we introduced a new numerical approach to the time dependent Ginzburg--Landau (GL) equations under the temporal gauge (zero electric potential gauge). The approach is based on a mixed formulation of the GL equations, which consists of two parabolic equations for the order parameter $\psi$ and the magnetic field $\sigma = \curl \, \bA$, respectively and a vector ordinary differential equation for the magnetic potential $\bA$. A fully linearized Galerkin FEM is presented for solving the mixed GL system. The new approach offers many advantages on both accuracy and efficiency over existing methods. In particular, the equations for $\psi$ and $\sigma$ are uniformly parabolic and therefore, the method provides optimal-order accuracy for the two physical components $\psi$ and $\sigma$. Since in the temporal direction, a fully linearized backward Euler scheme is used for $\psi$ and $\sigma$ and a forward Euler scheme is used for $\bA$, respectively, the system is fully decoupled and at each time step, the three variables $\psi$, $\sigma$ and $\bA$ can be solved simultaneously. Moreover, we present numerical comparisons with two commonly-used Galerkin methods for the GL equations under the temporal gauge and the Lorentz gauge, respectively. Our numerical results show that the new approach requires less iterations for solving the linear systems arising at each time step and the computational cost for the vector ODE seems neglectable. Several numerical examples with complex geometries are also investigated.

Meirong Zhang^{1}

^{1}Tsinghua University, Department of Mathematical Sciences, China

In this talk, I will report some recent results on the dependence of eigenvalues of Sturm-Liouville problems on potentials and weights. It will be shown that these eigenvalues are dependent on potentials and weights in a very strongly continuous way, namely, when the weak topologies are considered for potentials and weights in the Lebesgue spaces, as nonlinear functionals, eigenvalues are still continuous. Such a result can be called the complete continuity. If time is allowed, I will then report how we can apply such a complete continuity and the variational method to find some optimal lower and upper bounds of these eigenvalues.

This work is supported by the National Natural Science Foundation of China.

Roman Romanov^{1}

^{1}St Petersburg State University, , Russian Federation

We give an explicit necessary and sufficient condition for the resolvent of a singular canonical system to be of the Schatten--von Neumann class $ S^ p $, $ p &gt; 1 $, and more generally, for a wide class of ideals of compact operators properly containing the trace class, in terms of its Hamiltonian. The proof uses (a generalization of) Aleksandrov--Peller--Rochberg--Janson results on integral operators of the resolvent type, and an argument reducing the general case of the problem to the diagonal one. The talk is based on a joint work with H. Woracek.

Luiza Shapakidze^{1}

^{1}A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, Department of Elasticity Theory, Georgia

The effect of the direction of the azimuthal pressure gradient on the character of the instability and transitions to turbulence of heat-conducting diverging and converging flows in an annulus between two permeable cylinders is considered. The basic flow is maintained by the azimuthal pressure gradient in the transverse direction and by distribution of angular velocities under the rotation of two cylinders and radial flows. The methods of the nonlinear theory of bifurcations of hydrodynamic flows with cylindrical symmetry together with numerical calculations allow one to demonstrate that the magnitude and direction of the applied pressure gradient control the character of instability and transitions of the basic flow. The instability and transitions to chaotic regimes arising after the loss of stability of the main flow stability are investigated for different values of flow parameters and shown in the graphs.

Kaushik Ghosh^{1}

^{1}Vivekananda College, University of Calcutta, Physics, India

In this manuscript, we will discuss the construction of covariant derivative operator in a quantum theory of gravity. We will find it is more appropriate to use connection coefficients more general than the Levi-Civita connections in quantum gravity. We will demonstrate this for the canonical quantization procedure. We will find that quantization of gravity give an additional field besides metric. This field is associated with general connection coefficients and can be described by a third rank tensor which include torsion. This additional field is considered in the Palatini action. The complete tensor is not considered in quantum theories of gravity. General connection coefficients contribute a source term in the free theory of quantum gravity. They will also introduce additional particles besides graviton in quantum gravity. The energy-momentum conservation law no longer remains exactly valid when we do not use the Levi-Civita connections. We will also find that we have to extend the electric charge conservation law when we use connection coefficients more general than the Levi-Civita connections. General connection coefficients will become significant to solve cosmological problems.

Jagdish Prasad Maurya^{1}
, Sanjay Kumar Pandey^{2}

^{1}Indian Institute of Technology (BHU), Varanasi , Mathematical Sciences, India

^{2}Indian Institute of Technology (BHU), Varanasi, Mathematical Sciences, India

Abstract: Tornado vortex is a three dimensional nonlinear phenomenon. In this paper we investigate the dynamics of tornado like stationary vortex by considering the real inflow radial velocity distribution depends on both radial and vertical coordinates based on the experimental and observations data. We present an analytical model for steady, incompressible and viscous fluids. Although all the three components depend on radial and axial coordinates, viscosity affects merely the azimuthal velocity and the pressure. It is observed that the magnitude of the radial velocity increases to the maximum at the core but reverses the trend beyond and vanishes as it reaches the centre line. The magnitude reduces linearly with axial distance as per the supposition. At the core, larger the Reynolds number, lesser is the velocity for moderate Reynolds number. Insignificant impact is observed for very large Reynolds number. However, inside and outside the core, the trends are reversed. Radial pressure distributions for different axial positions are similar to theoretical, numerical and experimental observations. As we move outwards from the axis, pressure falls. Difference between the pressures at the axis and that in outward regions increases with height. Pressure falls with rising Reynolds number uniformly for all radial distances. This is an indication that quantitative difference in pressure is large between viscous and inviscid flows.

Olga Maltseva^{1}

^{1}Institute for Physics, Southern Federal University, Russia

Among the various methods to specify conditions of radio-waves propagation in the ionosphere empirical models play an important role. However, their accuracy in high-latitude region is not sufficiently estimated. In this paper, the International Reference Ionosphere model IRI is used. According to Russian ionospheric stations Gorkovskaya, Lovozero, Salekhard estimations of conformity of model values of parameters to experimental magnitudes are given from May 2017 till January 2018. The particular attention is paid to parameter MUF3000F2 for these stations and to coefficient M(D) for a path Gorkovskaya - Lovozero. It is shown, that these coefficients possess sufficient reliability. Adapting the model to the current diagnostic data has allowed to reduce root-mean-square deviation for MUF(D) almost two times from 31.4 % to 17.2 %. For the identification and localization of disturbances in high latitudes, it is possible to use measurements of the total electron content.

Authors express the gratitude to employees of the Arctic and Antarctic scientific research institute for providing data of vertical and oblique sounding on a site http://geophys.aari.ru/, Community Coordinated Modeling Center for granting of model IRI online. Work is supported by the Russian Foundation for Basic Research grant 18-05-00343А.

Alexander Galakhar^{1}

^{1}Bauman Moscow State Technical University, The School of Computer-driven Systems of Industrial Automation, Russia

The system of monitoring of operational risk of long distance pipelines would determine the actual safety factor for instant evaluation of accident risk in real time. The actual safety factor of a pipeline section is determined with not only technical condition of pipes in the section which is assessed using data of periodic inspections, but also with actual distribution of pressure of pipeline transported medium. The pipeline technical condition changes slowly due to degradation. Therefore, the actual safety factor of the operated pipeline depends mainly on loads exerted on the pipe, particularly the load from internal pressure which depends not only on the inlet pressure of the pipeline section, but also on the terrain where the pipeline lies, the actual position of the pipeline in the area and the actual varying conditions of heat transfer from the transported medium to surrounding environment through the pipeline wall. It is proposed to determine the actual pressure distribution in the pipeline section from the differential equation of motion of medium transported through the pipeline which is solved using non-uniform rational B-splines (NURBS) and the data of real time temperature measurements.

Eugenia Echkina^{1}
, Igor Inovenkov^{2}

^{1}Moscow State University, Computational Mathematics and Cybernetics, Russia

^{2}Moscow State University, Computational Mathematics and Cybernetics, Russia

A number of salient phenomena have been observed in the laser irradiation of cluster targets. These include the Coulomb explosion of the clusters, enhanced emission of x rays, generation of energetic electrons and energetic ions. There are indications that some strong nonlinear interaction of laser and clusters do occur upon the intense short pulse irradiation. The laser interaction with clusters is far stronger than that of the same intensity laser with other preparations of the same materials such as gas and/or usual plasma and solid with planer surface structure. The results of two - dimensional particle in cell simulations of the ultra short high irradiance laser pulse interaction with cluster targets are presented. We present the energy spectra of the fast ions generated during the Coulomb explosion of the multi-species clusters. The ion acceleration by the radiation pressure of a super-intense laser pulse is demonstrated. When the multi-cluster cloud is imbedded into an underdense plasma the laser radiation expels the electrons from the clusters and ejects them into the wake plasma wave generated by the ultrashort laser pulse. This provides a novel mechanism for the electron injection.

Timofey Tsvetkov^{1}
, Konstantin Khromov^{2}
, Alexey Kovalishin^{3}
, Valeriy Ryabov^{4}
, Vasily Velikhov^{5}

^{1}NRC Kurchatov Institute, Department of Nuclear Energy, Russia

^{2}NRC Kurchatov Institute, , Russia

^{3}NRC Kurchatov Institute, , Russia

^{4}NRC Kurchatov Institute, , Russia

^{5}NRC Kurchatov Institute, , Russia

We suggest a new atomistic method for efficient construction of edge and mixed type dislocations based on the introduction of four equispaced dislocations comprising a dislocation quadrupole. Contrary to all other currently used methods, our approach exactly preserves the topology of an ideal crystal with periodic boundary conditions even for small simulation volumes and, thus, restores the verity of the material frame indifference principle, broken in other techniques. A careful procedure for relaxation of atomic positions around a dislocation core has been developed which enables one to achieve arbitrary low dislocation densities characteristic for real crystals. As a demonstration of the method capabilities, we have constructed a sim- ulations volume with as low dislocation density as 1.5 · 1014m−2 , which is realistic in deformed crystals, and one can easily lower this value as desired. All details of dislocation construction process are explicitly specified, making it very easy to reproduce our results. FCC Al crystal is used as a test case.

Pelin Senel^{1}
, Munevver Tezer-Sezgin^{2}

^{1}Karadeniz Technical University, Department of Mathematics, Turkey

^{2}Middle East Technical University, Department of Mathematics, Turkey

In this study, the magnetohydrodynamic (MHD) flow is simulated in a circular pipe with slipping and arbitrarily conducting boundary. The electrically conducting fluid is interacted with the external magnetic field producing induced magnetic field in the pipe. The 2D governing coupled equations in terms of the velocity and the induced magnetic field are solved by the Dual Reciprocity Boundary Element Method (DRBEM). The MHD equations are transformed to boundary integral equations using the fundamental solution of Laplace equation, considering all the terms other than Laplacian as inhomogeneity. The discretized system of equations is solved in one stroke without introducing an iteration which reduces the computational cost. It is shown that, the flow decelerates, Hartmann layers enlarge through the top and the bottom of the pipe and induced current lines align as the wall conductivity or Hartmann number increases. An increase in the slip length accelerates the flow, shrinks the stagnant region, diminishes the boundary layers and retards the effect of the wall conductivity increase. The DRBEM is an advantageous method in solving MHD flow especially with slipping and arbitrarily conducting pipe wall, since it enables to insert both the unknowns and their normal derivatives with slip and conductivity wall conditions.

Munevver Tezer-Sezgin^{1}
, Cemre Aydin^{2}

^{1}Middle East Technical University, Department of Mathematics, Turkey

^{2}Middle East Technical University, Department of Mathematics, Turkey

In this study, the identification of the Hartmann number in the magnetohyrodynamic (MHD) flow equations is investigated through the inverse formulation in a duct with insulated but no-slip or variably slipping walls. The Cauchy MHD flow problem is constructed with the Dirichlet and Neumann boundary conditions for the velocity and the induced magnetic field, and also their internal values obtained from the direct solution corresponding to a specified Hartmann number. The solutions for direct and inverse problems are obtained through the dual reciprocity boundary element method (DRBEM), and the Tikhonov regularization with L-curve method is used for solving the ill-conditioned DRBEM discretized linear system of equations for inverse problem. The velocity and the induced magnetic field profiles are presented from the direct formulation for depicting the well-known MHD characteristics with increasing values of Hartmann number and slip length. The Hartmann number is reconstructed from the inverse problem with an accuracy of $10^{-6}$ in maximum error norm. The DRBEM discretizes only the boundary of the region and uses low number of arbitrary interior points. Thus, the important parameter, Hartmann number is regained at a small computational expense with the boundary and interior overdetermined information needed in the inverse MHD flow problem.

Vladimir Nefedov^{1}
, Igor Inovenkov^{2}
, Liubov Ponomarenko^{3}

^{1}Lomonosov Moscow State University, Computational Mathematics & Cybernatics, Russia

^{2}Lomonosov Moscow State University, Computational Mathematics & Cybernatics, Rwanda

^{3}Lomonosov Moscow State University, Computational Mathematics & Cybernatics, Russia

Within the framework of the two-dimensional spatial economy, various forms of urban structures are considered. Since the number of structurally stable urban configurations is rather limited, structurally unstable formations are of great interest. For example, there are city models that behave near unstable singular points like a «running» wave. In this paper we consider a system of quasi-linear parabolic equations for two unknown functions: population density and, so-called, quality of housing stock. After some transformations of the required functions and variables we seek a self-similar solution in the form of a running wave with an unknown amplitude and velocity. Further the stability of the obtained solution is analyzed. To determine the role of the obtained particular (self-similar) solution, the Cauchy problem for different initial conditions is also solved numerically. The results of computational experiments show us that the analytical wave-like solution can be interpreted is an intermediate asymptotic for the partial differential problem which is being considered this paper.

Olga Vasilyeva^{1}

^{1}Memorial University of Newfoundland, Grenfell Campus, Mathematics, Canada

We study the population dynamics of aquatic organisms in a river network. The habitat is viewed as a tree-like metric graph with the population density satisfying a reaction-diffusion-advection equation on each segment along with the appropriate junction and boundary conditions. In the case of a linear reaction term, the question of persistence in such models was studied by Sarhad, Carlson and Anderson. We investigate the case of a logistic reaction term and obtain necessary and sufficient conditions for the existence and uniqueness of a positive steady state solution for an arbitrary river network. The conditions are formulated in terms of the lengths of river segments, their cross-section areas, and the hydrological and biological parameters (growth rate, carrying capacity, diffusivity and advection speed, all assumed to be constant throughout the network).

The author is supported by a grant from the Natural Science and Engineering Research Council of Canada (NSERC).

Boris Sibiryakov^{1}

^{1}Novosibirsk State University, Geophysics, Russia

In this paper presents the elements of blocked media deforming theory. It means that these media have a specific surface and (related with it) average distance from one crack (pore) to another one. This way requires the creation a new model of continuum, which is contains the integral geometry of pore space. The equations of motion and equilibrium are equations of the infinite order due to infinite numbers of freedom degrees. Along the usual seismic waves, these equations describe very slow waves, not bounded below and, besides of it, they predict the instable solutions, due to parametric resonances in structures bodies. The number of instable solutions corresponds to seismological Gutenberg-Richter law. The dispersion of an average size of structure produces both the fast catastrophes (small dispersion) and slow catastrophes compare to Rayleigh wave velocity (high dispersion).

Valeriy Parkhomenko^{1}

^{1}Moscow State Technical University by N.E. Bauman (BMSTU), , Russia

Global warming climate changes are observed in recent decades. These changes largely associated with anthropogenic increases in greenhouse gases in the atmosphere. The problem and opportunity of the global climate stabilization at a current level by means of geoengineering methods are investigated. The study is based on a three-dimensional hydrodynamic global climate coupled model, including ocean model with real depths and continents configuration, sea ice evolution model and energy and moisture balance atmosphere model. The climate prediction calculations up to the year 2100, using different CO2 growth scenario are carried out on the first stage. A series of numerical experiments were carried out to assess the possibility of climate stabilization at the level of the year 2010 by controlling emissions into the stratosphere of aerosol, reflecting a part of the incoming solar radiation. Evolution and sedimentation equation for sulphur aerosol mass calculation is considered. A uniform and zonal stratospheric aerosol space distribution was investigated. It is shown that by this way it is impossible to achieve the space and seasonal uniform approximation to the existing climate, although it is possible significantly reduce the greenhouse warming effect. The Pareto optimal frontier was investigated and visualized for two parameters - atmospheric temperature MSD for the winter and summer seasons.

ARTUR DMITRENKO^{1}

^{1} National Research Nuclear University «MEPhI», Department of Thermal Physics, Russian Federation

In articles [1-15] the physical regularity of the equivalence of measures and the system of the stochastic equations for the determination onset of turbulence in isothermal and non-isothermal flows were presented. Using these new dependences the result of calculations of critical Reynolds number shows that there is the uncertainty ralation in turbulent processes. REFERENCES 1. Dmitrenko A.V. Equivalence of measures and stochastic equations for turbulent flows // Dokl. Phys. 2013.V.58 . № 6, P. 228–235. 2. Dmitrenko A.V. Calculation of pressure pulsations for a turbulent heterogeneous medium //Dokl. Phys. 2007. V.52 . № 7. P.384-387. 3. Dmitrenko A.V. Some analytical results of the theory of equivalence measures and stochastic theory of turbulence for nonisothermal fl ows//Adv. Studies Theor. Phys. 2014.V. 8, №. 25, P. 1101–1111. 4. Dmitrenko A. V. Analytical estimation of velocity and temperature fields in a circular tube on the basis of stochastic equations and equivalence of measures//, J. Eng. Phys. Thermophys. 2015.V. 88, . №6, pp. 1569–1576. 5. Dmitrenko A. V. Determination of critical Reynolds numbers for non-isothermal flows with using stochastic theories of turbulence and equivalent measures//Heat Transfer Res. 2016. V. 47, №. 1. P. 338–399. 6. Dmitrenko A. V. An estimation of turbulent vector fields, spectral and correlation functions depending on initial turbulence based on stochastic equations. The Landau fractal equation// Int. J. Fluid Mech. Res. 2016. V. 43, №. 3,P.82–91. 7. Dmitrenko A .V. The theory of equivalence measures and stochastic theory of turbulence for non-isothermal flow on the flat plate// Int. J. Fluid Mech. Res. 2016 V. 43, №. 2, Р. 182–187. 8. Dmitrenko A. V . Stochastic equations for continuum and determination of hydraulic drag coefficients for smooth flat plate and smooth round tube with taking into account intensity and scale of turbulent flow//Continuum Mech. and Thermodyn.V. 29, №1, Р. 1–9. 9. Dmitrenko A .V. Analytical determination of the heat transfer coefficient for gas, liquid and liquid metal flows in the tube based on stochastic equations and equivalence of measures for continuum //Contin.Mechan. Thermod. 2017.V. 29.№ 6.Р.1197-1206 10. Dmitrenko A .V. Estimation of the critical Rayleigh number as a function of an initial turbulence in the boundary layer of the vertical heated plate. //Heat Transf. Res. 2017.V.48. No. 13. P. 1195–1202. 11. Dmitrenko A.V. Determination of the Coefficients of Heat Transfer and Friction in Super-critical-Pressure Nuclear Reactors with Account of the Intensity and Scale of Flow Turbulence on the Basis of the Theory of Stochastic Equations and Equivalence of Measures //J.of Eng. Phys.and Thermophys.2017.V.90 .№6. P.1288-1294. 12. Dmitrenko A.V. 2013 Regular Coupling between Deterministic (Laminar) and Random (Turbulent) Motions-Equivalence of Measures Scientifi c Discovery Diploma No. 458 registration No. 583 of December 2 13. Dmitrenko A.V. Theory of Equivalent Measures and Sets with Repeating Denumerable Fractal Elements. Stochastic Thermodynamics and Turbulence. Determinacy–Randomness Correlator //Galleya-Print: Moscow.2013.226p. [ in Russian] 14. Dmitrenko A.V. Fundamentals of heat and mass transfer and hydrodynamics of single- phase and two-phase media.Criterial integral statistical methods and direct numerical simulation. // Galleya print: Moscow. 2008. 398p. 15. Dmitrenko A.V. Results of investigations of non-isothermal turbulent flows based on stochastic equations of the continuum and equivalence of measures// Journal of Physics: Conference Series.2018. 11th International Conference "Aerophysics and Physical Mechanics of Classical and Quantum Systems" (APhM-2017). IOP Conf. Series: Journal of Physics: Conf. Series 1009 (2018) 012017 doi :10.1088/1742-6596/1009/1/012017

This work was supported by the program of increasing the competitive ability of National Research Nuclear University MEPhI (agreement with the Ministry of Education and Science of the Russian Federation of August 27, 2013, Project No.02.a03.21.0005).

Igor Prokhorov^{1}
, Vladimir Kan^{2}
, Andrei Sushchenko^{3}
, Elizaveta Lyu^{4}

^{1}Institute of Applied Mathematics FEB RAS, , Russia

^{2}Far Eastern Federal University, Department of Information technologies, mathematical and computer modelling, Russia

^{3}Institute of Applied Mathematics FEB RAS, , Russia

^{4}Far Eastern Federal University, , Russia

Nowadays, studying of the world ocean remains a priority for the world community. Many research complexes are being developed to solve problems in the field of hydrography. Autonomous underwater robotic vehicles are equipped with remote sensing system. Today, the problem of mapping the ocean floor using the side scan-sonars set on the board of an autonomous unmanned submersible is very topical and perspective. The functioning of the sonar is based on periodic emission of the pulsed sound signals and detection of the echo signals reflected from the distant segments of seabottom. The authors have considered a kinetic model of the propagation of an acoustic signal. The model describes high-frequency acoustic wave propagation in a fluctuating medium. The process of propagation of acoustic radiation is described by the transfer equation: $$\frac{1}{c} \frac{\partial I}{\partial t} + k \cdot \nabla_r I(r,k,t) + \mu I(r,k,t) = \frac{\sigma}{4\pi} \int\limits_{\Omega} I(r,k',t) dk'+J(r,k,t),$$ here $r \in \mathbb{R}^3$, $t \in [0,T]$, and the wave vector $k$ belongs to the unit sphere $\Omega = \{k \in \mathbb{R}^3 : |k| = 1\}$. The function $I(r,k,t)$ is interpreted as the energy flux density of the wave propagating in the direction $k$ with the velocity $c$ at the time $t$ at the point $r$. Functions $\mu$ and $\sigma$ denote coefficients of the attenuation and the scattering, whereas $J$ describes the source of the sound field. Initial and boundary conditions for the transfer equation, respectively: $$I|_{t=0} = 0 \\ I(y,k,t)= \frac{\sigma_d}{2\pi} \int\limits_{\Omega_+(y)} |n(y)\cdot k'| I(y,k',t) dk', $$ here $n(y)$ denotes the exterior normal to the boundary surface $\gamma$. $\sigma_d$ is the bottom scattering coefficient. $ \\ $ The authors have derived the solution for radiative transfer equation with a single scattering approximation. The impulse source and receiver are supposed to be located at the same point. The boundary conditions have a diffuse reflection obeys Lambert's cosine law. In 2D case, the problem reduces to solving differential equation for function of a sea bottom profile. The article has included a numerical analysis of the solution for the equation is carried out. The dependence of the reconstruction of the lower surface on the curvature of the function describing the relief is shown.

The reported study was funded by RFBR according to the research project № 18-31-00050

M N S Qureshi^{1}

^{1}GC University, Lahore, Physics, Pakistan

Bipolar electric field solitary (EFS) structures are solitary structures in the electric field data observed frequently in space plasmas by various satellites. In this paper, we present a nonlinear fluid model to study ion-acoustic and electron acoustic waves in the presence of electrons that follow the generalized (r,q) distribution. The (r,q) distribution is the generalized form of kappa and Maxwellian distributions and can be reduced in both of these distributions in limiting cases. Non-Maxwellian distributions with high energy tail or flat shoulders at low energies are frequently observed in different regions of space plasmas and found to be best fitted only with the generalized (r,q) distribution function. Interestingly, it has been found that unlike Maxwellian and kappa distributions, the electrostatic waves admit not only density depletions but also allow the formation of density humps for positive values r which correspond to a flat-topped generalized (r,q) distribution. By studying the characteristics of bipolar EFS solitary structures, corresponding to density depletion and humps, we compare our numerical results with the observations from different regions of space plasmas and found good agreement.

Vasilij Tikhomirov ^{1}
, Vladimir Nefedov^{2}

^{1}Lomonosov Moscow State University, Computational Mathematics and Cybernatics, Russia

^{2}Lomonosov Moscow State University, Computational Mathematics & Cybernatics, Russia

The possibility of using the homotopy (deformation) method for studying the invariance of extremals for the generalized system of equations of relativistic electrodynamics is considered in this paper. This method makes it possible to investigate the stability of extremals of the action functional. It is known ([1-2]) that a generalized system of equations of relativistic electrodynamics can be obtained by solving the variational extremal problem for the sum of three action functionals, the first of which determines the freely moving charged mass of a particle in an electromagnetic field. The second functional gives the sum of the contributions determined by the four-dimensional electromagnetic Minkowski potential $$ A=(i \varphi, A_{r}) = (i \varphi, A_{1}, A_{2}, A_{3}) $$ and the third one takes into account the more complete dynamics of the electromagnetic (or gravitational) field. The solution of this variational problem allows to obtain a generalized (Hamiltonian) system of equations of the relativistic dynamics [1]. An analysis of the Lyapunov’s stability of solutions of this system is carried out by a deformation (homotopy) method for Hamiltonian systems [3].

Svetlana Zubkova^{1}

^{1}N. Frantsevych Institute for Problems of Materials Science, National Academy Sciences of Ukraine, Department of functional oxide materials, Ukraine

The theoretical investigation and ab initio calculations of the atomic and electronic structure of 4 variants of the 3C-SiC (111) (2√3 × 2√3) R30o Si- terminated surface: initial, relaxed, reconstructed and relaxed after reconstruction have been performed for the first time. In the layered superlattice approximation, the surface was modeled by a system of the thin films (slabs) with the thickness of 12 atomic layers (~ 1.6 nm) and separated by vacuum gaps of ~ 1.6 nm. To close the carbon dangling bonds on the opposite side of the film, 12 hydrogen atoms with a charge of 1 electron each were added. Ab initio calculations were carried out using the program QUANTUM ESPRESSO, based on the density functional theory In each of the variants, the equilibrium coordinates of the Si and C atoms of the upper four layers were determined. It was shown that the reconstruction leads to the splitting of layers. The results of our previous works and experimental data have shown that such splittings are inherent in the reconstruction of surface (111) in crystals with sphalerite structure. The band structures of 4 variants of slabs and full and layer-by-layer electronic densities of the states have been calculated and analyzed. It turned out that the real surface has a metallic conductivity.

Andrei Panteleev^{1}
, Valentin Panovskiy^{2}

^{1}Moscow Aviation Institute (National Research University), Mathematics and Cybernetics, Russia

^{2}Moscow Aviation Institute (National Research University), Mathematics and Cybernetics, Russia

Current work solves the problem of feedback control synthesis for a satellite using two-stage approach. During the first stage open-loop control determination task is formulated. It is being solved for various initial conditions via transformation to nonlinear programming problem and its further solution by metaheuristic optimization algorithms. Thus, as the result of the first stage it is possible to find optimal trajectories that correspond to obtained optimal open-loop controls. During the second stage previously gathered information is used to construct a regressor (as the solution of supervised learning problem) which can be treated as the approximation of the desired feedback controller for a satellite.

This work is supported by Russian Foundation for Basic Research (grant 16-07-00419 A).

Ding-wei Huang^{1}

^{1}Chung Yuan Christian University, Physics, Taiwan

In the academic publishing system, impact factor is a measure of citations to recent publications in a scholarly journal. A formulation known as the Lavalette distribution gave a fair description of the empirical data of journal impact factors. As the citation practices vary widely among different disciplines, impact factors can only be properly used to compare journals within the same discipline. Thus, mixing impact factors from different disciplines is not advocated. In some cases, however, the resultant distribution is smooth and conformed to the result within a single discipline, i.e., Lavalette distribution seems to provide a good fit when the data from two different disciplines are mixed together. We systematically explore the likelihood that the mixing of journal impact factors from different disciplines can be well described by a smooth distribution. We examine the capability of curve fitting of Lavalette distribution. We demonstrate that the combinations of two Lavalette distributions are not necessarily fitted well to a single Lavalette distribution. We suggest that the case of doublet, i.e., two identical datasets, can be taken as the criterion for the acceptance of goodness of fit. We distinguish two levels of fitting goodness: satisfied description and perfect description. Two parameters n and k are crucial. We use the approach of frequency distribution to show analytically that a satisfied description can be obtained when the parameter k from each distribution is close enough to one another. We also use the approach of curvature to show that the perfect description can be achieved when the difference in parameter k is small, the difference in parameter n is large, and a larger n is associated with a larger k. The satisfied description covers most of the empirical data. The perfect description is supported in some cases. We present both numerical and analytical results.

Arun Kumar Singh^{1}

^{1}Institute of Science, Banaras Hindu University Varanasi- 221005, Mathematics, India

Free convection of a micropolar ﬂuid is examined analytically in order to see the effect of heat and mass transfer in concentric vertical cylinders. The model of governing equations in non-dimensional velocity, microrotational velocity and temperature using the Boussinesq and Eringen equation with suitable boundary conditions are expressed in terms of cylindrical coordinate system and then exact solution one obtained. The influence of the non-dimensional physical parameters such as the vortex viscosity ratio, material and temperature ratio parameters on the velocity, microrotational velocity and the skin friction at the surface of cylinder has been studied by using graphs and table.

Mikhail Ulyanov^{1}
, Yuri Smetanin^{2}
, Mikhail Shulga^{3}
, Andrei Eserkepov^{4}
, Yuri Tarasevich^{5}

^{1}V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences and Higher School of Economics and Computational Mathematics and Cybernetics, M.V. Lomonosov Moscow State University, Program engineering , Russia

^{2}Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, , Russia

^{3}Higher School of Economics, Faculty of Computer Science, Russia

^{4}Astrakhan State University, Lab. Math. Mod., Russia

^{5}Astrakahan State University, Lab. Math. Modeling, Russia

During recent years, the attention of researchers has been drawn to the processes of self-organization in dense two-dimensional (2D) granular systems [1]. In particular, many experimental works have been devoted to the pattern formation in thin layers composed of elongated granules subjected to vibrations [2–4]. To model such the systems, it is convenient to use the lattice approach, i.e., a discrete space (square lattice) is used, the rod with the aspect ratio $k$ is represented as a linear $k$-mer (rectangle of size $1 \times k$ lattice units). In this case, the rod can have only two mutually perpendicular spatial orientations, and the coordinates of its angles can be only integer. The diffusion of $k$-mers on a square lattice has been studied using the kinetic Monte Carlo (MC) simulation [5–8]. Initial states with desired packing fraction of $k$-mers were produced using random sequential adsorption (RSA) [9]. Because the $k$-mers were considered as hard-core (completely rigid) particles, the mechanism inducing the force was necessarily entropic. Hard-core interaction between the rods means that energy of interparticle interaction, $U$, is defined as $$U = \begin{cases} 0, & \mbox{if } r>d, \\ \infty, & \mbox{if } r \leq d, \end{cases}$$ where $r$ is the distance between two particles and $d$ is the lattice constant. The Helmholtz energy, $F$, is defined as $F = U - TS,$ where $U$ is the internal energy, $T$ is the absolute temperature of the surroundings, and $S$ is the entropy of the system. At equilibrium, the Helmholtz energy of a system at constant temperature and volume is minimal. Due to hard-core interaction between the rods, the internal energy equals to zero. When the temperature is constant, any changes of the free energy can be induced only by changes of the entropy, i.e., $\Delta F = - T \Delta S.$ Pattern formation or phase transitions in such the system are said to be entropy-driven. In this case, an increase in macroscopic order is driven by an increase of microscopic disorder, i.e., a particle has more free volume to move in the final ordered state than it had in the initial disordered state [10]. The main attention was paid to isotropic systems, i.e., to systems with an equal number of vertical and horizontal particles. In dense systems, only translational diffusion of particles is possible, while rotational diffusion is completely suppressed. The system is isolated, i.e., neither matter nor energy can come in and go away. The influence of the aspect ratio of the particles, their concentration, the size of the system, and the type of boundary conditions on the formation of structures and the change in physical properties have been studied [5,6]. The configurational entropy has been applied to analyze entropy-driven phase transition in a system of long rods on a square lattice [11]. Recently, the orientational phase transitions that occur in the deposition of $k$-mers were characterized by information theory techniques [12]. Namely, the data recognizer ``word length zipper'' (wlzip) was created and utilized to find repeated meaningful information in any sequence of data. In spite of the fact that the change in various characteristics with time was investigated and reported [5,6], the key value---entropy---has not been investigated. Since the degree of randomness or orderliness of the system is characterized by its entropy, the study of this particular quantity is expected to be most important for understanding the details of diffusion-driven self-organization. In this paper, we propose to apply the entropy approach to the study of the process of self-organization, i.e. the study of the change in the entropy of such a system over time. Our goal is entropy changes in the dense 2D system of rods. References [1] I. S. Aranson and L. S. Tsimring, “Patterns and collective behavior in granular media: Theoretical concepts,” Rev. Mod. Phys., vol. 78, pp. 641–692, Jun 2006. [2] T. Borzsonyi and R. Stannarius, “Granular materials composed of shape-anisotropic grains,” Soft Matter, vol. 9, no. 31, pp. 7401–7418, 2013. [3] T. Muller, D. de las Heras, I. Rehberg, and K. Huang, “Ordering in granular-rod monolayers driven far from thermodynamic equilibrium,” Phys. Rev. E, vol. 91, p. 062207, Jun 2015. [4] M. Gonzalez-Pinto, F. Borondo, Y. Martinez-Raton, and E. Velasco, “Clustering in vibrated monolayers of granular rods,” Soft Matter, vol. 13, pp. 2571–2582, 2017. [5] N. I. Lebovka, Y. Y. Tarasevich, V. A. Gigiberiya, and N. V. Vygornitskii, “Diffusion-driven self-assembly of rodlike particles: Monte Carlo simulation on a square lattice,” Phys. Rev. E, vol. 95, p. 052130, May 2017. [6] Y. Y. Tarasevich, V. V. Laptev, A. S. Burmistrov, and N. I. Lebovka, “Pattern formation in a two-dimensional two-species diffusion model with anisotropic nonlinear diffusivities: a lattice approach,” J. Stat. Mech. Theory E., vol. 2017, no. 9, p. 093203, 2017. [7] Y. Y. Tarasevich, V. V. Laptev, V. V. Chirkova, and N. I. Lebovka, “Impact of packing fraction on diffusion-driven pattern formation in a two-dimensional system of rod-like particles,” J. Phys. Conf.e Ser., 2018, submitted. [8] S. Patra, D. Das, R. Rajesh, and M. K. Mitra, “Diffusion dynamics and steady states of systems of hard rods on a square lattice,” Phys. Rev. E, vol. 97, p. 022108, Feb 2018. [9] J. W. Evans, “Random and cooperative sequential adsorption,” Rev. Mod. Phys., vol. 65, pp. 1281–1329, Oct 1993. [10] D. Frenkel, “Order through disorder: entropy strikes back,” Physics World, vol. 6, no. 2, p. 24, 1993. [11] D. H. Linares, F. Roma, and A. J. Ramirez-Pastor, “Entropy-driven phase transition in a system of long rods on a square lattice,” J. Stat. Mech. Theory E., vol. 2008, no. 03, p. P03013, 2008. [12] E. E. Vogel, G. Saravia, and A. J. Ramirez-Pastor, “Phase transitions in a system of long rods on two-dimensional lattices by means of information theory,” Phys. Rev. E, vol. 96, p. 062133, Dec 2017.

The authors acknowledge that the reported study was funded by the Russian Foundation Basic Research according to research project No. 18-07-00343

Vyacheslav Kalnitsky^{1}

^{1}Saint-Petersburg state university, Higher geometry, Russian Federation

The concept of the tangent and cotangent bundles plays a central role in describing the dynamics of mechanical systems with singular configuration space. There is no any unified approach to the description of these objects at the present time. The reason for considering a particular construction is that the constructed model gives these bundles in the classical case. One example of such a generalization is the algebra of cosymbols of differential operators of the smooth functions algebra on a singular manifold in the category of geometric modules. This algebra has the natural structure of the Hopf algebra and the algebra dual to it in the classical case coincides with the cotangent bundle of a smooth manifold. Generalizing this example, we introduce the notion of a universal graded algebra for which we can define the structure of the Hopf algebra and the Poisson bracket is defined in a natural way on the dual algebra. This allows us to determine the evolution equation of the system.

Jamal Davoodi^{1}
, Soghra Mohammadzadeh^{2}
, Nader Mallih^{3}

^{1}University of Zanjan, Physics, Iran, Islamic Republic Of

^{2}University of Zanjan, Physics, Iran, Islamic Republic Of

^{3}University of Zanjan, Physics, Iran (Islamic Republic of)

The topic of wetting has received very much interest from both fundamental and applied points of view. It plays an important role in many industrial processes, such as oil recovery, liquid coating, lubrication, printing, and spray quenching. Boron sheets electron deficiency prevent it to form a graphene-like structure nevertheless it have been predicted that they are stable with triangular lattices and hexagonal vacancies and in recent studies of a cluster containing 36 boron atoms a planar C6v structure have been observed and named it borophene. After that, two-dimensional (2D) boron sheets (i.e., borophene) with a hexagonal structure has been grown on Ag(111) crystal. Previous studies on graphene shows hydrophobic with contact angles between 95° to 100° and because of graphene and borophene structure similarity it is expected that borophene will also exhibits hydrophobic behaviour. Therefore, we perform a molecular dynamics (MD) simulation to calculate the contact angle and define hydrophilic or hydrophobic characteristic of borophene surface with water. This was done by LAMMPS molecular dynamics simulator package. We use TIP3P model with long-range Columbic interaction for water molecules and ReaxFF force field to show interaction of boron atoms in borophene and also for interaction between surface and water nanodroplet. The obtained MD simulation results show that a borophene surface is hydrophobic. Moreover, the obtained MD results show that the contact angle of water nanodroplet on a borophene surface is considerably influenced by the temperature variations, i.e. as the temperature increases, the contact angle decreases and the surface becomes less hydrophobic. Finally, we show that contact angle of water nanodroplet on a borophene decrease with increasing nanodroplet size.

Vasily Lisenkov^{1}

^{1}Institute of Electrophysics UB RAS, Laboratory of Quantum Electronics, Russia

Nowadays, the generation of runaway electrons in high-pressure gases is one of the most studied topics in gas discharge physics. The most advanced method of modeling this process is the combination of the Monte Carlo method and the particle-in-cell method (PIC MC method). However, even in a two-dimensional formulation, this approach requires very large computational resources. This is explained by the fact that the numerical density of the runaway electrons is several orders of magnitude lower than the plasma (slow) electrons. Therefore, it is necessary to take into account a large number of particles. Using of a hybrid model can significantly reduce computing resources. In hybrid model, PIC MC method is used only for calculation of runaway electrons, and the standard hydrodynamic approach is used for calculation of plasma electrons. The paper analyzes the details of the application of the hybrid model for calculation of the formation of high-pressure gas discharge in conditions where the transition of electrons into runaway mode is possible. The results of calculation of kinetics of electrons emitted from a micro-spike on the cathode during the formation of the cathode layer of nanosecond and sub-nanosecond high-pressure gas discharge are presented. The conditions of transition of electrons into runaway mode at this stage and their influence on the further formation of the gas discharge are analyzed.

This work was supported by RFBR, Grant 16-08-00894

Franco Stoianoff Lindstron^{1}

^{1}UNLP, Matemáticas, Argentina

This paper is an application of the methods of Feynman and Kac for the study of the combinatorial factors of the Ising model.

Alexey Bondarev^{1}
, Igor Bataronov^{2}
, Irina Pashueva^{3}

^{1}Voronezh State Technical University, , Russia

^{2}Voronezh State Technical University, , Russia

^{3}Voronezh State Technical University, , Russia

Amorphous alloys based on rare-earth metals are of great interest due to their unique magnetic properties. In this work we report on the Monte Carlo simulation of magnetic properties of amorphous alloys of Re-Tb and Re-Gd systems in the wide compositional region. Using the Monte Carlo method in the frame of the Heisenberg model, the computer simulation of magnetic properties of Re-Tb and Re-Gd amorphous alloys was performed. For pure amorphous Tb and Re-Tb amorphous alloys the model Hamiltonian contained two terms responsible for nearest-neighbour exchange interaction between the Tb ions with the mean value J0 and for random single-ion anisotropy D. In the models of the Re-Tb amorphous alloys the spin-glass-like phase transition was also observed. With increasing concentration of Tb atoms, the transition temperature linearly increases, which is in a good agreement with the experimental results. The spin-glass transition is observed only above the percolation threshold in this system, i.e. at x>13 at. % Tb. For pure amorphous Gd and Re-Gd amorphous alloys the model Hamiltonian contained two terms responsible for ferromagnetic exchange interaction J1 between the nearest-neighbour Gd ions and for antiferromagnetic exchange interaction J2 between the Gd ions that were in the second coordination sphere. In the models of the Re-Gd amorphous alloys the spin-glass phase transition was also observed. With increasing concentration of Gd atoms, the spin-glass transition temperature linearly increases, which is in a good agreement with the experimental results. The magnetization relaxation after switching-off the external magnetic field at different values of D/J0 (for Re-Tb alloys) J1/J2 (for Re-Gd alloys) was studied. For amorphous Tb and Re-Tb amorphous alloys the magnetization relaxation goes on in two stages. In the first stage, the magnetization decreases abruptly by a definite magnitude ΔMZ. At this stage the magnetic moments reorients to the directions determined by the random anisotropy axes. At the second stage the magnetization decreases very slowly according to the logarithmic law. At this stage the magnetic moments of Tb atoms rotate by small angles due to competition of exchange interaction and random anisotropy. In the models of amorphous Gd and Re-Gd amorphous alloys, the jump of the magnetization after the switching-off the external magnetic field is not observed. Thus, in these systems the relaxation is one-stage. The time dependence of magnetization is not logarithmic, but is described by the lineal combination of two decreasing exponential functions.

Alexander Chebotarev^{1}
, Andrey Kovtanyuk^{2}
, Alena Astrakhantseva^{3}

^{1}Far Eastern Federal University, , Russia

^{2}Far Eastern Federal University, , Russia

^{3}Far Eastern Federal University, Informatics, mathematical and computer modelling, Russia

Endovenous laser ablation (EVLA) is a very effective minimally invasive therapy to manage leg varicosities. Modeling of radiative and thermal processes in the case of EVLA assumes taking into account the conduction of heat from the laser tip, thermal radiation from the laser tip, and thermal sources caused by the absorbed energy of the laser light. Moreover, at the blood fraction temperature close to 100$^\mathrm{o}$C, an important contribution to the heat transfer occurs due to the formation and convection of vapour bubbles. The paper of A.A. Poluektova et al (Laser Med. Sci., 2014) proposes to simulate the presence of bubbles by varying the coefficient of thermal conductivity. Namely, in the area inside a vein, if the temperature reaches 95$^\mathrm{o}$C or higher, the coefficient of thermal conductivity increases by 200 times. The last value is adjusted experimentally, which ensures a good consistency of the model considered in the work of A.A. Poluektova et al (Laser Med. Sci., 2014 ). Because of the spatially propagating sharp jump in thermal conductivity, the corresponding initial boundary value problem is a free-boundary one of the Stefan type. As usually, such a problem can be approximated by a relaxed phase field model with the smoothed interface. To describe the process of endovenous laser ablation, the system of heat and radiation transfer equations with moving source is chosen. The conventional non-stationary normalized $P_1$ approximation of the radiative and heat transfer model describing radiative, conductive, and convective contributions is considered in a bounded domain $\Omega \subset \mathbf{R}^3$: $$ \partial\theta/\partial t - \mathrm{div}\,(a\nabla\theta) + \mathbf v \cdot \nabla \theta - b\kappa_a \varphi = u_1,\; $$ $$ -\alpha \Delta \varphi + \kappa_a\varphi = u_2,\; x\in\Omega,\ t\in (0,T). $$ Here, $\theta$ is the normalized temperature, $\varphi$ the normalized radiation intensity averaged over all directions, $\textbf{v}$ a given divergence free velocity field, and $u_{1,2}=u_{1,2}(x,t)$ describe the intensities of heat and radiation sources moving along a given trajectory. Parameters $a$, $b$, $\kappa_a$, and $\alpha$ describe the radiation and thermal properties of the medium. Moreover, $a=a(x,\theta)$, $b= b(x)$, $\kappa_a= \kappa_a(x)$, and $\alpha= \alpha(x)$. Due to the dependence of the coefficient $a$ on the temperature, we will simulate a high thermal conductivity at the points of the blood fraction where temperature reaches 95$^\mathrm{o}$C The following boundary conditions on $\Gamma := \partial \Omega$ and the initial condition at $t=0$ are assumed: $$ a\partial_n \theta + \beta(\theta - \theta_b)|_\Gamma = 0,\;\; \alpha\partial_n \varphi + \gamma\varphi |_\Gamma = 0, \quad \theta|_{t=0} = \theta_0. $$ Here, the boundary functions, $\theta_{b}$, $\beta$, $\gamma$, and the initial function, $\theta_0$, are given. We note the essential nonlinearity of the problem caused by the dependence of the thermal conductivity on temperature. In the current work, theoretical and numerical analysis of this problem are fulfilled. The numerical examples simulated the process of endovenous laser ablation are presented.

Varsha Banerjee^{1}

^{1}Indian Institute of Technology, Delhi, Physics, India

Many magnetic and dielectric solids are well-represented by (i) the random field Ising model with conserved order parameter (C-RFIM) and (ii) the Ising model with dipolar interactions (IM+DI). We perform deep quenches to study the kinetics of domain growth in these spin systems. In the C-RFIM, (a) the correlation function exhibits dynamical scaling but is not robust with respect to the disorder $\Delta$; (b) at early times the domains follow algebraic growth with a disorder-dependent exponent: $L(t,\Delta)\sim t^{1/z(\Delta)}$. At late times, there is a cross-over to logarithmic growth $L(t,\Delta) \sim (\ln t)^{1/\varphi}$, where $\varphi$ is the disorder-independent logarithmic exponent. In the IM+DI, (a) there is emergence of columnar domains along $z$-axis (Ising axis) with a transient periodicity in the $xy$-plane; (b) there are anisotropic growth laws: $\ell_{\rho}(t) \sim t^{\phi}$; $\ell_z(t) \sim t^{\psi}$, where $\vec{\rho}= (x,y)$ and $\ell$ is the characteristic length scale. Our results explain experimental observations in a wide range of complex systems and have important technological implications.

Anton Beliaev^{1}
, Tatiana Beliakova^{2}
, Petr Chistyakov^{3}
, Alexander Inyukhin^{4}
, Liliia Kostyreva^{5}
, Pavel Mossakovsky^{6}

^{1}Lomonosov Moscow State Univercity, Mechanics and mathematics, Russia

^{2}Institute of Mechanics, Lomonosov Moscow State University, , Russia

^{3}Institute of Mechanics, Lomonosov Moscow State University, , Russia

^{4}Institute of Mechanics, Lomonosov Moscow State University, , Russia

^{5}Institute of Mechanics, Lomonosov Moscow State University, , Russia

^{6}Institute of Mechanics, Lomonosov Moscow State University, , Russia

The dynamic interaction of multilayer woven barriers of aramid fibers with various forms of impactors is considered, and the processes of energy dissipation are investigated. Experimental investigation of the interface friction coefficients is performed under the conditions of the transverse compression for different weaving types of fabrics. The required range for the compressive loads is estimated by the use of FE-modelling of impact loading of the multi-layer woven barriers. Static and dynamic friction coefficients are determined for all possible combinations of textile-textile interfaces with plain and four types of twill weaving patterns. It is obtained that the interface static friction coefficient depends on the weaving types of the fabric layers. Moreover, it cannot be always derived from the averaged values of friction coefficients for each weaving type on the interface. For the higher values of the transversal compression the experimental equipment is proposed and designed for pulling out a textile layer from a free multilayer structure of fabrics under the conditions of controllable transverse compression loading. The numerical modelling of the load distribution in the tested fabric layer is accomplished which allows us to estimate the irregularity of the transversal pressure in the specimen and to correct the experimental results for friction coefficients. The finite elements models of different complexity for fabrics of five weaving types and for their combinations are constructed. Advantages and disadvantages for different approaches to aramid fabric modeling are compared. The case of a three-dimensional plain-base weaving was considered separately. Verification of the models is carried out by full-scale ballistic experiments, and the simulation error is estimated. The contribution of various mechanisms to the energy dissipation of the impactor is estimated. The influence of the placement of layers with different weaving type in the woven barriers on the impactor velocity after the rupture is analysed numerically. The advantages of the particular combinations of weaving type placement in comparison with the uniform barriers with the same number of layers is shown. The research is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University. The work was carried out in accordance with the scientific research plan of the Institute of Mechanics Lomonosov Moscow State University.

Nikolai Magnitskii^{1}

^{1}Federal Research Center Informatic and Control, Institute for Sistems Analysis, Nonlinear and Chaotic Dynamics, Russia

In the paper, ether is considered as a dense compressible inviscid oscillating medium in three-dimensional Euclidean space, having a density of ether and a velocity vector of propagation of density perturbations at each instant of time. Ether can be described by two nonlinear equations, where the first equation is the continuity equation, and the second is the ether momentum conservation law. It is shown that the consequences of the system of these two equations are: a generalized nonlinear system of Maxwell-Lorentz equations that is invariant under Galilean transformations, the linearization of which leads to the classical system of Maxwell-Lorentz equations; laws of Bio-Savar-Laplas, Ampère and Coulomb; representations for Planck's and fine structure constants; formulas for the electron, proton and neutron in the form of wave solutions of the system of two ether equations for which the calculated values of their internal energies, masses and magnetic moments coincide, with an accuracy to fractions of a percent, with their experimental values, anomalous from the point of view of modern science.

Francisco Delgado-Cepeda^{1}
, Suset Rodriguez Aleman^{2}

^{1}Instituto Tecnologico y de Estudios Superiores de Monterrey, Physics and Mathematics, Mexico

^{2}Instituto Tecnologico y de Estudios Superiores de Monterrey, Physics and Mathematics, Mexico

Quantum Information is a quantum resource being advised as a useful tool to set up information processing. Physical components considered are normally two-level systems, but their combination and their entangling interactions become in a complex dynamics on $U(2^n)$ required to be modeled to reproduce certain processing operations useful to develop quantum algorithms. $SU(2)$ decomposition is a mathematical mechanism to translate the dynamics from $U(2^{2d})$ into $U(1)^{2(d-1)-1} \times SU(2)^{2(d-1)}$. Then, still the two-qubit case becomes relevant because it is the core of $SU(2)$ decomposition by Hamiltonians as: $$ H_h = \sum_{k=1}^3 J_k {\sigma_1}_k {\sigma_2}_k -{B_1}_h {\sigma_1}_h-{B_2}_h {\sigma_2}_h $$ (expressed for magnetic qubits despite other physical systems fits into equivalent Hamiltonians) into a direct sum of momentary two-level quantum information subspaces (with dynamics on $SU(2)$ group) as equivalent quantum resources to the direct physical elements (with dynamics on $U(4)$ group): $$ H_h(t) = \bigoplus_{k=1}^2 H_{k,h} \rightarrow U_h(t) = \bigoplus_{k=1}^2 {s_h}_{k} $$ where control is achieved as driven operations on physical elements of the setup, but reflected on certain pure quantum information states as a grammar. This work presents a comparative benchmark of numerical approaches to address these problems with time-dependent fields ${B_i}_h$ (more affordable technologically and free of resonant effects) together with some useful outcomes for the dynamics in the context of quantum information processing.

Robert Namm^{1}
, George Tsoy^{2}

^{1}CC FEB RAS, Laboratory of numerical methods in mathematical physics, Russia

^{2}CC FEB RAS, Laboratory of numerical methods in mathematical physics, Russia

A modified dual method for solving an elastic problem with a crack extending to the outer boundary is considered. The method is based on a modified Lagrange functional. The convergence of the method is investigated in detail under a natural assumption of $H^1$ - regularity of the solution to the crack problem. Basic duality relation for the primal and dual problems is proposed.

Pranesh Kumar^{1}

^{1}University of Northern British Columbia, Mathematics and Statistics, Canada

Copula functions are used as an alternative means of modeling and simulating multivariate data. Copula function is precisely a multivariate distribution function on marginally uniform random variables on the unit interval. Copula parameter measures both the linear and nonlinear dependence and provides an alternative to the dependence measures like linear correlation coefficient. Copula functions have played an important role in statistics, insurance, finance, economics, survival analysis, image processing and engineering applications. In this paper, we will describe the copula functions, their properties, simulations and examples of copula applications from different scientific areas.

Julio Guemez^{1}
, Manuel Fiolhais^{2}
, Luis Fernandez^{3}

^{1}University of Cantabria, Department of Applied Physics, Spain

^{2}University of Coimbra, Department of Physics, Portugal

^{3}University of Cantabria, Department of Mathematics, Statistics and Computation, Spain

This work is about relativistic rotations around a principal axis and results from a fruitful collaboration between physicists (M. Fiolhais and J. Güémez) and a mathematician (L.A. Fernández). The relativistic rotation, including the inertia and the moment of inertia of rotating bodies, and the rotation equation, has deserved some interest, certainly due to the qualitative differences in comparison with the classical situation. We describe the relativistic rotations through the relativistic rotation dynamical equations derived in the framework of the the Hamiltonian-Lagrangian formalism. We consider a rigid solid body of volume $\Omega,$ and constant density, $\rho,$ rotating with angular velocity $\omega$ around a principal axis. Differently from classical mechanics, now the inertia of the body is angular velocity dependent, and it is given by $$ {\cal M}(\omega) = \rho \int_{\Omega} \gamma\left( {\omega \, r(x)} \right) {\rm d}x\, , $$ where $r(x)$ is the distance from point $x$ to the rotation axis and $\gamma(v)= \left[ 1+\left( \frac{v}{c}\right)^2 \right]^{-1/2}$ is the usual relativistic factor. Similarly, the moment of inertia is also angular velocity dependent: $$ I(\omega) = \rho \int_{\Omega} r^2(x)\gamma\left( {\omega r(x)} \right) {\rm d}x\, . $$ As in non-relativistic mechanics, the angular momentum is $$ J=I \omega \, , $$ and $$ \frac{ {\rm d} J}{{\rm d} t} = \Gamma^{\rm ext} , $$ where the right hand side is the external torque. A central part of this work is to show that the following equality holds: $$ \frac{ {\rm d} \left[{\cal M} (\omega)c^2\right] }{ {\rm d}\, J } = \omega\, . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) $$ We present the mathematical proof of this equation which, physically speaking, relates the variation of energy with the angular momentum variation, throught the angular velocity. Though there is a counterpart in classical physics, Equation (1), to the best of our knowledge, has never been shown in the literature, until our derivation of it published in a recent issue of Europhysics Letters [1]. We show its usefulness by discussing some examples of relativistic rotating systems that illustrate how special relativity formalism, including Equation (1), deals with rotational processes of systems acted upon by external torques, including cases when dissipative forces are present. $$ $$ [1] J. Güémez, M. Fiolhais and L. A. Fernández, EPL 119 (2017) 10001 - doi: 10.1209/0295-5075/119/10001

Co-authors: J. Güémez (Department of Applied Physics, University of Cantabria - E-39005 Santander, Spain) and L. A. Fernández (Department of Mathematics, Statistics and Computation, University of Cantabria - E-39005 Santander, Spain)

Marcin Lawnik^{1}

^{1}Silesian University of Technology, , Poland

In the paper the combination of logistic and tent map is discussed. Some basic properties, like Lyapunov exponent and density of the iterated variable, of the dynamical system are analyzed. Furthermore, applications of the discussed model in chaos based cryptography are presented.

Andrei Melekhin^{1}

^{1}Federal state budget educational institution of higher education "MOSCOW STATE UNIVERSITY OF CIVIL ENGINEERING (NATIONAL RESEARCH UNIVERSITY)" , , Russia

The author have developed a mathematical model of process of heat exchange in heat exchange surfaces of apparatuses with the solution of multicriteria optimization problem and check its adequacy to the experimental stand in the visualization of thermal fields, an optimal range of managed parameters influencing the process of heat exchange with minimal metal consumption and the maximum heat output fin heat exchanger, the regularities of heat exchange process with getting generalizing dependencies distribution of temperature on the heat-release surface of the heat exchanger vehicles, defined convergence of the results of research in the calculation on the basis of theoretical dependencies and solving mathematical model.

Rustem Sakhapov^{1}
, Muhammat Gatiyatullin^{2}
, Nikolaeva Regina^{3}
, Marat Makhmutov^{4}

^{1}Kazan State University of Architecture and Engineering, "Road Construction Machinery", Russia

^{2}Kazan State University of Architecture and Engineering, Department of road-building machines, Russia

^{3}Kazan State University of Architecture and Engineering, Department of road-building machines, Russia

^{4}Kazan State University of Architecture and Engineering, Department of road-building machines, Russia

The article examines the issue of the effectiveness of goods shipment inside the country. Progress in the methods and principles of managing transportation processes allows to increase the speed of delivery, improve the quality of service, and regulate transport tariffs. Considering the current state of Russia with its vast territory, a wide variety of natural and climatic and economic-geographical conditions, it can be noted that it needs a clear logistics system to link production procedures. The aim of the study is the formation of a developed regional transport and logistics system. To achieve this goal, we consider two main tasks: the identification of regions in terms of the priority of design and location of logistics centers; optimization of logistic schemes of cargo delivery. Creation of a developed regional transport and logistics system is the most important condition for the economic development of the country, as well as the development of interregional economic interaction, which will increase the social and economic potential of the regions. Increasing the economic efficiency of commodity exchange is achieved both through the effective organization of a system of interregional logistics centers, and through the development of interregional economic interaction.

Sergey Maslov^{1}
, Vladimir Natyaganov^{2}

^{1}Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Russia

^{2}Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Russia

The work researches the influence of parent thundercloud charge structure and atmospheric electric field (AEF) perturbations on the process of tornadoes and downbursts formation on the basis of electrohydrodynamic equations system. The authors first have derived a one-parameter formula describing the process of thundercloud recharge, i.e. transition of dipole charge structure to tripole one. Thundercloud vorticity increase, i.e. transforming thundercloud into the tornado-cyclone, during the recharge process is shown to be possible due to electromagnetic factors, not Coriolis forces. Besides that, dipole and tripole structures of the cloud charge can create necessary conditions for correspondingly downburst and tornado formation. The authors have presented the electrohydrodynamic model of downburst formation from the dipole thundercloud, have determined velocity and AEF magnitude fields near the burst. The work researches the role of electrohydrodynamic pressure in the process of tornado formation from the tripole cloud. Using the formula of AEF perturbations, the authors have solved the problem of tornado funnel motion and theoretically explained frequently observed “hill” growth on the water surface toward the descending funnel. It is shown that strong AEF perturbations can lead to the funnel descent as well as its hang at some height above the ground.

Rustem Sakhapov^{1}
, Muhammat Gatiyatullin^{2}
, Marat Makhmutov^{3}
, Nikolaeva Regina^{4}

^{1}Kazan State University of Architecture and Engineering, "Road Construction Machinery", Russia

^{2}Kazan State University of Architecture and Engineering, Department of road-building machines, Russia

^{3}Kazan State University of Architecture and Engineering, Department of road-building machines, Russia

^{4}Kazan State University of Architecture and Engineering, Department of road-building machines, Russia

This article investigates the increase of traction-coupling properties of propellers, patency of machine-tractor units and decrease of soil compaction. For a propulsor equipped with anti-skid devices, the slippage process is formed due to factors of soil compression by the soil-tread and shear of the soil by tire hooks. With the decrease in the number of detachable hooks, the period when the first hook disengaged and the second has not yet entered in interaction with the ground, increases. At the moment, traction and coupling properties of the wheel are formed only due to the tire hooks. When the detachable hook engages with the soil, the traction capacity of the wheel is made up of the forces of shear of soil "bricks" sandwiched between tire hooks and the forces of soil deformation by detachable hooks. As a result of integrating the dependence of the shear stress and deformation of the soil, a formula was obtained to determine the tangential traction force of the tractor. If the skidding of the wheel assembly depends on the pulling force, then the drag force from the anti-skid device. Using the known dependencies of the slippage of wheeled propellers, dependences are obtained to determine the slippage of the propulsion unit with the anti-skid device. With the increase in the number of anti-skid devices on the wheel, slippage is reduced, however, according to the research, the time required for the assembly and disassembly of the device reduces the interchangeability of the wheel assembly. We calculated he number of anti-skid devices at which the maximum exchange capacity will be reached. The penetration depths at which the maximum efficiency of the wheel assembly running system will be reached are obtained and the results are tabulated.

Maxim V. Shamolin^{1}

^{1}Lomonosov Moscow State University, Institute of Mechanics, Russia

In many problems of dynamics, there appear mechanical systems with three-dimensional manifolds as position spaces. Tangent bundles of such manifolds naturally become phase spaces of such systems. For example, study of a four-dimensional generalized spherical pendulum in a nonconservative force field leads to a dynamic system on the tangent bundle of a three-dimensional sphere, and the metric of special form on it is induced by an additional symmetry group. In this case, dynamic systems describing the motion of such a pendulum possess alternating dissipation and the complete list of first integrals consists of transcendental functions that can be expressed in terms of a finite combination of elementary functions. The class of problems about the motion of a point on a three-dimensional surface is also known; the metric on it is induced by the Euclidean metric of the ambient space. In some cases of systems with dissipation, it is also possible to find a complete list of first integrals; the list consists of transcendental functions. The results obtained are especially important in the aspect of the presence of just a nonconservative force field in the system.

Sergey B. Levin^{1}

^{1}St.-Petersburg State University , Faculty of Physics, Department of Higher Mathematics and Mathematical Physics, Russia

We study the quantum scattering problem of three three-dimensional charged particles involving pair potentials of Coulomb attraction in the framework of the diﬀraction approach. The method was oﬀered earlier in the works [1]-[2] for the system of three one-dimensional neutral quantum particles and was developed further in the works [3]-[6] for the systems of few one-dimensional and three-dimensional like-charged quantum particles. In the present work we obtain the uniﬁed contribution of pair excited states to the asymptotics (at inﬁnity in conﬁguration space) of the three-body absolutely continuous spectrum eigenfunctions. The ﬁrst steps in the direction were already made in the work [7]. Note that it is the case that is most interesting and rich from the physical point of view, as it is connected in particular with re-clustering of charged quantum complexes. \vskip1cm References [1] V.S.Buslaev, S.P.Merkuriev, S.P.Salikov, On diﬀraction character of scattering in quantum system of three onedimensional particles. Problems of mathem. physics, Leningrad University, Leningrad, v.9, pp. 14–30, (1979) (in russian) [2] V.S.Buslaev, S.P.Merkuriev, S.P.Salikov, Description of the pair potentials in the system of three one-dimensional particles free of diﬀraction eﬀects. Boundary problems of mathematical physics and other questions in theory of functions, 11. Zap. Nauch. Sem. LOMI, v.84, pp.16–22, (1979) (in russian) [3] V.S.Buslaev and S.B.Levin, Asymptotic Behavior of the Eigenfunctions of the Many-particle Shroedinger Operator. I. One-dimentional Particles, Amer.Math.Soc.Transl., v.225, pp. 55–71, (2008) [4] V.S.Buslaev, S.B.Levin, Asymptotic behaviour of eigenfunctions of the three-body Schroedinger operator. II. One-dimensional charged particles. Algebra and Analysis, v.22(3), pp.60-79, (2010) (in russian) [5] V.S.Buslaev, S.B.Levin, The system of three charged quantum particles: asymptotic behaviour of continuous spectrum eigenfunctions at inﬁnity, Functional analysis and its applications, v.46(2), pp.83-89, (2012) (in russian) [6] S.B.Levin, On asymptotic behaviour of continuous spectrum eigenfunctions at inﬁnity for the system of three three-dimensional like-charged quantum particles, Journal of Mathematical Sciences, v.226(6), pp.744-768, (2017) [7] A.M.Budylin, Ya.Yu.Koptelov, S.B.Levin, Some aspects of the scattering problem for the system of three charged particles, Zap. Nauch. Sem. POMI, v.461, pp. 65-94, (2017) (in russian)

Adel Morad^{1}

^{1}Faculty of Science, Menoufia University, Mathematics and Computer Science, Egypt

In this paper, we explore theoretically the flow instabilities occur in thin liquid films of a perturbation flow using in Euler equation in Cartesian coordinate system. We derived the shallow water equations describing the motion of thin liquid film on the rotating cylinder surface. The governing equations have been constructed by using the usual multi-scale asymptotic expansion methods. The hodograph method is used to investigate the behavior of solutions of the liquid layer motion on the outer cylinder surface. It is found that, when the liquid layer located on the outer surface of the cylinder, the behaviors of solutions are periodic perturbations. The modelling results showed that the initial perturbations can develop in the unstable media. The diagrams are drawn to illustrate the physical properties of the exact solutions, and indicate their stability and bifurcation by using various techniques.

Polina Chirkova^{1}
, Oleg Shkolik^{2}
, Larisa Chirkova^{3}

^{1}Ural Federal University, Graduate School of Economics and Management, Russia

^{2}Centre of regional comparative research, Institute of economics of the Ural branch of Russian academy of sciences, Russia

^{3}Ural Federal University, Institute of Physics and Technology, Russia

Strong and stable financial sector is a primary element for the inflow of investments and increasing liquidity in the economic system of a state striving for a developed economy. The practice of several previous years has shown the presence of certain factors of the instability of the Russian financial sector. The most important among the fundamental factors of the instability of the national financial sector is the problem of ensuring the monetary obligations of financial institutions, which leads credit institutions and investment institutions at best to introduce external management from the Central Bank, and in the worst case to bankruptcy. Traditionally, the main focus in assessing the quality of assets of financial institutions is to determine the market (fair) value of tangible assets, namely financial assets and real estate. However, recent events that have occurred with large Russian credit institutions have revealed shortcomings in this approach. First of all, because the objects of intellectual property and other intangible assets are either undervalued or overvalued. Based on these facts, a team of researchers developed and presented a mathematical model that allows an alternative valuation of intangible intellectual assets in the ownership structure of financial institutions. The model was applied in the analysis of the activities of individual financial institutions operating within the national economic system.

Ahmed Abu-Nab^{1}

^{1}Faculty of Science, Menoufia University, Mathematics and Computer Science, Egypt

In this paper, the behavior of a gas bubble in a vertical cylindrical tube under the effect of peristaltic motion of long wavelength and low Reynolds number with two-phase density has been investigated. The mathematical model is solved analytically by using mass, momentum, and concentration equations to obtain the behavior of gas bubbles, concentration and velocity distributions. The growth of gas bubbles, concentration and velocity distribution are studied under the effect of some physical parameters, such as, the amplitude ratio, Grashof number, heat source parameter, volume rate, and density fraction. Several graphs for these results of physical interest are displayed and discussed in detail.

Sanjay Puri^{1}

^{1}Jawaharlal Nehru University, School of Physical Sciences, India

We discuss the dynamical behavior of freely-evolving granular gases, e.g., interstellar dust. This system loses energy (cools) continuously because of the inelastic collisions between particles. The gas initially cools in a homogeneous cooling state (HCS), but a clustering instability drives it into an inhomogeneous cooling state (ICS). We present results for the HCS and ICS of granular gases where (a) the restitution coefficient is constant; (b) the dissipation mechanism is friction. We discuss the analytical and numerical techniques used to study granular gases.

Dang Quang A^{1}
, Dang Long^{2}

^{1}Centre for Informatics and Computing, Vietnam Academy of Science and Technology, , Vietnam

^{2}Institute of Information Technology, Mathematical methods in IT, Viet Nam

In this work we propose a novel approach to investigate boundary value problems (BVPs) for fully third order differential equations which arise in various areas of applied mathematics and physics. It is based on the reduction of BVPs to operator equations for the nonlinear terms but not for the functions to be sought. By this approach we have established the existence, uniqueness and positivity of solutions and the convergence of the iterative method for approximating the solutions under some easily verified conditions in bounded domains. These conditions are much simpler and weaker than those of other authors when studying solvability of the problems by using different methods. Many examples illustrate the applicability and efficiency of the proposed approach.

This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 102.01-2017.306.

Tai-Ho Yu^{1}

^{1}National United University, Electronic Engineering, Taiwan

Exact solutions for flexural ridge wave propagation around a circular cylindrical tube have not previously been possible. Rather, such problem used to be solved numerically or using an empirical formula. This study investigates the dispersive properties of ridge waves traveling circumferentially around piezoelectric tubes, as well as their resonant modes. Based on the separation of variables, the displacements of ridge waves are represented as the product of a cross-sectional coordinate-dependent function and the propagator along the circumference of a tube. The dispersion equation of ridge waves is formulated by using Hamilton’s principle and the so-called bi-dimensional finite element method (Bi-d FEM). Furthermore, the dispersion curves of traveling waves and resonant frequencies corresponding to standing waves are solved numerically. Several applications, such as an optimal design for circular cylindrical ridge wave ultrasonic motors, are illustrated. The Bi-d FEM and the commercial finite element code ANSYS are used to determine the dispersion curves of circular cylindrical ridge wave ultrasonic motors. The responses of traveling waves around the stator are simulated by using a harmonic analysis of ANSYS. The optimal structural design is achieved and verified by using a laser Doppler vibrometer (LDV) and via impedance measurements made using modal sensors on the outer border of a stator with a network analyzer. The performance test results show that the prototype USM based on a flexural resonant mode of (m=1, n=4) achieves a maximum torque of 22.6 mN-m and the maximum revolution speed of 334 rpm. The validity of the presented numerical approach has thus been verified.

Hiroshi Koibuchi^{1}
, Chrystelle Bernard^{2}
, Jean-Marc Chenal^{3}
, Gildas Diguet^{4}
, Gael Sebald^{5}
, Jean-Yves Cavaille^{6}
, Toshiyuki Takagi^{7}
, Laurent Chazeau^{8}

^{1}National Institute of Technology, Industrial Engineering, Japan

^{2}Tohoku University, , Japan

^{3}Universite de Lyon Batiment B. Pascal, , France

^{4}Tohoku University, , Japan

^{5}Tohoku University, , Japan

^{6}Tohoku University, , Japan

^{7}Tohoku University,, , Japan

^{8}Universite de Lyon Batiment B. Pascal, , France

A mathematical modeling, the Finsler geometry (FG) technique, is applied to study rubber elasticity. The existing experimental data of stress-strain diagrams, which are highly non-linear, are numerically reproduced. The rubber elasticity is called entropy elasticity and still insufficiently understood from the microscopic perspective. In the FG modeling for studying the mechanical property of materials, we modify the geometric structure inside the material simply by replacing the Euclidean metric with Finsler metric, and as a consequence an interaction energy between the directional and positional degrees of freedom of polymers is effectively or implicitly introduced in the Hamiltonian (2014, Physica A ${\bf 393}$, 37-50). It must be emphasized that the interaction is automatically introduced in this modeling. Therefore, this technique is new and completely different from the ordinary modeling techniques, in which an interaction energy must be introduced explicitly in the Hamiltonian for the statistical mechanical treatment. Most interestingly in the FG modeling, several different phenomena can be identified in the sense that the models share the same mathematical structure. Indeed, the FG moeling technique was already successfully applied to several interesting phenomena: The so-called soft elasticity as a response to external mechanical forces, the elongations under the temperature change, and those by external electric fields observed in the liquid crystal elastomers are analyzed by 3D FG model (2017 Polymer ${\bf 114}$, 335-369). It was also shown that the J-shaped stress-strain curve of biological materials such as animal skin and muscle is calculated by 2D FG model (2017 Phys. Rev. E ${\bf 95}$, 042411(1-11)). Moreover, the origin of the line tension energy, which plays an important role in the model of two-component membranes, is explained by a 2D FG model (2016 Polymers ${\bf 8}$, 284(1-18); 2017 Axioms ${\bf 6}$, 10).

Peter Weisenpacher^{1}
, Jan Glasa^{2}
, Lukas Valasek^{3}

^{1}Institute of informatics, Slovak Academy of Sciences, Department of Parallel Computational Methods and Algorithms, Slovakia (Slovak Republic)

^{2}Institute of Informatics, Slovak Academy of Sciences, , Slovakia

^{3}Institute of Informatics, Slovak Academy of Sciences, , Slovakia

Fires in road tunnels are extremely destructive events destroying irreversibly not only tunnel facilities but also concrete tunnel walls. Therefore it is almost impossible to perform full-scale fire tests of realistic fire scenarios in particular tunnels in operation. From these reasons, specific testing aerosol is used for road tunnel ventilation tests to represent steady-state movement of fire smoke downstream of the fire and to avoid the damage of the tunnel. The testing aerosol is generated by a chemical reaction of 50 times less heat release rate (HRR) than is the HRR of modelled fire; however, it is assumed that the optical density of aerosol in a sufficiently long distance from the modelled fire source corresponds to the optical density of the modelled fire smoke. This study investigates the spread of aerosol in a 240 m long part of a road tunnel by computer simulation using the well-known FDS simulation system and compares it with the fire smoke spread. Conditions under which optical density for both cases is similar are determined, mainly the mass flux of aerosol needed to represent the smoke produced by fire and the distance at which steady-state flow of smoke is formed. The influence of the fire HRR and air flow velocity on the determined aerosol mass flux is studied as well. The determined value of mass flux increases with flow velocity and decreases with fire HRR. The results indicate that the aerosol can be used to represent the optical density of steady-state fire smoke spread downstream of the fire in tunnel ventilation tests.

Olga Trofimova^{1}
, Andrey Kovtanyuk^{2}
, Igor Prokhorov^{3}

^{1}FEFU, Informatics, mathematical and computer modeling, Russia

^{2}Far Eastern Federal University, , Russia

^{3}Far Eastern Federal University, , Russia

Mathematical modeling of passing the laser radiation through biological tissue is important for estimating the effectiveness of laser thermotherapy of tumors. It is important to determine the optimal parameters of laser irradiation to provide a sufficient temperature in the tumor to damage it, and to minimize damage to surrounding tissue. The mathematical model of passing the laser radiation through biological tissue bounded by domain $G$ includes the radiation and the heat equations: $$ \dfrac{1}{c}\frac{\partial I(r, \omega, t) }{\partial t} + \omega \cdot \nabla I(r, \omega, t) +(\mu_a+\mu_s)I(r, \omega, t) = \frac{\mu_s}{4 \pi} \int\limits_{\Omega} P(r, \omega \cdot \omega', t)I(r, \omega', t)d\omega',$$ $$\rho C_p \frac{\partial T }{\partial t} - \nabla \cdot (k\nabla T ) = \beta (T_b - T ) + \frac{\mu_a}{4 \pi} \int\limits_{\Omega}I(r, \omega, t)d\omega.$$ Here, $I(r, \omega)$ is the intensity of radiation at the point $r \in G$ and in the direction $\omega \in \Omega$, $\Omega$ is the unit sphere, $T$ is the temperature, $\mu_a$ the absorbtion coefficient, $\mu_s$ the scattering coefficients, $c$ is the speed of light in the tissue, $P(r, \omega \cdot \omega')$ the phase function of scattering, $\rho$ the tissue density, $C_p$ the specific heat capacity, $k$ is the termal conductivity, $\beta$ the blood perfusion coefficient, and $T_b$ the blood temperature. At the boundary $\Gamma$ of the domain $G$, we set the following boundary conditions: $$I(r,\omega, t)= h(r,\omega, t), \;\; r \in \Gamma, \;\; \omega \cdot n < 0, \;\;\;\; k \partial_n T + \gamma(T_c - T)|_{\Gamma} = 0, $$ where $n$ is the outward normal, $\partial_n$ is the normal derivative, $\gamma$, $T_c$, and $h$ are given functions. The initial condition is given by $$I(r, \omega, 0) = 0,$$ $$T(x, 0)=T_0.$$ In the current work, an iterative algorithm to find a solution of the initial-boundary value problem is constructed and implemented. The numerical experiments simulated the pulse laser irradiation of the skin are conducted. The temperature fields at different moments of time are determined.

Darya Gurina^{1}

^{1}G.A. Krestov Institute of Solution Chemistry of RAS, -, Russia

The p-coumaric acid (pCA) is a representative of a wide class of phenolic compounds, namely, hydroxycinnamic acids. pCA exhibits such useful activities as antioxidant, anti-inflammatory, antiplatelet, anti-melanogenic and so on. Due to its high physiological activity p-coumaric acid has a great potential for biomedical and industrial applications. It was shown that pCA can be extracted from plant sources by means of supercritical (SC) fluid extraction process [1] with CO2 as a solvent. Solubility of pCA in SC CO2 is very low that is why using small amount of a polar cosolvent (such as alcohol or acetone) is a common practice. The reasons for increasing the solubility of bioactive molecules in modified SC CO2 are probably related to intermolecular interactions between the solute and the cosolvent. In order to study in details intermolecular interactions in such systems the classical and ab initio molecular dynamics simulation of triple mixtures (pCA-methanol-SC CO2) has been performed. Firstly classical molecular dynamics (MD) simulation was carried out with Gromacs-5.0.7 software package (GPU-accelerated). NVT cubic box with periodic boundary condition contains 10342 CO2 molecules, 320 methanol molecules, and 1 pCA molecule. Constant temperature 328 K was maintained by Nose-Hoover thermostat, box length was 10.0045 nm which corresponded to a system density of 772 kg/m3 and experimental pressure of 21.3 MPa. After the equilibration, a production time was 5 ns with a time step 0.1 fs and collecting time each 0.1 ps. Statistical analysis of obtained data allowed defining the composition of the pCA first solvation shell. Preferential solvation of pCA by methanol molecules was observed. From 3 to 6 methanol molecules can be found around pCA simultaneously. One of them forms a stable hydrogen bond (HB) with H atom of a hydroxyl group of pCA and two molecules form HBs with H atom of –COOH group and carbonyl oxygen. Based on the classical MD data we have chosen an initial configuration of the first solvation shell for further simulation by ab initio molecular dynamics. Car-Parrinello MD simulation was carried out with help of CPMD-3.13.2 program in NVT-ensemble (1 solute, 6 molecule of methanol, 50 CO2 molecules, T=328 K). BLYP exchange-correlation functional and ultrasoft Vanderbilt pseudopotential with plane wave basis set were used. The production time was 20 ps with time step 0.121 fs. Structural properties, namely atom-atom radial distribution functions, coordination numbers, geometric parameters of solute-cosolvent hydrogen bonds were obtained and analyzed. Vibrational densities of states (VDOS) were calculated from velocity autocorrelation function. The peak positions were compared with the peak positions of IR spectrum presented in literature for pCA in gas phase. The redshift in the frequency of the OH stretching mode for hydroxyl and carboxyl groups of pCA involved in HBs with methanol was observed. 1. Paviani, L. C., Saito, E., Dariva, C., Marcucci, M. C., Sánchez-Camargo, A. P., & Cabral, F. A. (2012). Supercritical CO2 extraction of raw propolis and its dry ethanolic extract. Brazilian Journal of Chemical Engineering, 29(2), 243-251.

The reported study was funded by RFBR according to the research project No.16-33-60060_mol_a_dk

Elena Koryukina^{1}

^{1}National Research Tomsk State University, Plasma Physics Department, Russian Federation

A study of the influence of electric fields on the emission spectra of atoms is a topical problem of modern theoretical and experimental physics. Electric fields lead to changes in all spectral characteristics of atoms, and an analysis of these characteristics enables us to understand processes taking place in emitting media. In this work, the software package StarkD for calculating the emission spectra of atoms in an alternating circularly polarized electric field is presented. The emission spectra are calculated by the numerical method based on diagonalization of the energy matrix of an atom in an electric field. This method is free from limitations of perturbation theory and valid in wide ranges of the electric field frequency and strength. The package StarkD allows us to calculate the positions of the Stark components of atomic spectral lines as well as the transition probabilities, lifetimes, line intensities, and profiles of spectral lines of atoms in the electric field. The package capabilities are demonstrated by the example of calculating the emission spectra of He, Ne, Ar, and Kr atoms in the electric field. The calculations have allowed us to establish a number of interesting theoretical regularities and to analyze mechanisms of the formation of atomic spectra in the electric field. The package StarkD can be used in solving problems of gas discharge physics, low-temperature plasma, laser physics, and in technologies of creation of new radiation sources.

Leonid Kagan^{1}
, Peter Gordon^{2}
, Gregory Sivashinsky^{3}

^{1}Tel Aviv University , Exact Sciences, Israel

^{2}Kent State University, Department of Mathematical Sciences, United States

^{3}Tel Aviv University, Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Israel

Leonid Kagan$^1$, Peter Gordon$^2$, Gregory Sivashinsky$^1$ $^1$ Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel.\\ $^2$ Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA. Successful numerical simulations of deflagration-to-detonation transition began fifteen years ago. Since that time many theoretical problems of the transition have been solved. However, numerical studies per se suffer from a serious unsolved issue relating to resolution. It is generally agreed that a finer grid yields more accurate results. Yet, a finer grid demands a more powerful computational facility with larger internal memory as well as CPU-time. The question is how fine the grid should be for the results to be reliable. The single way to answer this question is to conduct resolution tests. Generally, one repeats simulations with a grid twice finer than the original one. If the results obtained differ less than 2-3% one assumes that the resolution is acceptable. This approach however does not guarantee the convergence of the numerical scheme employed. The present study is concerned with the following practical points: (i) how to prove the convergence of the solution (ii) how to estimate the resolution required, (iii) why the degree of convergence differs from the degree of approximation.

Igor Derevich^{1}
, Anastasiya Panova^{2}

^{1}Moscow State Technical University by N.E. Bauman (BMSTU), Faculty of Fundamental Sciences, Department of Applied Mathematics, Russia

^{2}Moscow State Technical University by N.E. Bauman (BMSTU), Faculty of Fundamental Sciences, Department of Applied Mathematics, Russia

Influence of random migration on population growth of biological system I.V. Derevich, A.A. Panova Faculty of Fundamental Sciences, Department of Applied Mathematics, Moscow State Technical University by N.E. Bauman (MSTU), 2-ya Baumanskaya Str. 5 (main Building), Moscow, 105005, Russian Federation A new approach is proposed to analyze the effect of random noise on biological systems with explosive behavior. The growth of the population of biological object, for example, a viral infection as a result of migration to an region with favorable conditions for reproduction of the virus, is considered. In this case, the initial concentration of the virus in the considered region can be equal to zero. The rate of reproduction of the population is proportional to the square of the concentration, and the rate of extinction is proportional to the concentration of the population. The diffusion of microorganisms from the environment takes place in this region. On the boundary of the region we establish condition for mass transfer. As a result of averaging over the volume of the restricted region, we obtain an equation for the average concentration of microobjects within the region. In the obtained equation migration flux is proportional to the concentration difference in the region and environment. We obtained an analytical solution of the concentration equation, in which we taking into account the migration of microorganisms across the boundary of the region. It is shown that there is a critical concentration beyond the boundary of the region, exceeding which causes an explosive growth in the concentration inside the region. In real biological systems there is no complete determinism in their behavior. There are always fluctuations in the parameters of vital activity of organisms, or random movement of the microobjects. The effect of fluctuations in the concentration of microorganisms in the environment is investigated as a result of two approaches. In the first case we carry out an analytical study based on the probability density function of the concentration of microorganisms in the restricted region. On the basis of the method of functional differentiation a closed equation for the density distribution function of the probability of the concentration inside the region is obtained. On the basis of a closed system of equations for the first and second moments the effect of concentration fluctuations outside the region on the dynamics of population growth is studied. The qualitative difference in the behavior of a system with an explosion in a random environment is illustrated. Concentration fluctuations outside the region can lead to explosive growth even if the averaged concentration outside the region is noticeably below the critical level. In the deterministic case, the concentration in this situation is always finite and reaches a stationary value. The second approach to the study of the influence of random fluctuations of concentration outside the region is based on a direct numerical solution of the system of stochastic ordinary differential equations. The numerical algorithm is based on the Euler-Maruyama integration. It is shown that there are actual realizations in which a random concentration fluctuates near a stationary value found from an analytical solution. On the other hand, random trajectories in the concentration space are observed, which lead to an explosive character of population growth. Thus, we illustrate the principal possibility of explosive behavior of a dynamical system in a random environment, when in a deterministic case the explosion is not realized. This work was supported by RFBR project 17-08-00376.

This work was supported by RFBR project 17-08-00376.

Ashraf Zaher^{1}

^{1}American University of Kuwait, Electrical and Computer Engineering, Kuwait

Recently, many continuous-time chaotic systems were synthesized using microcontrollers and FPGAs. This requires applying mathematical discretization to convert integration into recursion. Depending on the approximation algorithms, the speed of the numerical processors, and the number of bits used to represent data, different accuracies and stabilities could be obtained. This article explores the conditions necessary to faithfully generate signals that reflect the true behavior of the chaotic systems, while maintaining the same values for their Lyapunov exponents. This is very important for chaos control and synchronization, especially for applications in secure communication that rely on digital cryptography. The Lorenz system and the Duffing oscillator are investigated to illustrate the effect of having autonomous versus nonautonomous structures. In addition, the Nosé-Hoover dynamical model is investigated to detect the relationship between agility and numerical accuracy. The obtained results prove that identical deterministic chaotic systems can behave differently, for the same set of initial conditions, depending on the discretization algorithm. This added sensitivity requires careful design of the mathematical models required for digital implementations of chaotic systems. The article concludes with useful recommendations for best practices in designing, synthesizing, and implementing digital chaotic systems, while commenting on the best compromise between mathematical complexity and numerical accuracy.

Ashraf Zaher^{1}

^{1}American University of Kuwait, Electrical and Computer Engineering, Kuwait

The design of controllers for nonlinear systems has always been a challenge due to their complex dynamic structures that have ill-defined models with inherent uncertainties. This article investigates the implementation of PID controllers, using a gain scheduling technique that allows dynamic tuning of their gains. The methodology introduced in this paper is model-based, where sub-models for the system to be controlled are created, based on the output, as a scheduling parameter. Dominant dynamics of the nonlinear models are encapsulated into a linearized version that has a dynamic nature to cover the whole range of operation. A pole-placement technique is used to design the PID dynamic gain. The proposed technique is shown to be effectively applied to robotic manipulators, process industry, and automotives.

Mayken Espinoza-Andaluz^{1}
, Martin Andersson^{2}

^{1}Escuela Superior Politécnica del Litoral, Centro de Energías Renovables y Alternativas, Ecuador

^{2}Lund University, Department of Energy Sciences, Sweden

The Poiseuille flow has been widely analyzed from analytical and experimental point of view with different approaches. Although the Lattice Boltzmann method (LBM) has proven to be a suitable tool for mimicking the fluid flow behavior not only for complex but also for simple geometries such as lid driven cavity and channel flow, the required computational size or characteristic length to obtain an acceptable approximation of the described physical phenomena has to be defined. The aim of this study is to evaluate the impact of considering different width-sizes of a channel flow when the velocity profiles are obtained. Channel flow with several width-sizes are evaluated while keeping the Re and aspect ratio as a constant.

This research is financially supported by ESPOL and FIMCP-CERA-05-2017 (Åforsk).

Aleksei Ageev^{1}
, Alexander Osiptsov^{2}

^{1}Institute of Mechanics, Lomonosov Moscow State University, Laboratory of Mechanics of Multiphase Media, Russia

^{2}Institute of Mechanics, Lomonosov Moscow State University, Laboratory of Mechanics of Multiphase Media, Russia

Within the Stokes approximation, a steady-state pressure-driven flow of a viscous fluid in a plane microchannel with one or two striped superhydrophobic walls containing rectangular microcavities partially filled with gas bubbles is studied numerically. The most general case is considered when the cavities on the bottom and top walls of the microchannel are located either symmetrically or asymmetrically with respect to the centerline of the selected flow period. The mathematical problem corresponding to the considered geometry of the flow is solved by the boundary element method. This method makes it possible to reduce the initial differential problem with mixed boundary conditions on the flow boundaries to a system of boundary integral equations and thus to reduce the dimension of the original problem by one. An effective numerical procedure based on the boundary element method is developed to calculate the flow patern in the chosen periodic domain. A parametric study of the flow is performed for a microchannel containing microcavities partially or fully filled with gas bubbles on the bottom and/or top channel walls. The value of the effective slip length entering in the averaged Navier slip boundary condition is found using the calculated fluid velocity field. The reduction of the pressure difference in the microchannel due to the presence of the gas bubbles trapped in the microcavities is estimated. The method developed and the results obtained can be used for the opimization of the design of striped superhydrophobic surfaces in microfluidic applications.

The work is supported by RFBR (project No. 17-01-00057).

Diana Rakhmatullina^{1}
, Elvira Akhmetshina^{2}
, Alina Safiullina^{3}

^{1}Kazan Federal University, the Institute of Management, Economics and Finance, Department of Economic Methodology and History, Russia

^{2}Kazan Federal University, the Institute of Management, Economics and Finance, Department of Economic Methodology and History, Russia

^{3}Kazan Federal University, the Institute of Management, Economics and Finance , Department of Economic Methodology and History, Russia

The content and the structure of a personal consumption is an important factor determining the dynamics of economic and social indicators, since consumer expenditure accounts for more than half of total expenditures and determines the amount of investment into the human capital that, in the conditions of a postindustrial society, plays a decisive role in shaping sustainable economic growth rates and realizing a socially oriented national development strategy. The article analyses the peculiarities of the personal consumption process in the conditions of the modern welfare economy and its determinants, there is made an attempt to systematize and identify the interrelations between them, which will help in developing an objective forecast of GDP dynamics, on the basis of which state programs for social and economic development are formed.

Nail Khabeev^{1}

^{1}Bahrain university, mathematics department, Bahrain

In acoustics and vibration theory the basic equation are usually linearized. The condition of existence of a nontrivial solution of the linear equations leads to the characteristic transcendental equation with respect to H. Here H is the complex number imaginary part of which is natural frequency of oscillations of the system. Real part is describing the rate of damping of oscillations or instability (in case if real part is positive). The structure of roots of such equations is studied by using argument principle for some specific applied problem from fluid mechanics. It is shown that dispersion equation has two complex-conjugate roots and infinite number of real roots. All roots lie in the left complex half-plane, providing damping of oscillations.

Vitaly Borisov^{1}
, Anton Ivanov^{2}
, Boris Kritskiy^{3}
, Igor Menshov^{4}
, Mukamay Ramazanov^{5}
, Evgeny Savenkov^{6}

^{1}Keldysh Institute of Applied Mathematics (Russian Academy of Sciences), Numerical methods and mathematical modelling, Russia

^{2}KIAM RAS, 3, Russia

^{3}Keldysh Institute for Applied Mathematcis RAS, , Russia

^{4} Keldysh Institute for Applied Mathematics , Russian Academy of Sciences , Russia

^{5}Keldysh Institute for Applied Mathematics RAS, Numerical methods and mathematical modelling, Russia

^{6}Keldysh Institute of Applied Mathematcis RAS, Computational methods and mathematical modelling, Russia

In this work we present a mathematical model and simulation technique for the analysis of hydraulic fracture propagation (HF) in fully 3D setting and accounting for all major flow and geomechanical effects. The approach can be used for hydraulic fracture design as well as for analysis of such processes as production/injection stress re-orientation, spontaneous fracture growth during reservoir development etc. The reservoir is described by 3D Biot poroelastic model which couples flow and mechanical processes in medium. The fracture is described as an arbitrary sufficiently smooth surface with boundary. The flow in fracture is described by 2D Reynolds lubrication equation. Only simple Newtonian flow model will be considered but generalizations to more realistic rheological models are straightforward. These (sub)models (for the fracture and for the reservoir) are coupled by appropriate interface conditions which ensure continuity of mass and momentum fluxes at the fracture/reservoir interface. The complete model is fully coupled and self consistent. The numerical approach is based on finite elements method. To account for discontinuous displacement and pressure fields an eXtended Finite Element (X-FEM) method is used [1]. In contrast to most X-FEM implementations we do not use level set approach for fracture representation and evolution [2,3]. Instead the so called closest point projection method is utilized for both solution of the fracture flow Reynolds lubrication equations [4] and fracture geometry representation and evolution in the X-FEM context. The vector version of J-integral for poroelastic medium was used for describing pointwise direction field of fracture evolution. The unified way of HF representation provides flexibility and efficiency of the complete coupled algorithm. The complete system of equations is solved in a partitioned way which provides flexibility for applications. The applicability of developed techniques is demonstrated by a number of simulation results. The research was supported by RSF (project No. 15-11-00021). REFERENCES [1] Moës, N., Dolbow, J., Belytschko, T. A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Engng. 1999. 46 (1): 131–150. [2] Moes N., Gravouil A., Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets—Part I: Mechanical model. Int. J. Numer. Meth. Engng. 2002.; 53:2549–2568. [3] Gravouil A., Moes N., Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets—Part II: Level set update. Int. J. Numer. Meth. Engng. 2002. 53:2569–2586. [4] Ruuth, S.J., Merriman, B. A simple embedding me thod for solving partial differential equations on surfaces. J. Comput. Phys., 2008. 227(3):1943–1961

The research was supported by RSF (project No. 15-11-00021).

Yohei Nishidate^{1}

^{1}University of Aizu, Computer Science and Engineering, Japan

Electroluminescence (EL) spectra from light-emitting diode (LED) with vertical stack of two sets of quantum wells (QWs) for blue and green emission and mesh-like top electrode revealed equal peak intensities of blue and green emission lines and shallow trough between them. These spectral peculiarities were associated with the influence of mesh-like electrode. Spatially nonuniform potential distribution created by the patterned electrode along the QW-plane was assumed to result in spatially nonuniform compensation of quantum-confined Stark effect (QCSE) which manifests itself as spatially-dependent EL blue-shift. In this paper we verify the assumption of spatially nonuniform compensation of QCSE by performing analysis of the EL-spectrum of LED with blue-green emission by its decomposition. We applied spectrum decomposition procedure in two steps. For this purpose EL-spectrum is presented as a set of sampling points ($\lambda_i$, $g_i$), where $\lambda_i$ and $g_i$ are the wavelength and the intensity of i-th sampling point. Then the EL-spectrum is assumed to be a sum of N Gaussian functions. To find the coefficients for all Gaussian functions we need to minimize the total residual by solving a system of nonlinear equations for these coefficients. After first decomposition the Gaussian functions for blue and green emission lines were obtained with FWHM 23.5 and 40.6 nm, respectively. At the second step we expanded each of blue and green Gaussian functions independently into sets of Gaussian function with smaller FWHM. The refined spectra of blue and green emission lines can be interpreted as a manifestation of spatially-dependent compensation of QCSE resulted in position dependent blue shift of EL.

Valeriy Ryabov^{1}

^{1}NRC Kurchatov Institute, , Russia

The theory of continuum and statistical mechanics in terms of molecular variables “coordinate-impulse-force” instead of tensor variables “stress-strain”is developed. The concept is based on the representation of classical mechanics in T6 manifold. Six true topological dimensions of the torus surface are represented by six scalars: three relative strains and three shear angles. The separation of spaces, which address the deformation and the coordinates of the particles greatly simplifies the whole computational basis of continuum and statistical mechanics. Instead of the stress-strain ratio, the new governing equations contain the dependence of the generalized tension forces acting on each atom (or small mass element) on the scalar deformation variables. The notions of surface and surface forces can be excluded from statistical and continuous mechanics within new concept. This concept gives rise to a new type of ensemble with a constant tension force NfE. A much simpler principle of virtual work provides a number of advantages over the widespread isostress ensemble NtE in molecular dynamics. In addition, the actual equality of the internal tension forces to the traction ones allows a fully atomistic interpretation of the boundary value problems. In the thermodynamic limit, instead of pressure and volume as state variables, this ensemble uses the tension forces measured in energy units and the stretch ratios. Thus, the changes might be also applied to the formulas in the textbooks. The governing equations of nonlinear elasticity theory for inhomogeneous medium are formulated. Unlike the standard theory, it is not necessary of the compatibility conditions or Saint-Venant principle for solving boundary value problems. The new method offers a completely different strategy in continual mechanic’s computations. The usual calculation algorithms assume the use of the finite element method for constitutive equations. In contrast, the discretization scheme in scalar deformation variables is analogous to the optimization procedure applied in molecular dynamics. This means that the continuum mechanics ceases to be an independent brunch of theoretical physics, becoming a part of many particle mechanics with an effective interaction depending on the deformation variables in the extended space. Several key examples illustrate the implementation of the new theory in calculations of statics and dynamics and thermodynamics of deformed solids. Referencies 1. V.A. Ryabov, Mechanics of deformations in terms of scalar variables. Cont. Mech. Thermodyn. Springer, 29, 3, 715 (2017). 2. V.A. Ryabov, Implementation of isotension ensemble in molecular dynamics. Comp. Meth. Appl. Mech. and Eng., Elsevier, accepted for publication (2018).

Roman Rezaev^{1}
, Katerina Smirnova, A.S. Gogolev, S.G. Chistyakov, N.A. Filatov, R. Rezaev^{2}

^{1}TPU, Physics and Technology, Russia

^{2}Tomsk polytechnic university, Physics and Technology, Russia

The uncertainty quantification technique – forward uncertainty propagation based on stratified sampling is applied to find the probability density functions of the noise standard deviations of the refraction angle $σ_{(α,T)}$ and $σ_{(α,TL)}$ in X-ray differential phase contrast imaging systems are derived. Here subscripts $T$ and $TL$ correspond to Talbot (one source of radiation with final focal spot is considered) and Talbot-Lau (absorbing grating forms an array of sources) type of interferometers and the expressions for the $σ_{(α,T)}$ and $σ_{(α,TL)}$ read as
$$σ_(α,T)=(γ_1 (k_2,τ_2)γ_2 (p_1,p_2,σ_s))/(4a√2),$$
$$σ_(α,TL)=(γ_1 (k_0,τ_0)γ_1 (k_2,τ_2)γ_3 (p_0,p_2))/(4a√(2I_0 )),$$
where
$$γ_1 (k,τ)=√(τ+k(1-τ))/(k(1-τ) sinc(k)),$$
$$γ_2 (p_1,p_2,σ_s )=(p_2^2)/(p_2-p_{11} ) (exp(2〖(π σ_s (p_2-p_{11})/(p_2 p_{11} ))〗^2))/√(ϱ σ_s ),$$ $$ p_{11}=p_1⁄η$$
$$γ_3 (p_0,p_2 )=p_0+p_2,$$
$p_i$- the grating period, $k_i$ - the duty cycle, $τ_i$ - transmission factor, $η={1,2}$ - phase gating type (π/2 shift phase grating, π- shift phase grating) are the parameters of the grating $G_i,i=0,1,2,$ $a$ – a pixel size of the detector, $ϱ$ – a dimension coefficient, $I_0$ – the intensity of the radiation, which is proportional to the source size, $σ_s$ is the standard deviation of the source size.
The parameters of geometrical and material characteristics of interferometers’ components are assumed to be Gaussian-like distributed over the corresponding range of values that results in the change of the design working flow to reach the high-performance grating-based interferometer [1]. In particular, the optimal values of geometrical parameters determined in [1, 2] are shifted by more than 10% as compared to the deterministic approach.
**Reference**

[1] P Modregger, BR Pinzer, T Thüring, S Rutishauser, C David, and M Stampanoni. Sensitivity of X-ray grating interferometry. Optics Express, 19(19):18324–18338, 2011.

[2] W Yashiro, Y Takeda, and A Momose. Efficiency of capturing a phase image using cone-beam x-ray Talbot interferometry. Journal of the Optical Society of America. A, Optics, image science, and vision, 25(8):2025–2039, 2008

This study was supported by the Federal Targeted Program of the Ministry of Education and Science of the Russian Federation agreement no. 14.578.21.0198 (RFMEFI57816X0198).

Moses Gasparyan^{1}

^{1}Moscow Institute of Physics and Technology, General and Applied Physics, Russia

Separation of a binary mixture of gases in a modern version of Knudsen compressor is studied. The device that operates at rarefied flow regime consists of 18 narrow tubes connected with a large tube is shown in figure 1 and 2 together with a graphic of temperature distribution along the tube walls. The numerical simulation of the flow is made by finite-difference solution of a system of 2 Boltzmann kinetic equations with application of the conservative projection method for evaluation of the collision integrals. The special discretisation is made with the use of GMSH software. Parallel implementation of the code uses MIP technology. The main features of the gas separation process are obtained and presented in graphics. The computations are performed at “Complex of modelling and data processing of mega-class research facilities” of NIC Kurchatov institute.

The work was supported by RFBR (18-08-00523, 18-08-00291, 18-08-00132, 18-08-00495, 18-08-00192)

Anatoly Korolev^{1}
, Boris Sorokin^{2}

^{1}Lomonosov Moscow State University, Faculty of Physics, Photonics and Microwave Physics, Russia

^{2}Lomonosov Moscow State University, Faculty of Physics, Photonics and Microwave Physics, Russia

Fast evolution and implementation of new mobile communication standards require more accurate estimations of electromagnetic field strength for the purpose of checking electromagnetic compatibility of different systems. This paper discusses result of application of the method based on Kirchhoff integral for estimation of the electromagnetic field scattered by Earth’s surface. Reasons of higher accuracy of this method in comparison with geometrical optics are discussed.

Georgiy Nujdov^{1}
, Valeriy Krivorotov^{2}
, Sirojiddin Mirzaev^{3}

^{1}The Institute of ion-plasma and laser technologies, Thermophysics of multiphase states, Uzbekistan

^{2}The Institute of ion-plasma and laser technologies, Thermophysics of multiphase states, Uzbekistan

^{3}The Institute of ion-plasma and laser technologies, Thermophysics of multiphase states, Uzbekistan

The results of quantum-chemical calculations of ionic motion in the LaF3 lattice are presented. The dependence of the values of these parameters on the number of the nearest defects such as the site vacancy-interstitial ion is established. The dynamics of the energy parameters during the transition of a nanolattice from a dielectric state to a superionic state is shown.

Yuriy Dimitrienko^{1}
, Elena Gubareva^{2}

^{1}Bauman Moscow State University, Computational Mathematics and Mathematical Physics, Russia

^{2}BAUMAN MOSCOW STATE TECHNICAL UNIVERSITY (National research university of technology), Department of computational mathematics and mathematical physics, Russia

The paper proposes a new model of mathematical knowledge representation based on the hierarchical neural network structure of the links between various elements of knowledge. The concept of the level of knowledge and the minimum element of knowledge- «the quantum of knowledge» are introduced. On the base of the developed model, the structuring of mathematical knowledge corresponding to the basic mathematical training of engineers was carried out. The information and educational environment Nomotex was developed for mathematical training of engineers based on the proposed model of knowledge representation and a unique collection of 3D animation of all basic mathematical knowledge, as well as examples of the use of knowledge in engineering. The information and educational environment allows implementing individual educational trajectories in which the educational process is automatically formed from the neural network structure of knowledge according to the specified output competences of the engineer. A specialized software for visualizing educational trajectories was developed. Informational and educational environment allows to realize the multilevel and continuity of mathematical preparation, taking into account the nature of the activity of engineers in their mathematical preparation, taking into account the specifics of engineering training in mathematical courses. The information and educational environment allows implementing distance education, creating online courses, video lectures and automated testing. The introduction of the information and educational environment provides an increase in the attractiveness of the learning process through the extensive use of 3D visualization and animation, as well as computer game elements in the process of mathematical learning. Information and educational environment NOMOTEX, provides the possibility of implementing a systematic approach in the study of mathematical knowledge. The results of the experimental implementation of the information and educational environment in the educational process of engineers at the Bauman MSTU are presented.

Ioan Bica^{1}

^{1}West University of Timisoara, Faculty of Physics, Romania

Magnetorheological bio-suspensions (MRSBSs)-based membranes are fabricated using a microfibre cloth soaked with a mixture of honey, turmeric powder (TP) and carbonyl iron (CI) microparticles. Plane capacitors are manufactured by introducing the membranes between two parallel Copper plates. The electrical conductivity and the relative dielectric permittivity of the membranes are measured in a static magnetic field superimposed on an electric field with fixed frequency. The results show that the membranes can be used for fabrication of devices which require a controlled release of the bio-active components found in honey and TP, by fixing the value of magnetic field density.

The work has been performed within the project PN-III-P1-1.2-PCCDI-2017-0871 (CNDI-UEFISCDI) and the collaboration protocol between JINR (Russia) and West University of Timisoara (Romania).

Nikolay Evstigneev^{1}

^{1}Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Institute for System Analysis, Russia

We wish to solve the problem of linear stability analysis of stationary solutions for a fluid dynamics problem governed by Navier Stokes equations. Let $\mathtt{A}$ be a linear operator of discrete approximation of the Navier-Stokes equations linearised on some stationary solution. The discrete problem is of order $\mathcal{O}(10^6)-\mathcal{O}(10^7)$ so we use parallel computations to perform linear analysis. We aim at the heterogeneous computational architecture based on central (CPU) and graphical (GPU) processing units. There are many methods exist that recover the desired part of the spectrum, e.g. power method, inverse power method, Jacobi-Davidson methods, Krylov methods, methods based on Newton iterations. However, there are no implementations of these methods aimed at parallel heterogeneous architecture that take into account parallel computational difficulties. In the papaer we use the implicitly restarted Arnoldi method (IRAM) to solve the eigenvalue problem of recovering the most dangerous eigenvalues, i.e. eigenvalues closest to the imaginary axis. The IRAM has poor convergence without specific transformation of the matrix spectrum, e.g. exact matrix exponential $\exp(\mathtt{A})$, shift-invert $(\mathtt{A}-\sigma \mathtt{E})^{-1}$, Cayley transformation $(\mathtt{A}-\sigma_1 \mathtt{E})^{-1}(\mathtt{A}-\sigma_2 \mathtt{E})$, inverse Kronecker sum $2(\mathtt{A} \otimes \mathtt{E}+\mathtt{E} \otimes \mathtt{A})^{-1}$, where $\mathtt{E}$ is an identity matrix and $\sigma_j$ are real valued parameters. All these methods require either long computations (matrix exponential) or inverse of the modified matrix. This can be troublesome in fluid dynamics problems where matrix is not available explicitly (only matrix vector product is available). In addition the matrix is non-symmetric, ill-conditioned and can be full (for spectral approximation). This is a difficult problem to solve, especially for heterogeneous architecture with multiple GPUs. We propose the following inexact matrix exponential transformation that can perform shift-inverse using only matrix-vector product: $$\left( \left( \mathtt{E}+\mu \mathtt{A} \right)^m - \sigma \mathtt{E} \right) \mathbf{u}^{n+1}=\mathbf{u}^{n},$$ where $\mathbf{u}$ are Krylov space basis vectors used in the IRAM. Here $\mu$ is the compression parameter, $m \sim \mathcal{O}(1)$, $\sigma$ is the shift and $n$ is the number of the Krylov basis vector. The left part of the system is freely available as a time stepper part of the Navier Stokes solver code. The inverse is performed using Krylov methods that require only matrix-vector product. We prove convergence bounds for the GMRES method that is used to recover $\mathbf{u}^{n+1}$ in the Arnoldi process and show that this type of matrix transformation greatly accelerates GMRES convergence. We demonstrate application of the method in recovering eigenvalues close to the imaginary axis in some model problems on the following matrices from the Matrix Market: GT01R, rdb450 and S80PI_n1. We also demonstrate the application of the method for fluid dynamics on incompressible (Kolmogorov 3D flow, Rayleigh-Benard convection) and compressible (Kelvin-Helmholtz instability) flow problems. We demonstrate convergence, acceleration and efficiency on the multiple GPU cluster.

Hrvoje Dodig^{1}

^{1}University of Split, Faculty of Maritime Studies, Department of naval electronic and information technology, Croatia

In this paper, we derive the expression for magnetic field of the point charge moving along the straight line directly from Coulomb's force law using the previously unknown mathematical equation that we have recently derived. This mathematical equation shows that the gradient of electrostatic potential function (Coulomb's law) can be expressed as the curl of vector function (vector potential). From this mathematical equation, it is straightforward to derive the expression for magnetic field of the moving point charge directly from Coulomb's law. Furthermore, Coulomb's law and the Newtonian gravity have the same mathematical form up to the constant, therefore the same derivation is carried out from Newton's gravity law. Therefore, we mathematically prove that moving mass produces gravitomagnetic field, the mathematical form of which is exactly the same as the mathematical form of magnetic field of the moving point charge. Furthermore, we theorize that two masses moving with high velocities interact in the same way as predicted by Lorentz force law.

Maria Begoña García-Ramiro^{1}
, J Arrue^{2}
, M Illarramendi^{3}
, Felipe Jimenez^{4}
, Joseba Zubia^{5}

^{1}Faculty of Engineering in Bilbao (University of the Basque Country UPV/EHU), Applied Mathematics, Spain

^{2}University of the Basque Country (UPV/EHU), , Spain

^{3}University of the Basque Country (UPV/EHU), , Spain

^{4}University of the Basque Country (UPV/EHU), , Spain

^{5}University of the Basque Country (UPV/EHU), , Spain

Abstract: By means of a mathematical model, we carry out a theoretical study of light emission in poly (methyl methacrylate) polymer optical fibers (PMMA POFs) doped with a typical coordination complex, or chelate, of Eu3+. Our mathematical and compu-tational results are intended to provide an insight into the influence of some important design parameters in the field of europium-chelate-doped POF lasers and amplifiers. The main interest of this type of dopant, as compared to other ones proposed for active POFs, such as organic dyes, is that it serves to avoid photodegradation effects. In this respect, highly photostable and strongly luminescent europium-chelate-doped POFs have been proposed as new probes for optical sensing of temperature, and also as solar collectors. In addition, the use of light-emitting POFs enables the achievement of low-coherence, speckle-free light, which is advantageous for optical probes in Medicine, for the manufacture of fiber gyroscopes, etc. Moreover, the emission of Eu3+ in the red re-gion of the spectrum coincides with one of the attenuation windows of typical POFs, thus enabling the design of optical amplifiers for these fibers. One of the reasons why europium ions alone are not employed as dopant is that europi-um is not directly soluble in PMMA. Besides, the absorption cross section of these ions would be too low for the manufacture of compact devices. Both problems are solved if each ion is attached to a chelate, which contains a europium ion and various absorbing ligands, as is the case of the analysed dopant. The set of relevant energy levels and en-ergy transitions of all these chelates are depicted in Fig. 1. We have modelled the system by means of four rate equations that describe the tem-poral and spatial evolution of the generated power, of the pump power and of the num-ber of molecules per unit volume in the excited state corresponding to the europium ion and to the set of ligands. One of the parameters analysed has been the optimum fiber length and its dependence on the pump power. For this purpose, the rate equations have been solved in stationary state by using the fourth-order Runge-Kutta method. Addi-tionally, we have derived a simplified analytical expression that serves to model the evo-lution of the generated power as it propagates while the pump power is still non-negligible. The attenuation of the generated light in the green region is also taken into account, and its effect is studied as well. We also show why and under what conditions the optimum length for maximum generated power is about 10 times shorter when the concentration is 10 times higher, i.e. the factor by which the concentration is multiplied is similar to the factor by which the optimum length is divided.

Carlos Frajuca^{1}

^{1}IFSP, Mechanics, Brazil

An experiment to measure the speed of gravity is being planned. For this purpose, a numerical method was developed for the optimization of a composite quadrupole mass at very high-speed rotation. The calculation aims to optimize a quadrupole mass which must generate a periodic gravitational signal in a very high frequency (3200 Hz) with the maximum amplitude, taking into account geometric features and mechanical properties of the component materials. Using the brazilian gravitational wave detector as the signal receiving device, an estimate was obtained for the distance between the emiter and the detector is 100 m. A simplified modeling of the emitter-detector indicates that the gravitational signal amplitude decreases proportional to r^(-5), where r is the emitter-detector distance.

Acknowledgements to CNPq (Brazil).

Carlos Frajuca^{1}

^{1}IFSP, Mechanics, Brazil

The work presents a study focused on a specific type of inertia wheel, known as Flywheel, capable of accumulating kinetic energy and so, can be used as electromechanical battery. Computational analyzes were performed on conjugate materials for the rotor, the materials chosen were steel for the core and carbon fiber for reinforcement. This study aims at analyzing a flywheel at a rotational speed of up to 170,000 rpm, using different proportion of reinforcement material but keeping the outer dimension constant. The analyses was made by Finite Element Method, with the intention of verifying the behavior of the tensile strength of the material at its high speed of rotation. With this data the storaged energy for each case was calculated and the results are shown.

The authors acknowledges CNPq (Brazil).

Yuriy Dimitrienko^{1}
, Elena Gubareva^{2}
, Anna Pichugina^{3}

^{1}Bauman Moscow State University, Computational Mathematics and Mathematical Physics, Russia

^{2}BAUMAN MOSCOW STATE TECHNICAL UNIVERSITY (National research university of technology), Department of computational mathematics and mathematical physics, Russia

^{3}BAUMAN MOSCOW STATE TECHNICAL UNIVERSITY (National research university of technology), Fundamental Sciences, Russia

A new theory of thin multilayer anisotropic elastic shells is proposed, based on the application of the asymptotic analysis over small geometric parameter to the general 3-dimensional equations of the elasticity theory for curvilinear coordinates. In deriving the equations of the shell theory no assumptions are made concerning to the distribution of displacements, deformations, or displacements trough thickness. Recurrent consequences of local problems of the elasticity theory for shells are formulated and an analytical solutions are obtained. The averaged equations of the asymptotic theory of shells that are of the same type as the classical equations of the Kirchhoff-Love theory of shells are derived. It is shown that the asymptotic theory makes it possible to obtain in explicit form the distributions of all 6 components of the stress tensor over the thickness of the shell. An example of the calculation of a cylindrical shell under axisymmetric bending is considered. The case of monoclinic shell layers, which have no more than 13 independent elastic constants, is considered. An algorithm is proposed for obtaining explicit analytical equations for calculating the distribution of components of the total stress tensor over the cylindrical shell. The effect of the geometric dimensions of the shell on the character of the distribution of displacements and stresses in the shell is analyzed.

Evgeniya Shapovalova^{1}
, Julia Shapovalova^{2}
, Svetlana Kurushina^{3}

^{1}Samara National Research University, physics, Russia

^{2}Samara National Research University, Physics, Russia

^{3}Samara National Research University, physics, Russia

We investigate the influence of additive and multiplicative noises on the order-disorder phase transition process. The two-component spatially extended chemical reaction model – brusselator, in which at first multiplicative noise was added, and then additive noise was added, is used for study. Single-site two-dimensional nonlinear self-consistent Fokker-Planck equation in the mean field approximation in Stratonovich interpretation for the system under study was derived in two variants. In the result of numerical simulation in the definite parameter regions, which are different, the intermittency of ordered and disordered phases was discovered. We observed increasing of the ordered and disordered phases frequency intermittency at the receding from the bifurcation point and at the increasing of the noise intensity. We found, that existence duration of ordered phase increases with noise intensity increasing. In our research we showed how disorder originates and grows inside ordered phase under the influence of the additive or multiplicative noise.

JESUS GONZALEZ^{1}
, José Escobar^{2}
, Henry Sánchez^{3}
, Jesús Beltrán^{4}
, Jhon De la Hoz^{5}
, Liliana Selene Martínez Chima 57439768^{6}

^{1}Universidad del Magdalena, Magdalena, Colombia

^{2}Universidad del Magdalena, Facultad de Ciencias Básicas, Colombia

^{3}Universidad del Magdalena, Facultad de Educación, Colombia

^{4}Universidad del Magdalena, Facultad de Ingeniería, Afghanistan

^{5}Universidad Cooperativa de Colombia, Facultad de Educación, Colombia

^{6}Universidad del Magdalena, Magdalena, Colombia

Direct laboratory exercises and observations are a fundamental part of science courses, but the concepts such as distance learning, and open universities, are now becoming more widely used for teaching and learning. However, due to the nature of the subject domain, the use in science and engineering are relatively behind when using online distance learning. We propose a solution developing several advanced virtual laboratories of electromagnetism which is a replica the real lab fully software-based, using a mobile game development with Unity 3D. Unity is a game development ecosystem: a powerful rendering engine fully integrated with a complete set of intuitive tools and rapid workflows to create interactive 3D and 2D content that is considered a tool of artistic expression which joint the art and technology. We argue for this solution since it offers advantages over real labs, which will be elaborated further in this paper.

Universidad del Magdalena (FONCIENCIAS).

Andrey Dmitriev^{1}

^{1}National Research University Higher School of Economics, School of Business Informatics, Russia

The present paper is a continuation of our studies (see Studies in Computational Intelligence, vol. 689, 2018, pp. 390-400). As a result, it has been recognized that the probability density function for a Twitter time series is a q-exponential (Tsallis) distribution. This phenomenon has not received the theoretical explanation. Therefore, the purpose of the presented study is to construct and analyze a model that explains this phenomenon. Our model is based on the statistical theory of hierarchical networks. Based on the generalized Fokker-Planck equation, we established that the evolution of the microblogging network represents anomalous diffusion over hierarchical levels. In the model, hierarchically subordinate users of the social network form an ultrametric space. At the lower levels of the Cayley tree are the followers of the followers of the social network. In the process of diffusion, a stationary q-exponential distribution is established. Thus, the ultrametric diffusion model is the adequate model of the microblogging networks (such as, for example, Twitter).

The work was supported by the Russian Foundation for Basic Research (grant 16-07-01027).

Katerina Smirnova, A.S. Gogolev, S.G. Chistyakov, N.A. Filatov, R. Rezaev^{1}

^{1}Tomsk polytechnic university, Physics and Technology, Russia

The uncertainty quantification technique – forward uncertainty propagation based on stratified sampling is applied to find the probability density functions of the noise standard deviations of the refraction angle $σ_{α,T}$ and $σ_{α,TL}$ in X-ray differential phase contrast imaging systems are derived. Here subscripts $T$ and $TL$ correspond to Talbot (one source of radiation with final focal spot is considered) and Talbot-Lau (absorbing grating forms an array of sources) type of interferometers and the expressions for the $σ_{α,T}$ and $σ_{α,TL}$ read as
$$σ_{α,T}=\frac{γ_1 (k_2,τ_2)γ_2 (p_1,p_2,σ_s)}{4a\sqrt 2},$$
$$σ_{α,TL}=\frac{γ_1 (k_0,τ_0)γ_1 (k_2,τ_2)γ_3 (p_0,p_2)}{4a\sqrt{2I_0 }},$$
where
\begin{align}
& γ_1 (k,τ)=\frac{\sqrt{τ+k(1-τ)}}{k(1-τ) sinc(k)},\\
&γ_2 (p_1,p_2,σ_s )=\frac{p_2^2}{p_2-p_{11} }\frac {\exp(2 (π σ_s \frac{p_2-p_{11}}{p_2 p_{11}})^2)}{\sqrt{ϱ σ_s }}, p_{11}=p_1⁄η,\\
&γ_3 (p_0,p_2 )=p_0+p_2,
\end{align}
$p_i$ - the grating period, $k_i$ - the duty cycle, $τ_i$ - transmission factor, $η={1,2}$ - phase gating type ($π/2$ shift phase grating, $π$- shift phase grating) are theparameters of the grating $G_i, i=0,1,2,$ $a$–a pixel size of the detector, $ϱ$ – a dimension coefficient, $I_0$– the intensity of the radiation, which is proportional to the source size, $σ_s$ is the standard deviation of the source size.
The parameters of geometrical and material characteristics of interferometers’ components are assumed to be Gaussian-like distributed over the corresponding range of values that results in the change of the design working flow to reach the high-performance grating-based interferometer [1]. In particular, the optimal values of geometrical parameters determined in [1, 2] are shifted by more than 10% as compared to the deterministic approach.

**Reference**

[1] P Modregger, BR Pinzer, T Thüring, S Rutishauser, C David, andM Stampanoni. Sensitivity of X-ray grating interferometry.OpticsExpress, 19(19):18324–18338, 2011.

[2] W Yashiro, Y Takeda, and A Momose. Efficiency of capturing a phaseimage using cone-beam x-ray Talbot interferometry. Journal of the OpticalSociety of America. A, Optics, image science, and vision, 25(8):2025–2039, 2008

This study was supported by the Federal Targeted Program of the Ministry of Education and Science of the Russian Federation agreement no. 14.578.21.0198 (RFMEFI57816X0198).

Fatma Sidre Oglakkaya^{1}
, Canan Bozkaya^{2}

^{1}Middle East Technical University, Department of Mathematics, Turkey

^{2}Middle East Technical University, Department of Mathematics, Turkey

This study examines the natural convection flow and the heat transfer of a water-based ferrofluid consisting $\ce{Fe_3O_4}$ nanoparticles in a semi-annulus cavity with a circular outer and a sinusoidal inner wall under the effect of a non-uniform magnetic field produced by multiple nodal magnetic sources. The fluid flow under consideration is two-dimensional, viscous, steady, laminar and incompressible. For the mathematical formulation of the problem both magnetization and electrical conductivity effects on ferrofluid are taken into account and thence the combined principles of magnetohydrodynamics and ferrohydrodnamics are considered. The governing equations are numerically solved using the dual reciprocity boundary element method in which the equations are transformed into integral equations on the boundary by using the fundamental solution of the Laplace equation and treating all other terms as nonhomegeneity through radial basis function approximation. Thus, the resulting discretized system is small in size compared to the ones obtained by domain discretization techniques, and hence the solution is obtained with a less computational cost. The numerical simulations are carried out to investigate the effect of Rayleigh, Hartmann and magnetic numbers, solid volume fraction of nanoparticles and amplitude of sine waves on the flow and heat transfer characteristics. It is observed that adding nanoparticles to the base fluid and increasing the amplitude of sine waves enhance the average Nusselt number, which indicates that the heat transfer inside the enclosure can be controlled by the shape of the computational domain under various combinations of physical parameters.

Arman Kussainov^{1}

^{1}al-Farabi Kazakh National University, Physics and Technology, Kazakhstan

We have written, optimized and tested the C/C++ project designed to simulate an arbitrary 3D radiological phantom in computer tomography experiment. The simulation code includes various filtering techniques, multithreaded by using OpenMP directives, has capability to create the arbitrary shaped and structured radiological phantoms in various testing chambers conditions and configurations including mammography case.

This research was commissioned and funded by the ADANI Company, producing and manufacturing Medical, Security, Spectroscopy and Inspection X-ray Systems, Minsk, Belarus.

Dina Shmeliova^{1}
, Alexandre Zakharov^{2}
, Sergey Pasechnik^{3}

^{1}Moscow Technological University (MIREA), Moscow 119454, Russia , Problem Laboratory of Molecular Acoustics, Russia

^{2}Saint Petersburg Institute for Machine Sciences, The Russian Academy of Sciences, Saint Petersburg 199178, Russia, , Russia

^{3}Moscow Technological University (MIREA), Moscow 119454, Russia, Problem Laboratory of Molecular Acoustics, Russia

The thermally excited vortical flow in a microsized liquid crystal (LC) volume has been investigated theoretically based on the nonlinear extension of the Ericksen-Leslie theory, with accounting the entropy balance equation. Analysis of the numerical results show that due to interaction between the gradients of the director field $\nabla{\hat{\bg n}}$ and temperature $\nabla T$, caused by the focused laser radiation, the thermally excited vortical fluid flow is maintained in the vicinity of the heat source. Calculations have shown that the features of the vortex flow is influenced not only by the power of the laser radiation, but also by the angle at which the radiation is directed.

This work is supported by the Ministry of Education and Science of the Russian Federation (project ID RFMEFI58316X0058)

Olga Ivanova^{1}

^{1}IT Decision CJSC, --, Russia

Free oscillations of a one-dimensional Cosserat continuum model are considered. The model is built on the base of Ilyuishin’s mechanical modeling approach. It consists of a beam supplied by rigid massive inclusions periodically placed along the longitudinal line of the beam. Those inclusions are connected with their nearest neighbors by belt drives. Bending-tension motion of this construction is considered in one plane. Model behavior is elastic, except for the moment action of inclusions on supporting beam elements which is viscoelastic (Kelvin-Voigt model is used). The linearization of the such model motion equations is made. The problem of free oscillations is considered for the linearized model. The fundamental difference between the system of equations for this model and for the one with fully elastic behavior is mentioned. The general solution of the problem of free oscillations is examined on the assumption with the special form of solution. The “antenna type” construction with known stress-strain properties is taken as an example. The computational solution of the problem of free oscillation is obtained for such construction. It is found that for each oscillation mode there exist exactly two forms of motion. The rate of decay became apparent to depend on viscosity value and oscillation mode. Some graphs are given to demonstrate the dependence.

Md. Atiqur Rahman Patoary^{1}
, A K Fazlul Haque^{2}
, M Alfaz Uddin^{3}

^{1}University of Rajshahi , Department of Physics, Bangladesh

^{2}University of Rajshahi, Department of Physics, Bangladesh

^{3}University of Rajshahi, Department of Physics, Bangladesh

We propose an extension of the modified Vaisburd and Evdokimov model (MVE) [Russ. Phys. J. $\textbf{46}$, 1160 (2003)] to investigate the $L1$-, $L2$- and $L3$-subshells as well as total $L$-shell ionization cross-sections of atomic targets with atomic numbers 18 $\leq Z \leq $ 92 for incident energies upto 1 GeV. The present MVE model combines the Lorentz expression, Bethe term and Gryzinski's relativistic factor. We compare our results with the available experimental data as well as with the theoretical results of XMUIBED, XMCN, GKLV, MUIBED, XCVTS, RPWBA , PWBA, SDWBA and SCADW models. Our model describes satisfactorily most of the experimental data and produce more or less reasonable agreement with the other calculations apart from some differences at the high energy regions for some of the elements considered. This analysis reveals that this model can be used to calculate the inner-shell ionisation cross-sections for any element at any energy in the above range. This simple model may be used for applications in science, technologies and industries where first generation of reasonable cross-sections is needed.

Igor Menshov^{1}
, Pavel Pavlukhin^{2}

^{1} Keldysh Institute for Applied Mathematics , Russian Academy of Sciences , Russia

^{2}RDI KVANT, , Russia

We develop the finite-volume method for solving compressible non-stationary Navier-Stokes equations in geometrically complex domains with using simple unfitted Cartesian grids. This method belongs to the class of immersed boundary methods when an extended region including both the fluid flow domain and the solid geometry is discretized in space with a simple, in most cases Cartesian mesh that is unfitted to the solid surface. To treat numerically the inner boundary conditions on that surface in cut cells, the method named as Free Boundary Method (FBM) has been proposed in [1-3] for the Euler model of inviscid gas dynamics equations. The method is based on the Godunov approach with the numerical flux approximation based on the exact solution to the Riemann problem. The calculation is executed over the set of actual mesh cells containing both fluid and cut cells in a manner as there would be no presence of solid surface in cut cells. The effect of the solid surface is taken into account by means of the compensating flux applied on the element of solid surface in the cut cell. This flux is introduced to compensate losses in mass, momentum, and energy which occur when we virtually remove the solid element from the cut cell. It can be proven that the solution of the initial value (Cauchy) problem with the compensating flux embedded in the governing equations in the extended domain is equivalent to the solution of the conventional boundary value problem in the fluid flow subdomain. This approach is extended to the Navier-Stokes equations. We show how the viscous compensating flux should be introduced to assure the adequate solution in the flow domain, discuss its discrete approximation, and propose the numerical method for solving the system of discrete equations. The calculation algorithm is homogeneous; all computational cells are treated in a unique manner. The only difference in computing fluid and cut cells is adding the compensating flux in summation of conventional fluxes. Calculation of the compensating flux requires a few data about the solid geometry in the cut cell, namely fluid volume fraction, area of the solid element and its unit normal, and baricentric coordinates of the fluid sub-element. All the data can be obtained from the level set method for setting the solid geometry. Numerical results concern testing the FBM on the solution of several benchmark viscous problems of aerodynamics. [1] Menshov, I. S. and Kornev, M. A. Free_Boundary Method for the Numerical Solution of Gas_Dynamic Equations in Domains with Varying Geometry. Mathematical Models and Computer Simulations (2014) 6(6): 612–621. [2] Menshov, I. S. and Pavlukhin, P. V. Efficient Parallel Shock-Capturing Method for Aerodynamics Simulations on Body-Unfitted Cartesian Grids. Computational Mathematics and Mathematical Physics (2016) 56( 9): 1651–1664. [3] Menshov, Igor and Pavlukhin, Pavel. Highly scalable implementation of an implicit matrix-free solver for gas dynamics on GPU-accelerated clusters. J Supercomput (2017) 73:631–638.

This research was supported by the grant 17-71-30014 from Russian Scientific Fund

Pavel Pavlukhin^{1}
, Igor Menshov^{2}

^{1}RDI KVANT, , Russia

^{2} Keldysh Institute for Applied Mathematics , Russian Academy of Sciences , Russia

Free Boundary Method (FBM, [1]) which we develop as extension to the Navier-Stokes equations requires high mesh resolution for problems with complex geometry. In case of Cartesian grids it leads to unnecessarily huge increasing of computational costs and memory consumption whereas high resolution is required only near solid surfaces and for high-gradient regions. So one needs to exploit some kind of adaptive mesh refinement (AMR) approach. The most suitable and robust approach for our methods is AMR based on octree graph representation of 3-dimensional Cartesian grids with recursive multi-level cell subdividing. One of our main goals is to implement highly-scalable framework for solving CFD problems on HPC clusters with massively-parallel accelerators (such GPU and Xeon Phi) so we deal with domain decomposition and dynamic load balancing challenges. Initially domain is discretized with coarse Cartesian grid called base grid and then necessary refinements are performed so each base grid cell becomes root of corresponding octree for its subcells. Z-like space filling curve (SFC) over all octrees is used for domain decomposition on distributed-memory systems with the only restriction: any octree (and corresponding base grid cell) is entirely located in one process i.e. splitting of octree over processes is forbidden. It considerably decreases complexity of mesh and octrees inter-process management with small limitation on dynamic load balancing: minimal portion for exchange between processes is a base grid cell with its whole octree. Parallel implementation for the matrix-free LU-SGS method [2] used in iterative solver for discrete equations of implicit scheme leads to graph coloring problem. If grid is colored such any two neighboring (by face) cells are of different colors than all cells of the same color could be processed simultaniously in parallel keeping strong correctness of the LU-SGS method. In case of Cartesian grids only two colors with "chess"-like pattern are needed to implement highly efficient CFD solver with CUDA and MPI [3]. In case of 3-dimensional 2:1 balanced octree grids one needs allready four colors, correspond coloring algorithm presented in [4]. One its important property is simple local recoloring in case of mesh coarsening and refinement that permits to effectively solve problems with implicit time steps even in case of dynamically modified grids strictly conforming to LU-SGS method over whole distributed mesh. Applying this coloring scheme we achieve good scalability with thread- and process-level parallelism. Our approaches to AMR handling on octree grids are similiar to ones used in p4est [5] but this library lacks of thread-level parallelism (it exploits only MPI) and doesn't provide API for low-level data manipulation which is necessary for implementing such type of parallelism and graph coloring algorithms. CFD tests were performed to evaluate performance of solver, mesh manipulation procedures and overheads at CUDA- and MPI-levels. [1] Menshov, I. S. and Kornev, M. A. Free_Boundary Method for the Numerical Solution of Gas_Dynamic Equations in Domains with Varying Geometry. Mathematical Models and Computer Simulations (2014) 6(6): 612–621 [2] Jameson A, Turkel E (1981) Implicit schemes and LU decomposition. Math Comput. doi: 10.2307/2007433 [3] Pavlukhin P, Menshov I (2015) On implementation high-scalable CFD solvers for hybrid clusters with massively-parallel architectures. In: Parallel Computing Technologies: 13th International Conference, PaCT 2015, Petrozavodsk, Proceedings. [4] Adamy U., Hoffmann M., Solymosi J., Stojaković M. (2004) Coloring Octrees. In: Chwa KY., Munro J.I.J. (eds) Computing and Combinatorics. COCOON 2004. Lecture Notes in Computer Science, vol 3106. Springer, Berlin, Heidelberg [5] Carsten Burstedde, Lucas C. Wilcox, and Omar Ghattas, p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees. Published in SIAM Journal on Scientific Computing 33 no. 3 (2011), pages 1103-1133

This research was supported by the grant 17-71-30014 from Russian Scientific Fund

Salah Haggag^{1}

^{1}Egyptian Russian University, Basic Sciences, Egypt

It is shown that some problems of Einstein's field equations of General Relativity can be modeled as optimal control problems. The advantages of adopting such an approach are explained. Some examples are given for problems in relativistic astrophysics and cosmology. Extension to other problems is explored, and difficulties in the formulation of optimal control problems in General Relativity are indicated.

Alexander Bushuev^{1}
, Mikhail Ivanov^{2}
, Dmitry Korotaev^{3}

^{1}Bauman Moscow State Technical University, Computational Mathematics and Mathematical Physics Department, Russia

^{2}JSC Military Industrial Corporation NPO Mashinostroyenia, Bauman Moscow State Technical University, Propulsion Systems Department, Computational Mathematics and Mathematical Physics Department, 0

^{3}Bauman Moscow State Technical University, Computational Mathematics and Mathematical Physics, Russian Federation

Hydraulic synchronization methods are widely used in various technical systems to ensure simultaneous movement of two or more actuators, which are structurally coupled to gas-fluid power cylinders and perform translational or rotational movements. As a rule, the actuators operate in the presence of external force impacts being different in magnitude and direction. These impacts affect dynamic characteristics of the entire synchronization system. Under these conditions, the task of maintaining the simultaneous movement of several actuators can be solved using working fluid flow stabilizers. However, there are tasks in the practice of designing synchronization systems, and external impacts are given in a rather narrow range in these tasks. In this case, the design of the synchronization system can be simplified – throttle washers are used instead of flow stabilizers; the washers are installed in a fluid feed line of each power cylinder. At the same time, there is the need for an optimal choice of flow rate parameters of the throttle washers in order to minimize a mismatch time of the movement of the actuators. An efficient numerical method of a multidimensional optimization based on a Hooke-Jeeves method was proposed, a binary coded genetic algorithm was considered, and a software package was developed. They allow calculating the optimum values of the flow rate coefficients and the geometric characteristics of throttle washers during the computer modeling of a nonstationary process of the synchronization of two power cylinders, the actuators of which are affected by force impacts being constant in modulus and different in sign. A compute core of the software product comprises a dynamic mathematical model of the operation of said mechanical system based on Lagrange’s equations of the second kind and fundamental balance equations of continuum mechanics, for the solution of which the third-order numerical finite-difference Runge-Kutta method was used. The software product was developed using the Qt cross-platform framework. The product offers a user-friendly interface for performing computer-aided design calculations. The performed series of computational experiments demonstrated a quantitative agreement of the obtained results of the solution of the task with the use of these optimization methods.

Vesna Spasic Jokic^{1}

^{1}Faculty of Technical Sciences, University of Novi Sad, Power, Electronic and Communication Engineering, Serbia

The objective of this research was to determine the absorbed dose to the organs that were likely to receive significant doses from the capsule 131I for treatment of hyperthyroid and thyroid cancer patient. This was achieved by using computer codes that were executed with Monte Carlo N-particle-4b which gives the possibility of studying parameters that cannot be measured experimentally. A key factor in retaining the confidence of the referring clinician, the patient and the public, is to maintain and up-to-date knowledge of the radiation risks associated with their procedures. A method for quantifying the risk to the patient is described in terms of estimating the effective dose. Administered radioactivity presents a risk to the patient who should be balanced against the benefit from obtaining a diagnosis or carrying out the treatment. According to standard therapy routine twenty four administrative activities between 185 MBq and 6105 MBq were studied. Organs of interest were bladder, bone surface, colon, liver, gonads, skin and stomach. Significant risks were observed only in doses received by the stomach wall. The calculated risk for 15 minutes is lowest for 185 MBq (0.340 x10-3), and the highest one is for 6105 MBq (11.237 x10-3).

Valery Fedorov^{1}

^{1}North-Eastern Federal University, Reseach Mathematical Institute, Russia

Boundary value problems for some higher order nonclassical equations of mathematical physics were studied, for example, in [1-3]. In this work, using the stationary Galerkin method, the solvability of boundary value problem in cylindrical domain for a higher order equation with changing time direction in weighted Sobolev space is proved, under certain conditions for coefficients and the right hand of equation. In terms of the eigenvalues of the spectral problem for a quasi-elliptic equation the error estimate for approximate solutions of this problem is obtained. The eigenfunctions of this spectral problem are a special basis for approximate solutions. The statement of the problem under consideration was proposed in [2], where I.E. Egorov, using the nonstationary Galerkin method and the regularization method, investigated the boundary value problem for this equation in case when the leading coefficient has another sign than in our case, on the bases of cylinder. In [3] the error estimate of the nonstationary Galerkin method is obtained, for the problem in [2].

The research was supported by the Ministry of Education and Science of the Russian Federation (Grant No 1.6069.2017/8.9)

Valery Fedorov^{1}

^{1}North-Eastern Federal University, Reseach Mathematical Institute, Russia

Boundary value problems for some higher order nonclassical equations of mathematical physics were studied, for example, in [1-3]. In this work, using the stationary Galerkin method, the solvability of boundary value problem in cylindrical domain for a higher order equation with changing time direction in weighted Sobolev space is proved, under certain conditions for coefficients and the right hand of equation. In terms of the eigenvalues of the spectral problem for a quasi-elliptic equation the error estimate for approximate solutions of this problem is obtained. The eigenfunctions of this spectral problem are a special basis for approximate solutions. The statement of the problem under consideration was proposed in [2], where I.E. Egorov, using the nonstationary Galerkin method and the regularization method, investigated the boundary value problem for this equation in case when the leading coefficient has another sign than in our case, on the bases of cylinder. In [3] the error estimate of the nonstationary Galerkin method is obtained, for the problem in [2].

The research was supported by the Ministry of Education and Science of the Russian Federation (Grant No 1.6069.2017/8.9)

Anastasiya Fedosimova^{1}

^{1}Institute of Physics and Technology, Satbayev University, , Kazakhstan

This paper presents the results of modeling the features of the formation of fractal structures in thin films of tin dioxide SnO2 obtained by the sol-gel method. The process of hydrolysis of tin tetrachloride SnCl4 leads to the formation of tin acid Sn(OH)4, which has a gel-like structure. Unstable tin acid decomposes to form water and tin dioxide. The formation of cluster structures is considered depending on the "competition" of these two processes. The nonlinear evolution of sol-gel stochastic processes leads to a fractal structure of the film clusters. The peculiarities of the formation of fractal clusters and the morphology of a thin film are estimated depending on the control (technological) parameters.

The work is supported by MES RK (grant No. АР05134263)

Peter Hockicko^{1}

^{1}University of Zilina, Department of Physics, Slovakia

The paper deals with analysing of ultrasonic measurements realised at a constant frequency 13 MHz and the temperature above 300 K – acoustic attenuation of metaphosphate ion conductive glasses. Attenuation measurements have revealed that relaxation peaks from experimental studies of glasses are much broader than those from a Debye process. So, for description of loss peak and distribution of relaxation time, which varies with activation energy were used Double Power Law (DPL) and Gaussian functions. From both of approaches were estimated the value of activation energies of relaxation processes, which are comparable.

This study was supported by project VEGA No. 1/0510/17 and ITMS: 26220120046 co-funded from EU sources and European Regional Development Fund.

Kwok Yip Szeto^{1}

^{1}Hong Kong University of Science and Technology, Physics, Hong Kong

Zhijie Feng, Wenshuo Guo, and Kwok Yip Szeto* \\ Department of Physics, Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, HKSAR, China *Corresponding author: phszeto@ust.hk \\ We investigate the importance of time scale in winning an election with a well-defined deadline $t_D >0$ for counting the vote. We use a symmetric Watts-Strogatz network to model the voting population and consider the election game between two parties, red and blue, modeled by Monte-Carlo simulation with the Ising model. We further study the effect of spies implanted in the enemy’s camp based on the physics of damage spreading in Ising model. We introduce two counterbalancing local disturbances at two different times during the voter interaction. We consider the first agent chosen from the red party. This agent first performs simple random walk on the social network of voters, searching for the optimal site in terms of maximum damage to the blue party. After a preset time $0\le t_1 \le t_D$ of search, the red agent will choose a site (voter) who is originally loyal to the blue party, and influence this voter so that he will change his mind forever to vote for the red party. Similarly, a second agent will do the similar damage to the red party at $0\le t_1 \le t_2 \le t_D$ . This second agent chooses a site(voter) who is originally loyal to the red party and influence this voter so that he will change his mind forever to vote for the blue party. The two ratios of time scale, $\tau_1 = {t_1 \over t_D}$ and $\tau_2 = {t_2\over t_D}$, are important parameters in terms of the probability of winning the election for the two parties. We perform numerical analysis of this model for different sets of parameters for the social network, different deadline values, as well as different critical values (>0.5) defined for a win of the election. A party will win if the percentage of vote surpass the critical value at the deadline. We investigate the relationship between probability of winning and the independent variables $\tau_1, \tau_2$. We also investigate the optimal strategy of the later agent for maximized winning probability using genetic algorithm in the search for the optimal sites based on the topological features of the chosen node during the random walk search by the agents.

Feng Zhijie and Guo Wenshuo acknowledge the support of UROP-HKUST

Foma Fedorov^{1}
, Nikifor Pavlov^{2}
, Oksana Buskarova^{3}
, Sargy Potapova^{4}

^{1}North-Eastern Federal University, Scientific Research Institute of Mathematics, Russia

^{2}North-Eastern Federal University, , Russian Federation

^{3}North-Eastern Federal University, , Russian Federation

^{4}North-Eastern Federal University, Scientific Research Institute of Mathematics, Russia

It is known that infinite systems of linear algebraic equations had been applied extensively in solving problems of mechanics and physics. Since the beginning of the last century and up to the present time, infinite systems have been used to solve various problems of the static theory of elasticity. The various problems of the theory of diffraction, the theory of electric circuits, the theory of waveguides, the variety of problems in natural science and technology, for example, in the theory of cybernetic systems, in queuing theory, in quantum chemistry, etc., are solved using infinite systems. But the further broad practical application of infinite systems is greatly limited by the insufficient development of the theory of these systems. In particular, the question of the uniqueness of the solution of infinite systems remains open. This is because that solving the corresponding homogeneous system is a much more difficult task than solving the inhomogeneous system. The solving of homogeneous infinite systems is connected with the study of special infinite systems, the so-called quasihomogeneous systems. Such systems are found, for example, in the theory of electrical circuits. An infinite system of linear algebraic equations is quasihomogeneous, if the finite amount of numbers on the right hand side are not equal to 0, and the infinite amount of them are equal to 0. In this paper, we develop a numerical algorithm for finding the solution of quasihomogeneous infinite systems. In particular, it has been shown that such systems can behave both as "purely"\,inhomogeneous, and as almost homogeneous systems depending on the coefficients of the system.

The research has been supported by the Ministry of Education and Science of the Russian Federation (Grant No. 1.6069.2017/8.9)

JESUS GONZALEZ^{1}
, Fernando Durán^{2}
, Jader González^{3}

^{1}Universidad del Magdalena, Magdalena, Colombia

^{2}Universidad Pontificia Bolivariana-Bucaramanga, , Colombia

^{3}Universidad Pontificia Bolivariana-Bucaramanga, , Colombia

Advances in nanofabrication techniques have made it possible to make ultra-thin layer superconductors in which a superconducting sample are connected to non-superconducting component, but also self-assembly of magnetic molecules on solid surfaces that allow to create a new two-dimensional (2D) hybrid superconducting system. Here, we study properties of superconducting vortices in the thin layer of lead (Pb) with inclusion of central stripe of niobium (Nb) in the the presence of external magnetic field. Our primary findings include a substantial controllability of the fundamental superconducting parameters through changing the wide of central stripe that allows the manipulation of magnetic flux quanta.The present study opens a route for designing and creating exotic 2D superconductors with an atomic-scale precision.

Universidad Pontificia Bolivariana-Bucaramanga and Fonciencias-Universidad del Magdalena

Yuriy Dimitrienko^{1}
, Andrey Zakharov^{2}

^{1}Bauman Moscow State University, Computational Mathematics and Mathematical Physics, Russia

^{2}Bauman Moscow State Technical University, Computational Mathematics and Mathematical Physics, Russian Federation

The paper covers the area of block-structured adaptive grid generation for finite-difference, finite-element, and finite-volume methods. It deals with a problem of development of information technologies and algorithms for generation of such grids in complex geometries. Its key idea is to bring together the different existing techniques for computer-aided geometric design of domains, grid generation, and subdivision. The methods used for the construction of domain geometries and working with block-structured quasi-continuous grids are discussed. The results in the generation of three-dimensional adaptive grids for the domain of external flow near a hemisphere, flow around a surface of the high-speed aircraft of the Falcon HTV-2 type and for the domain of the gap between a wheel and body of an aircraft are analyzed. The presented methods and algorithms for grid generation has the following advantages: (1) the grids are structured and mesh cells are hexahedra in the entire domain, (2) the grids can be applied for finite-difference methods, (3) the grid lines are adjusting with the flow in the whole domain.

Dmitry Fetisov^{1}

^{1}Bauman Moscow State Technical University, Mathematical Modeling, Russian Federation

State feedback linearization is well-known as an effective tool to solve various problems for nonlinear control systems. An affine system is called to be state feedback linearizable if there exist smooth invertible changes of the state and the inputs which transform the system into a linear controllable system. If an affine system is not state feedback linearizable one can try to use orbital feedback linearization. An affine system is called to be orbital feedback linearizable if there exists time scaling which transforms the system into a state feedback linearizable system. As usual for affine systems, time scaling is considered to be depending only on the state. Recently, it has been shown that if time scaling depends both on the state and on the inputs, then it becomes possible to linearize affine systems which are not orbital feedback linearizable. In this paper, while considering such transformations, we suggest the new sufficient condition for linearizability of single-input control affine systems. We derive a system of partial differential equations which has to be solved in order to find an appropriate time scaling. We provide en example how the proposed approach can be applied to solve a terminal problem.

This work was supported by the Russian Foundation for Basic Research under grant no. 17-07-00653.

Shun Kuroki^{1}

^{1}Chiba University, Dept. of EEE, Japan

In order to realize energy-saving motor drive systems, the energy consumed by the motor drive must be reduced as more as possible. For this purpose, the use of magnetic bearing (MB) and low inertia shaft are two promising options. MB is friction-free due to its noncontact support property. Meanwhile, the reduction of shaft inertia leads to low vibration frequencies (flexible mode velocities) which imposes a great challenge for the control system. There are two kinds of vibrations in high-speed rotation: one is the bending vibration caused by the decentering of shaft, the other is the torsional vibration caused by disproportion among rotation systems. When driving the motor beyond the flexible mode velocity, it is necessary to model the motor drive system considering the flexible vibrations. In particular, at high-speed rotations these two motions couple with each other. Though the modeling of shaft's torsional vibration is well studied, the bending vibration was ignored up to now such as in the studies on magnetic bearing system. In this paper, we propose a new and integrated model for motor drives including both the bending vibration and the torsional vibration as well as the gyroscopic effect. Special attention is paid to the coupling of these two kinds of vibrations. The approach is based on Lagrangian mechanics and the precise computation of the energies in the bending vibration and the torsion vibration from the viewpoint of flexible material mechanics. Numerical simulations validated that the behavior of the derived model is consistent with the physics. In particular, it was shown that when the motor speed got closer to the critical speed, the coupling between the bending vibration and the torsional vibration was excited, which must be considered in the control design. This model forms the first step toward the construction of a safe, energy-saving and high performance motor drive control system for high-speed and high-power motors.

Stanislav Budzinskiy^{1}
, Tatiana Romanenko^{2}

^{1}Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics, Russia

^{2}Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics, Russia

We study self-organization in a nonlinear optical system with a feedback loop, where light is delayed and undergoes a spatial transformation within the thin annulus aperture. The dynamics of the system are modelled by a quasilinear functional differential diffusion equation on a circle that tracks the effects of diffraction by means of a coupled linear Schrodinger equation. We prove the existence of rotating wave solutions and study their stability by constructing the Faria normal form of a Hopf bifurcation.

The reported study was funded by RFBR according to the research project № 18-31-00236.

Leonid Igumnov^{1}
, Vladimir Metrikin^{2}
, Mikhail Zaythev^{3}

^{1}National Research Lobachevsky State University of Nizhni Novgorod, Research Institute for Mechanics, Russia

^{2}National Research Lobachevsky State University of Nizhni Novgorod, Research Institute for Mechanics, Russia

^{3}National Research Lobachevsky State University of Nizhni Novgorod, Research Institute for Mechanics, Russia

The paper investigates the dynamics of a frictional system, consisting of a rough body located on a rough belt moving at a constant speed. The vibrations of the body are limited by an elastic obstacle. The characteristic of dry friction is chosen, according to A.Yu. Ishlinsky and I.V. Kragelsky, in the form the hereditary type friction with a static friction coefficient (SFC) in the form of a monotonically nondecreasing continuous time function of relative rest. A mathematical model, which is a system with variable structure, is constructed, the structure of its phase space with nonclassical boundaries of slip-motion plates is studied. The study is carried out using the numerical- analytical approach employing the mathematical method of point mapping. The equations of point mapping of the Poincaré surface are given in a parametric form. As a result of the study, the dependencies of dynamic characteristics on the parameters of the system (tape speed, the form of the functional dependence of the SFC, the rigidity of the elastic obstacle, etc.) are obtained. The developed software allowed to search for complex periodic motion modes, as well as to carry out calculations of bifurcation diagrams, according to which the main reorganizations of the motion regimes from periodic to chaotic are established (the period doubling scenario).

Nicolae IACOBICI^{1}
, Flaviu Mihai FRIGURA-ILIASA^{2}
, Mihaela FRIGURA-ILIASA^{3}
, Sorin MUSUROI^{4}
, Petru ANDEA^{5}
, Florin Balcu^{6}

^{1}POLITEHNICA University Timisoara , Power Systems, Romania

^{2}National Institute for Research and Development in Electrochemistry and Condensed Matter Timisoara, Power Systems, Romania

^{3}POLITEHNICA University Timisoara , Power Systems, Romania

^{4}POLITEHNICA University Timisoara, Academy of Romanian Scientists , Electrical Engineering, Romania

^{5}POLITEHNICA University Timisoara, Academy of Romanian Scientists , Power Systems, Romania

^{6}National Institute for Research and Development in Electrochemistry and Condensed Matter Timisoara, LERF, Romania

Operating at absolute minimum cost can no longer be the only criterion for dispatching electric power due to increasing concern the environmental consideration. The environmentally constrained economic dispatch problem which accounts for minimization of both cost and emission is a multiple objective function problem. In this paper, tabu search is used to solve the economic dispatch problems for a couple of reasons. The modified tabu search algorithm uses a real-valued solution vector and adaptive mechanism for producing neighbors. However, the classical TS algorithm uses a binary solution vector. The neighbors are produced by adding an adjusted coefficient at each iteration. This neighbor production mechanism enables us to find the most promising region of the searched space. The proposed method has achieved efficient and accurate solutions for power systems with 3, 6, 10, 20, 40, 80, 120, 160 and 240 units respectively. The results of the proposed method are compared with those of an improved Hopfield neural network approach, an advance engineered-conditioning genetic approach, an advance Hopfield neural network approach and a fuzzy logic controlled genetic algorithm. It will demonstrate that the proposed method is superior in providing optimal allocation of generation units which minimizes the total fuel cost. Simulation results show that the proposed method yields better solutions when compared to the alternate techniques.

Peter Dymochkin^{1}
, Igor Inovenkov^{2}
, Vladimir Nefedov^{3}
, Liubov Ponomarenko^{4}

^{1}Lomonosov Moscow State University, Computational Mathematics & Cybernatics, Russia

^{2}Lomonosov Moscow State University, Computational Mathematics & Cybernatics, Rwanda

^{3}Lomonosov Moscow State University, Computational Mathematics & Cybernatics, Russia

^{4}Lomonosov Moscow State University, Computational Mathematics & Cybernatics, Russia

In recent years diverse mathematical physics models became widely used in practice as a great implement for describing economical and physical phenomena. Modeling of the market of derivatives, in particular case, computation of European option rational price, is carried out with Black-Scholes equation, which in terms of constant volatility and non-dividend situation has the well-known analytical solve. Several generalizations of Black-Scholes model were suggested to reach better conformity between the results of mathematical modeling and the real market figures. In these generalized models volatility depends on the desirable function in a rather complicated way. Surveyed are Leland model, Frey-Patie model and also the model of indefinite volatility, which vary from minimum to maximum value. In order to calculate the numerical solution of this mixed task the scheme with scales, which has the second approximation order on the coordinate and time, is used for the equation of parabolic type. Series of computing experiments which allow to compare the results of analytic and numerical analysis were conducted for the different range of parameters. According to the results of computer simulation, generalized non-linear Black-Scholes equation describes the real market simulation describes the real market situation with a very close approximation.

Kamila Jarczewska^{1}
, Ryszard Hołubowski^{2}
, Wojciech Glabisz^{3}

^{1}Wrocław University of Science and Technology, Civil Engineering, Poland

^{2}Wrocław University of Science and Technology, , Poland

^{3}Wrocław University of Science and Technology, , Poland

In this study, the critical buckling load and natural vibration frequency of Euler-Bernoulli single nano-beams based on Eringen’s nonlocal elasticity theory are investigated. The cantilever nano-beams with attached spring mass are subjected to a concentrated and distributed follower forces. The influence of nonlocal scale of single nano-beams with attached spring mass on critical load is consider for both cases of analyzing loads and the obtained results are compared with the results from local theory. The proposed algorithm is based on exact solutions of motion equations for segments separated in the beam with boundary and continuity conditions. The effect different value of the nonlocal parameter, tangency coefficient, spring coefficient and distributed of mass on stability are discussed.

Natalya Kandaurova^{1}
, Vladimir Chekanov^{2}
, Vladimir Chekanov^{3}
, Mikhail Shevchenko^{4}

^{1}Moscow Technological University, (MIREA) , , Russia

^{2}North-Caucasus Federal University, , Russia

^{3}Moscow Technological University, (MIREA), , Russia

^{4}LLC "Optosystems", , 0

The near-electrode thin layer of a magnetic fluid is an active medium in which an autowave process was experimentally observed in an electric field. Authors propose a system of autowave process equations, based on the FitzHugh-Nagumo model: $$\dot{V} = D\Delta V + \frac{(\alpha - V)(V - 1)V - W}{1 + 2\varphi} \\ \dot{I} = \varepsilon (1 + \frac{\varphi}{1.5})( \beta V - \gamma I - \delta)$$ where V is a function that depends on the field strength in a thin near-electrode layer of a magnetic fluid (activator), I is the function associated with a charge change (inhibitor), the coefficients are responsible for the properties of the oscillating medium: α is the excitation limit, ε is the excitability. Coefficient φ (x, y), defined by authors, is the deceleration factor – it allows the properties of the medium modification near the obstacle to be included into the model. A solution of the system in COMSOL Multiphysics was obtained. Phase portraits and zero isoclines for the activator and inhibitor are constructed, special points and stationary solutions are found. The typical autowave effect – obstacle bending – is investigated in details. It is established that autowaves change its speed near the obstacle. The solution obtained allows us to identify areas with different periods of refractoriness, depending on the geometry and dimensions of the obstacle. A similar situation can actually occur in the heart muscle, when the period of refractoriness of myocardium individual parts increases due to the necrosis of the tissue in a heart attack. This is the practical significance of this work.

The work was performed as the element of the basic part of the state task Project №3.5385.2017/8.9 for the implementation of the project on the topic: "Experimental research and mathematical modeling of interphase and near-surface phenomena in a thin membrane of a nanostructured magnetic fluid" Moscow Technological University (MIREA)

Nikolay Evstigneev^{1}
, Oleg Ryabkov^{2}

^{1}Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Institute for System Analysis, Russia

^{2}Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Institute for System Analysis, Russia

We are considering a numerical method that is applied in rectangular domains to find numerical solution of the initial-boundary value problems for the incompressible Navier-Stokes equations. It uses nodal Finite Element approach to reconstruct pressure and Finite Difference method for other parts of the equations. This approach benefits from usage of simple finite difference schemes for linear parts (compact difference schemes) and WENO-type schemes for the the nonlinear part. Finite element is formulated in terms of pressure values in vertexes of elements to form matrix equations. This is a combination of finite element method and difference methods. Let us consider the Navier-Stokes system for velocity vector function $\mathbf{u}$, pressure scalar function $p$ and external force vector function $\mathbf{f}$ in rectangular domain $\Omega$: $$\partial_t \mathbf{u}+(\mathbf{u}, \nabla) \mathbf{u}+\nabla p=\mu \Delta \mathbf{u}+\mathbf{f},$$ $$\nabla \cdot \mathbf{u}=0,$$ with appropriate initial and boundary conditions. We use projection method to recover pressure and enforce incompressibility. Let us introduce rectangular cuboids $W_{jkl}$ that from a 3D tessellation of a rectangular domain $\Omega=\bigcup_{\mathbf{j}}^{}W_{\mathbf{j}}$, where $\mathbf{j}$ is a multi index in 3D. We introduce another set of tessellation $U_{\mathbf{k}}$ that is constructed from swapping central nodes and vertexes, thus each vertex of $W_{\mathbf{j}}$ becomes a geometric center for $U_{\mathbf{k}}$ and vice versa. We define trilinear or spline basis functions in an element $Q_{\mathbf{k}}$, where $Q$ can be $W$ or $U$. Now we consider a set of points $\mathbf{j}$ formed by the centers of $W_{\mathbf{j}}$ or vertexes of $U_{\mathbf{k}}$. Such elements can be considered as finite volumes, i.e. $\overline P_{\mathbf{j}}=\int_{W_{\mathbf{j}}} P(\mathbf{x}) d\mathbf{x}$. We now define Laplace operator $\mathtt{L}$ and gradient operator $\mathtt{Q}$ in the space of nodal finite elements $W$. We also define Laplace operator $\mathtt{A}$, divergence operator $\mathtt{G}$ and nonlinear operator $\mathtt{B}(\cdot,\cdot)$ in the space of finite volumes $W$ (using finite difference schemes). We also define projection of nodal operator to central finite volume point as $\mathcal{W}$ and projection of central finite difference operator to nodes as $\mathcal{U}$. The first operation is performed using finite element expension provided above with $Q:=W$ and the second operation is the inverse of the expansion with $Q:=U$. In this case we introduce temporal grid with grid slices denoted $l$ and time step $\Delta t$ and propose the following scheme: $$ \left \{ \begin{matrix} \mathbf{u}^*=\mathbf{u}^l+\Delta t \mathbf{f}^l + \Delta t \mathtt{B}(\mathbf{u}^l,\mathbf{u}^l) + \Delta t \mathtt{A} \mathbf{u}^*- \beta \Delta t \mathcal{W}\mathtt{Q}p^l,\\ \mathtt{L} \phi=-1/\Delta t \mathcal{U} \mathtt{G} \mathbf{u}^{*},\\ \left \{ \begin{matrix} \mathbf{u}^{l+1}=\mathbf{u}^*-\Delta t \mathcal{W} \mathtt{Q} \phi,\\ p^{l+1}=\beta p^{l}+\phi. \end{matrix} \right. \end{matrix} \right .$$ Here $0 \leq \beta \leq 1$ is the relaxation parameter. For the scheme we can prove the following $$\textbf{Proposition} \textit{The Ladijenskaya-Babuska-Brezzi (LBB)}$$ $$\textit{$inf$-$sup$ condition for the presented scheme is satisfied.}$$ Using the proposed scheme we can show that the approximation is of order $\mathcal{O}(\Delta t)$ for velocity terms. We improve temporal approximation by using either explicit Runge Kutta 3-d order method or Diagonally Implicit Runge Kutta method. The method is implemented under the heterogeneous computational architecture aimed on the utilization of graphical processing units (GPUs). We show parallel efficiency of the method and demonstrate space-time convergence of the method using exact Navier-Stokes solutions (Taylor-Green vortex and Shapiro solutions). We also demonstrate the method capabilities in performing direct numerical simulation of the Rayleigh - Benard convection problem in a cubic domain.

Yuliya Ryzhikova^{1}
, Iuliia Mukhartova^{2}
, Sergey Ryzhikov ^{3}

^{1}Lomonosov Moscow State University, Department of Optics, Spectroscopy and Nanosystems Physics, Russia

^{2}Lomonosov Moscow State Uninersity, Faculty of Physics, Mathematics, Russia

^{3}Lomonosov Moscow State University, Department of General Physics, Faculty of Physics, Russia

The present work is dedicated to the solving the fundamental scientific problem of modern fractal optics. It is associated with the establishing of general regularities that determine the relationship between the features of the structure of fractal-like objects and their spectral characteristics. Finding these laws will enable to develop new diagnostic tools for research of fractal-like objects and processes for their formation. Nanocluster systems and multilayer structures, including metamaterial inserts, were chosen as the objects of research. The choice of a wide range of objects of different physical nature is due to the penetration of fractal optics into a system of interdisciplinary technologies and theoretical concepts. Particular attention is paid to the study of the scaling of the analyzed objects structures and their optical characteristics. On the basis of theoretical studies of the distribution of the amplitude and phase of light waves that diffract or interfere on the analyzed objects the features of scaling in light fields are determined. When analyzing the obtained modeling results, a comparison between the features of the structure of the analyzed objects with the scaling parameters in diffraction patterns and transmission spectra is made.

The reported study was funded by RFBR according to the research project № 18-01-00723 a.

Hovik Matevossian^{1}
, Anatolii Vestyak^{2}

^{1}Russian Academy of Sciences, Federal Research Center "Computer Science and Control", Russian Federation

^{2}Moscow Aviation Institute (National Research University), Mathematical Modeling, Russia

We study the asymptotic as $t\to\infty$ behavior of solutions $u(x,t)$ of the initial boundary value problems for a second-order hyperbolic equation with periodic coefficients on the semi-axis. In the case of non-homogeneous equation, initial and boundary data are zero, and the right-hand side of the equation is of the form $f(x)exp(-i\omega t)$, where $\omega >0$ is real.

asymptotic behavior, hyperbolic equation, initial boundary value problem, periodic coefficients

Fedor Potemkin^{1}
, Evgenii Mareev^{2}
, Boris Rumiantsev^{3}
, Anton Bychkov^{4}
, Alexander Karabutov^{5}
, Elena Cherepetskaya^{6}
, Vladimir Makarov^{7}

^{1}Lomonosov Moscow State University, Faculty of Physics & International Laser Center, Russia

^{2}Lomonosov Moscow State University, Faculty of Physics & International Laser Center, Russia

^{3}Lomonosov Moscow State University, Faculty of Physics & International Laser Center, Russia

^{4}Lomonosov Moscow State University, Faculty of Physics, Russia

^{5}Lomonosov Moscow State University, International Laser Center, Russia

^{6}The National University of Science and Technology MISiS, LUNDT Laboratory, Russia

^{7}Lomonosov Moscow State University, Faculty of Physics & International Laser Center, Russia

Femtosecond laser filamentation is a result of dynamic balance of Kerr-induced self-focusing and laser-induced plasma defocusing. The filament is a coupled system of laser plasma (the core) and electromagnetic radiation around it (the reservoir). The energy in the reservoir is used to form the core, and if the reservoir of the filament is blocked (e.g. using a diaphragm), the filament can no longer propagate. While several purely optical methods of filament investigation have been developed, there is a lack of methods for direct reservoir imaging. Photoacoustic tomography is a well-developed imaging method based on the local non-stationary thermoelastic expansion of the medium and generation of acoustic pressure waves. Since both the filament core and the reservoir create a thermal source due to plasma recombination and optical absorption respectively, photoacoustic tomography can be a promising method for identification of the location and the size of the core as well as for reservoir imaging. In order to preserve the wide bandwidth of acoustic signals, broadband (~100 MHz) piezoelectric crystals (e.g. lithium niobate) should be used to record them. Moreover the acoustic attenuation and finite size of the piezoelectric sensor limit the spatial resolution of the optoacoustic imaging, and at the "water-sensor" interface the refraction of acoustic waves occurs due to the speed of sound mismatch. In this work, the semi-analytical model for fast calculation of the acoustic signals from the filament (forward problem) is presented. The model uses the approximation of geometrical acoustics to account for the refraction at a plane "water-sensor" interface and a representation of a filament that allows for a semi-analytical solution accounting for acoustic attenuation. The approximation of geometrical acoustics is used to modify the 2D back-projection algorithm for photoacoustic image reconstruction (inverse problem). The results of numerical simulation are compared with the experimental results. The model is intended to help optimize the experimental setup for photoacoustic imaging of femtosecond laser filaments in water.

The reported study was funded by RFBR according to the research project № 18-302-00001.

Mihaela FRIGURA-ILIASA^{1}
, Flaviu Mihai FRIGURA-ILIASA^{2}
, Marius MIRICA^{3}
, Stefan NOVACONI^{4}
, Lia DOLGA^{5}
, Hannelore Filipescu^{6}

^{1}POLITEHNICA University Timisoara , Power Systems, Romania

^{2}National Institute for Research and Development in Electrochemistry and Condensed Matter Timisoara, Power Systems, Romania

^{3}National Institute for Research and Development in Electrochemistry and Condensed Matter Timisoara, LERF, Romania

^{4}National Institute for Research and Development in Electrochemistry and Condensed Matter Timisoara, LERF, Romania

^{5}POLITEHNICA University Timisoara , Mechatronics, Romania

^{6}POLITEHNICA University Timisoara , Mechatronics, Romania

The main goal of this article is to conceive an algorithm for a software application designed to solve many power engineering optimization problems applying the critical path method (CPM). The classical solution for these problems is found via the schedule graph (the Gantt method), which presents a series of shortcomings: it does not highlight clearly the interdependencies between operations, it does not explain the temporal coincidences, it does not indicate the alternatives of declaring the various operations, it does not have a rigorous mathematical foundation, it does not allow any studies of optimization. All the shortcomings are cleared when using the representation of the program via a graph, where the values of the arcs are the durations of the component operations, and applying the critical path method (CPM), which mainly consists in determining a path of a maximum value between two peaks of that graph. Using CPM offers a great number of advantages: it offers a clear image of the evolution in time of the program, it allows the decreasing of the total duration of realization for the program without condensing the component operation, it highlights the operations that directly determine the duration of realization of the program, as well as those that allow the redistribution of the resources and the reduction of cost; it offers the alternative of rapid evaluation of the consequences of certain delays in realizing certain operations (without totally rebuilding the graph and the calculations); it can be easily implemented on a computer, having a solid mathematical base. Applying CPM and optimizing the program is achieved via the following steps: a) setting the list of the component operations, of their characteristics and representing the program via graphs; b) setting the graphs in order; c) determining the critical path (CP) and its value; d) calculating the time stocks related to the realization of the component operation (using the Ford and the Bellman-Kalaba algorithms); e) optimizing the program as from the length, the cost and the necessary resources points of view. For illustrating the algorithm and the computing program we propose an application from power engineering: a 400 kV electrical overhead line section construction. In the first part of the paper we present the application as a critical path problem. In the second part, we determine the critic path in a program graph and time reserves. In the third part, we present a representative numerical application. In the fourth part, it is described the computing program.

Andrey Bulygin^{1}

^{1}V.E. Zuev Institute of Atmospheric Optics SB RAS (IAO SB RAS), Laboratory of Nonlinear Optical Interactions, Russia

The propagation of laser radiation in a turbulent medium (atmosphere) is considered on the basis of the stochastic Schrödinger equation with higher nonlinearities in terms of statistics. This problem is a stochastic problem with multiplicative noises. A general method is proposed that allows us to reduce this class of problems to an equivalent quantum field model. On the basis of this method, an equivalent quantum-theoretical model is formulated that makes it possible to obtain the Fokker-Planck equation for the system under consideration, as well as the expression for the equilibrium state of the physical system under consideration. On the basis of the numerical approach, the stationary solution is investigated from the point of view of searching for the statistics of extreme values of the field.

The reported research was funded by Russian Foundation for Basic Research and the government of the region of the Russian Federation, grant № 18-41-703004»

Sergey Gerasimov^{1}
, Igor Inovenkov^{2}

^{1}Lomonosov Moscow State University, Faculty: Computational Mathematics and Cybernetics, Department: Automation for Scientific Research, Russia

^{2}Moscow State University, Computational Mathematics and Cybernetics, Russia

Urban problems have become very complicated as a result of technological progress and change in the behavior of human beings. The urban systems of our epoch are characterized by increasing spatial and temporal variation. Many urban models have been suggested to explain and forecast urban pattern formation in urban economics, regional science and geography. Within the framework of the spatial economy model, equilibrium states of commodity flows and labor, which are described by two-dimensional vector fields, are considered. These configurations are obtained on the assumption of the validity of divergent and gradient laws, which characterize the influence on the commodity flows and labor by external factors, such as the distance from the places of habitation of labor to industrial objects, the local costs of transportation and so on. The structural stability of such a system for various types of configuration perturbations is investigated. It is very difficult to obtain general characteristics of structurally stable systems, but in the two-dimensional case it turns out to be possible to draw precise picture of structurally stable flows and spatial organization of the economy corresponding to such flows. In the paper possible flow structures for a hyperbolic umbilical for a cubic potential function that depends on three parameters are constructed numerically in accordance with the concept of catastrophe theory. Bifurcation varieties are found in the parameter space, flux fields for various parameter combinations are constructed. A qualitative change in the character of the flow structure in the neighborhood of a bifurcation manifold is demonstrated.

Vadim Matveev^{1}

^{1}Midas marketing UAB, Vilnius, Research and Development Group, Lithuania

Purely by mathematical means a theorem of the finiteness of the velocity of material bodies is proved without reference to the special theory of relativity. If in material bodies there are always present massless interaction signals and carrier particles that are propagated at a velocity, V, greater than any achieved velocity of the material bodies moreover the massless interaction signals and carrier particles traveling the mean free path, λ, collide with elements of these bodies committing interactions and continuously initiating processes in all bodies then the velocity, V, of massless interaction signals and carrier particles at the distance L > λ is finite. Leaving out of account the limitation of velocity coming from the special theory of relativity we a priori admit the existence of unlimited velocities, v, of material bodies and particles that may have any value. Nevertheless, the velocities of the material bodies are assumed to be less than the signal and carrier particles velocity, V, i.e. v < V. We prove logically that according to the condition of the theorem there is a limit to the allowable achieved velocity of material bodies, admitting that without the condition there is no such limiting velocity. First we ask ourselves question: “What does the concept “instantaneously” mean”? In a metrological sense “instantaneously” means a vanishing time interval between emission and return of the body. In a physical sense “instantaneously” means that no physical process or change of a system can occur between these events. The emission time and the return time must merge together in this instance. We considered a situation with a material body at rest at point, O, and a massless signal travelling a distance, 2L, from point, O, to point, A, and back. While the massless particle outside the body travels the distance, 2L, then inside the material body at rest n signals are committing a certain kind of interaction, and there occur 2nL/λ events in the body, where λ is the average free path of the signals committing this kind of interaction within the body, and it is assumed that λ < L. Since there are interactions and processes in the body, it will change in time, and therefore an arbitrarily accurate clock will show a non-vanishing time interval during the travel of the signal.

We would like to thank Dr. Øyvind Grøn (the University of Oslo, Norway) for help with the formulation in English of this paper.

Veronika Kopylova^{1}
, Stanislav Boronovskiy^{2}
, Yaroslav Nartsissov^{3}

^{1}Institute of cytochemistry and molecular pharmacology, Mathematical modeling and statistical analysis of results, Russia

^{2}Institute of cytochemistry and molecular pharmacology, Department of mathematical modeling and statistical analysis of results , Russia

^{3}Institute of cytochemistry and molecular pharmacology, Department of mathematical modeling and statistical analysis of results , Russia

The transport of oxygen and the essential metabolites throughout the body and the supply of tissues is the main function of the circulatory system. Convection in blood flow and diffusive exchange between blood and surrounding tissues provides effective organ perfusion. For correct functioning of the system rigid constraints should be imposed on the structure of the vasculature. In particular, the delivery of oxygen and other metabolites by diffusion to all cells of the body depends significantly on the spatial arrangement of the vessels, while the network topology and the diameters and lengths of the vessels strongly influence the blood flow distribution. In the present study we have proposed the algorithm for construction of the arterial system of the rat brain, which describes both the main arteries and smaller vessels. In the proposed approach the vasculature is divided into two parts with different modeling methods. The first one includes the main arteries forming the Circle of Willis, which were realized as an ordered set of cylinders. The radii and coordinates of the cylinders were assigned. The second part is a stochastic structure that describes smaller vessels branching from the main arteries, which were modeled as a binary tree according to the physiological laws. The distribution of the bifurcation level follows the gamma distribution with the small deviations in the region corresponding to the vessels of the basic structure. The modal level of the branch order which is necessary for effective perfusion is achieved when the bifurcation exponent is higher than 2.9. Despite the fact that the lower limit of symmetry is not changed with an increase of the bifurcation exponent, analysis of the vasculature symmetry showed that the density of vessels with a symmetry value above 0.5 increases significantly.

Aleksandr Bobylev^{1}

^{1}Lomonosov Moscow State University, Department of Theory of Elasticity, Faculty of Mechanics and Mathematics, Russia

Both plane and axisymmetric wear contact problems for elastic semi-infinite bodies with a thin non-uniform coating and finite rigid punches are considered. The coating is modelled by an elastic layer of the Winkler type. Only nominal contact zones are assumed to be known. Real contact areas that may change due to wear are unknown and have to be defined. The variational approach is used to solve the wear contact problems. The variational formulation of the problems in the form of a system of an evolutionary variational inequality and an ordinary differential equation of the first order is obtained. An explicit Euler scheme is used for the time discretization. A spatial discretization of the problems is made using spaces of the integrated fundamental solutions. The boundary element method is applied to construct the spaces. As a result, it is necessary to solve a square programming problem with restrictions in the form of equalities and inequalities on each temporary layer. The linear transformation of variables allowing simplification of a type of restrictions is offered. A modification of the conjugate gradient method considering specifics of restrictions is used for the numerical solution of the problems. Wear processes of wave surfaces with a thin non-uniform coating by finite rigid punches is investigated. The distribution of the wear function in the contact zone is determined and discussed.

Aleksandr Grigoriev^{1}
, Evgeny Shilko^{2}
, Sergey Psakhie^{3}

^{1}Institute of strength physics and materials science SB RAS (ISPMS SB RAS), Laboratory of computer-aided design of materials, Russia

^{2}Institute of strength physics and materials science SB RAS (ISPMS SB RAS), , Russia

^{3}Institute of strength physics and materials science SB RAS (ISPMS SB RAS), , Russia

The numerical simulation is a promising tool for the study of material behavior under dynamic loading. The widespread numerical methods for the simulation of material response to dynamic loading are finite element and finite difference methods. The mathematical formalisms of these methods make possible numerical solution of the complex problems of solid mechanics, including dynamic one. At the same time application of these methods to dynamic problems meets difficulties at the simulation of multiple fractures accompanied by mass mixing and mass transfer. An effective tool for the numerical solution of the problems related to complex fracture of solids is the discrete element method (DEM). Despite the intensive use of DEM for the numerical study of deformation and fracture of brittle materials and media, mathematical formalism of this method is limited by the quasi-static models of material behavior. Therefore the fields of application of the DEM are limited by the strain rate interval <10 s-1. Therefore the aim of this work is the development of the mathematical formalism of the DEM to simulate the behavior of brittle solids under dynamic impact. We developed the model of the dynamic behavior of brittle solids based on the kinetic theory of strength, which was first proposed by Zhurkov and developed in the works of Morozov and Petrov. The model takes into account strain rate sensitivity of inelastic and strength parameters and allows modeling of the dynamic behavior of brittle solids at <103 s-1. The developed model is the modification of the classic “static” Nikolaevsky’s plasticity model and the failure model with a two-parameter criterion of Drucker and Prager. A feature of the proposed dynamic model is the use of the “stress” form of the yield and failure criteria instead of the integral form that is traditionally used in the kinetic theory of strength. Dynamic behavior of material is conventionally described through the dependences of model parameters on strain rate. Note that the strain rate is a technical parameter of loading, which characterizes average (smoothed) deformation of a sample. At the same time there is a physical parameter, which characterizes the process of degradation/fracture of material at the considered spatial scale under applied loading. This is the stress relaxation time. Typically, the relaxation time is a period during which nucleation and growth of the main crack in the sample take place. Therefore the relaxation time as physical parameter of degradation/fracture is preferable to be used as a key parameter of proposed dynamic behavior model. The main issue that arises when applying this model to solving applied problems is the determination of the dependences of model parameters (strength and cohesion) on the time of generation of discontinuities of the corresponding scale. Ideally, a series of laboratory dynamic tests is required to determine these dependencies for considered material. We propose an alternative way to solve this problem. In particular, we propose to estimate the required dependencies with use of the available databases of experimental data about the strain rate dependences of the mechanical characteristics of various brittle materials. The approach to dynamic models developing is relevant for solving applied problems associated with natural and man-caused dynamic impacts on structures of artificial building materials, including concretes, elements of ceramic structures and rock materials. In particular, the implementation of the proposed approach within the framework of the particle-based method makes it possible to solve the problems of the estimating of fracture time of structures and brittle materials with complex internal structure as a function of impact amplitude and strain rate, taking into account the structural features of the materials in question. At the same time the solution of the inverse problem provides to obtain estimation of the characteristic size of the elements of material internal structure which determine the fracture process.

The work was carried out in the framework of the Fundamental Research Program of the State Academies of Sciences for 2013–2020 and under financial support from Russian Science Foundation (project No.17-11-01232).

Roman Semenko^{1}
, Alexander Blokhin^{2}

^{1}Novosibirsk State University, Mechanics and Mathematics, Russia

^{2}Novosibirsk State University, Mechanics and Mathematics, Russia

We studied the discontinuous stationary solutions for the rheological mesoscopic modified model of Pokrovskii-Vinogradov, which describes the dynamics of liquid polymers. The Rankine-Hugoniot conditions for the model were introduced. We justified the existence of stationary solutions with flat surface of strong discontinuity for the case of constant velocity direction across the discontinuity and for the case with change of direction (rotating discontinuity). The stability of such solutions was also considered. For linearized equations of the model we posed the eigenvalue problem for partial solutions with unlimited grow in time. It was shown that such solutions exists in anisotropic case wich means the stationary solutions with flat discontinuity are unstable within the given model.

This work is supported by RFBR, project No 17-01-00791 А and by the program of fundamental research SB RAS No I.1.5., project No 0314-2016-0013.

Kwok Yip Szeto^{1}

^{1}Hong Kong University of Science and Technology, Physics, Hong Kong

Qi Bu, Juntao Wang, and Kwok Yip Szeto* Department of Physics, Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, HKSAR, China *Corresponding author: phszeto@ust.hk ---------------------------- Perforated embedding is a powerful tool searching for optimal Markov approximations of time series data. When a Markov model of order m is assumed, the knowledge of the past values of the time series earlier than m steps has no contribution to the decrease of the uncertainty in the future. A common method for a given conditioning set approximates the information entropy by Grassberger-Procaccia correlation sum, which searches for nearest neighbors with different resolutions, and then finds optimal perforated embedding sets by minimizing both the modeling and statistical errors. However, when time steps in the earlier history are considered, the number of possible conditionings increases exponentially with m. Besides, the complexity of nearest neighbor searching also increases when the time series is embedded into higher dimensions. We propose an efficient alternative solution by using genetic algorithm to find the optimal conditioning sets without testing all candidates. Furthermore, we introduce the horizontal visibility graph algorithm to map the time series to a complex network, which is more efficient. This mapping between time series and complex networks prove to be very useful for searching the optimal conditioning sets. In our application of search for the optimal conditioning sets with genetic algorithm, the fitness criterion is similar to previous methods but we compute the mutual information between time steps using the degree distribution of nodes in horizontal visibility graphs. We illustrate this method of finding the optimal conditioning set in a Markov model of time series on several standard examples.

Wang Juntao acknowledge the support of UROP-HKUST

Antoni Grau^{1}
, Yolanda Bolea^{2}
, Alberto Sanfeliu^{3}
, Sergio Moyano^{4}

^{1}Technical University of Catalonia, Automatic Control Department, Spain

^{2}Technical University of Catalonia, Automatic Control Department, Spain

^{3}Robotics Industrial Institute, , Spain

^{4}Technical University of Catalonia, Automatic Control Department, Spain

In this work, a multi-model of a quadrator is developed in order to control this system. The kinematic model of each part of the quadrotor will be derived using the Euler angles, and also the dynamics model of the quadrotor will be calculated based on the first principles of a rigid body using the Newton-Euler formulation. Furthermore, the following assumptions are used :1) The structure is completely rigid and perfectly symmetric. 2) The center of mass is in the origin of the quadrotor fixed frame. 3) The thrusts are proportional to the square of the motors rotational speed. An state-space model (kinematics and dynamics) is developed by physical laws. But, this deduced model presents several no linearities that are produced by three factors: the orientation ( Pitch, Roll and Yaw ), the control action and the angular velocities. To be able to control the quadrator system in simple, linear and manageable way, it is necessary to linearize the system. Two method are possible: a classical linearization around several set-points and a multi-model linearization. In this case, a multi-model linearization is proposed due to the obtained control model will be used to compute a multi-model controller using fuzzy techniques. Fuzzy control techniques are suitable for linear parameter varying systems with no linearities, as our quadrator.

Kwok Yip Szeto^{1}

^{1}Hong Kong University of Science and Technology, Physics, Hong Kong

Juntao Wang, Ka Ki Lau, and Kwok Yip Szeto* Department of Physics, Hong Kong University of Science and Technolgy Clear Water Bay, Hong Kong, HKSAR, China *Corresponding author: phszeto@ust.hk Abstract Evolving complex system such as the stock market provides a very challenging problem for physicists. By focusing on a particular feature such as the management of a portfolio of stocks, we use evolutionary computation to optimize the Sharpe ratio of the portfolio. Among numerous techniques in this approach, we use genetic algorithm to address the adaptive approach for the portfolio management in the asset management involving two stocks and cash. We analyse two stocks selected from Hang Seng Index constituents that satisfy two conditions: they are highly anti-correlated during some periods of time, and with more than 5000 historical data on closing prices. The strategy comprises three parts: conditions and action with selected fraction for investment. By quantization of the closing prices and trading volumes of the historical data of the two stocks and the encryption of the conditions of the strategy into a binary string, we form a population of investment strategies. The population of these strategies form a population of chromosomes, which evolve according to the Darwinian principle of the survival of the fittest. We use a particular form of genetic algorithm with mutation matrix to perform the search for a prediction rule that has sufficient high prediction accuracy as well as high probability of application. The optimal strategy for managing the portfolio is one that achieves the largest Sharpe ratio. By employing this dynamic set of strategies, we show that in testing set, the Sharpe ratio of the investment can be increased and outperform the index.

Wang Juntao and Lau Ka Ki acknowledge the support of UROP-HKUST

Andrey Bulygin^{1}

^{1}V.E. Zuev Institute of Atmospheric Optics SB RAS (IAO SB RAS), Laboratory of Nonlinear Optical Interactions, Russia

The propagation of high-power laser radiation in a medium with saturable cubic self-focusing on the basis of the nonlinear Schrödinger equation is studied. It is shown that under certain conditions this system exhibits superfluidity properties. Numerical schemes are considered for numerical realization of this effect. It is shown that numerical schemes based on the discrete difference approximation of the NLS equations possess the best properties from the point of view of the conservation laws. An effective parallel algorithm for solving the nonlinear Schrödinger equation on an no-uniform grid is realized. Accurate numerical integration is employed to study the general nonlinear evolution of the system from the unstable stationary solutions to the formation of stable vortex patterns.

The reported research was funded by Russian Foundation for Basic Research and the government of the region of the Russian Federation, grant № 18-41-703004»

Andrey Bulygin^{1}

^{1}V.E. Zuev Institute of Atmospheric Optics SB RAS (IAO SB RAS), Laboratory of Nonlinear Optical Interactions, Russia

The properties of the motion integral and evolution of the eective radius of a light beam are analyzed within the framework of the stationary model of the nonlinear Schrdinger equation describing lamentation. A completely conservative numerical scheme for the nonlinear Schrdinger equation is developed. Within the framework of this model, it is shown that lamentation is limited only by dissipative mechanisms. An inverse variational problem is solved for the nonlocal nonlinear Schrdinger equation used in simulation of lamentation in various nonlinear media. The corresponding integral relations, which generalize the conservation laws for the non-conservative case, are derived.

«The reported research was funded by Russian Foundation for Basic Research and the government of the region of the Russian Federation, grant № 18-41-703004»

Rostislav Gerasimov^{1}

^{1}Bauman Moscow State Technical University, Basic Sciences, Russia

R. Y. Gerasimov, Y. V. Gerasimov, G. N. Fadeev Bauman Moscow State Technical University The proposed new method based on the spectra of radio wave radiation of the microwave range wideband scanning receiver. It is experimentally proved that this method has a high accuracy in the determination of the frequency spectrum. This allows to reach a sensitivity of 5-6 Hz/nm. The obtained spectra are analyzed for the particular program hardware-software complex. Practice has shown that the proposed method allows not only to detect the difference of the substances qualitative composition and concentration, but also to determine the presence of heavy and superheavy hydrogen isotopes in water. The method for determining the presence of substances in nanoscopic amounts is based on the registration of radio-wave emission spectra in the microwave region using a scanning wideband receiver [1]. This radiation arises owing to the excitation of alternate high-frequency displacement and conduction currents in an object subjected to testing between the flexible plates of a capacitive working sensor. The results from measurements (the obtained spectra) are analyzed using a hardware–software complex. SUMMARY 1. The method of radioscopy in the microwave range, allows you to explore the fluid system, by comparing the spectra of the pure solvent (reference samples) and solutions. 2. In this work, the parameters of the spectra of the standard with bidistilled water and aqueous solutions: individual substances in the analyzed water; the difference of the concentrations of dissolved substances; evaluation of isotopic composition in the system of hydrogen isotopes. 3. It was established experimentally that this method can detect the presence of small quantities of heavy (D) and superheavy (T) hydrogen isotopes in ordinary (H2O) and heavy (D2О) water. REFERENCES 1. R.Yu. Gerasimov, G.N. Fadeev, Yu.V. Gerasimov, E.A. Kondrakova. // Doklady Physical Chemistry, 2015, Vol. 464, Part 1, pp. 219–221. 2. R.Yu. Gerasimov, G.N. Fadeev. // VI Vseros. konf. “Neobratimye protsessy v prirode i tekhnike” (VI All_ Russian Conference “Irreversible Processes in Nature and Technics”)), Moscow: MGTU, 2011, vol. 2, pp. 284–285. 3. R.Yu. Gerasimov, G.N. Fadeev, Yu.V. Gerasimov, E.A. Kondrakova. // Russian Journal of Physical Chemistry A, 2016, Vol. 90, No. 9, pp. 1785–1788 4. Bovenko V N and Gerasimov R Yu // Nonliner world 2011 № 6 pp 369–377 5. Gerasimov R Yu., Fadeev G N, Gerasimov Yu V and Kondrakova E A 2016, J of Phys: Conf Ser731 (IOP Publishing) 012005 6. Gerasimov R Yu., Gerasimov Yu V and Fadeev G N 2017, J of Phys: Conf Ser918 (IOP Publishing) 012011 7. Fadeev, G.N., Ermolaeva, V.I., Boldyrev, V.S., Sinkevich, V.V. // Russian Journal of Physical Chemistry A, 2016 , Vol. 90, No. 9, pp. 1724–1728

Vladislav Balashov^{1}
, Evgeny Savenkov^{2}

^{1}Keldysh Institute of Applied Mathematics RAS, Computational methods and mathematical modelling, Russia

^{2}Keldysh Institute of Applied Mathematcis RAS, Computational methods and mathematical modelling, Russia

The present paper is devoted to numerical simulation of two-phase two-component isothermal viscous compressible microflows with surface tension and wetting effects within complex voxel-based flow domains. As a basic mathematical model the Navier-Stokes-Cahn-Hilliard (NSCH) equations are utilized. To use easy-to-implement explicit finite difference approximations we take advantage of regularized (within quasi-hydrodynamic framework) version of NSCH system. The simulation results demonstrate that the developed algorithm for the NSCH system is correct and suitable for solution of industrial level problems thus being a promising technique for simulation of multiphase microflows within complex geometries, including geophysical <

The research has been supported by Russian Science Foundation, grant No. 17-71-30014.

Mikhail Alekseev^{1}
, Evgeny Savenkov^{2}
, Fedor Voronin^{3}

^{1}Keldysh Institute of Applied Mathematics (Russian Academy of Sciences) , Numerical methods and mathematical modelling, Russia

^{2}Keldysh Institute of Applied Mathematcis RAS, Computational methods and mathematical modelling, Russia

^{3}Keldysh Institute of Applied Mathematics (Russian Academy of Sciences), , Russia

Recently it has been paid a substantial attention for simulation of high velocity and penetration problems using Eulerian formulations for solid mechanics with finite strain. The main advantage of such approach is possibility to handle extreme deformations when for the application of the more traditional arbitrary Eulerian-Lagrangian formulations it is difficult. The natural mathematical model for the considered class of applications has a form of the first-order hyperbolic systems of conservation laws in the Eulerian reference frame [1]. This model is proposed by Godunov and Romenski [2] and has a number of characteristic features from more traditional second-order systems employed in elasticity. The most important one is the presence of a complex structure of contact, shock and rarefaction waves and their interactions. Numerical solution of such problems is a difficult task due to a complex wave structure of the solution. Recently a number of efficient approaches were presented to handle the problem. Most of them are based on linearized HLL type solvers and high order WENO reconstruction techniques to obtain accurate fronts resolution in order to resolve complex solution structure. In this paper we apply high order RKDG method as an alternative to WENO schemes to solve the problem. Up to 5-th order reconstruction is considered with different limiting techniques and 3-rd order timestepping scheme. As a trial problem a one dimensional hyperelastic formulation is presented similar to [1] for a number of test case. The developed computer code utilizes automatic differentiation (AD) approach which allows for an easy switch of underlying EOS and strain measure. The presented results will cover discussion of the problem statement, corresponding RKDG approximations and simulation results which demonstrate possibilities of the RKDG approach applied for the hyperbolic hyperelastic formulations in the Eulerian reference frame. This work is supported by the Russian Science Foundation under grant №17- 71-30014.

This work is supported by the Russian Science Foundation under grant №17- 71-30014

Nadejda Rautian^{1}

^{1}Lomonosov Moscow State University, Mechanics and Mathematics, Russian Federation

The main goal of our work is studying of the asymptotic behavior of the solutions of Gurtin-Pipkin type integro-differential equations on the base of spectral analysis of their symbols. The Gurtin–Pipkin equation arises in such domains of mechanics and physics as heat transfer theory, theory of viscoelastic media, and kinetic theory of gases. Since the Gurtin-Pipkin type integro-differential equations arise in numerous applications, it is reasonable and natural to study integro-differential equations with unbounded operator coefficients in a Hilbert that are the operator models of such type partial integro-differential equations and have the following type $$ \frac{{du(t)}}{{dt}} + \int_0^t {\mathcal K(t - s)A^2 u(s)ds} = f(t), \quad t\in \mathbb R_+, $$ where $A$ is a self-adjoint positive operator acting in the separable Hilbert space $H$ and having a compact inverse operator. Assume that kernel $\mathcal K(t)$ is scalar function that determined empirically and admits the representation $$ \mathcal K(t) = \int_0^\infty {\frac{{e^{ - t\tau } }} {\tau }d\mu (\tau )}, $$ where $d\mu$ is a positive measure corresponding to an increasing right-continuous distribution function $\mu$. The integral is understood in the Stieltjes sense. The correct solvability of initial-value problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces is determined. A spectral analysis of the operator functions, which are the symbols these integro-differential equations is provided. Strong solutions of these equations are represented as a sum of terms corresponding to the real and nonreal parts of the spectrum of the operator functions that are the symbols of these equations. The resulting representations are new for the given class of integro-differential equations.

The work was supported by the Russian Federation President grant for the state support of the leading scientific schools of the Russian Federation (project NSh-6222.2018.1).

Alain Haraux^{1}

^{1}CNRS, Mathematics, France

We discuss some global properties of the semi-flow generated by the scalar ODE $$u’’ + cu’ + g(u) = f(t)$$ where $c>0$, $f$ is a bounded signal and $g(u)$ is an odd polynomial of degree 3 or a more general function with similar properties in a sense to be specified. When $f(u) = au+ bu^3$ with $a, b>0$ (the so-called single well Duffing ODE), many interesting questions have been investigated in the literature as early as the middle of twentieth century, among which rather sharp estimates of the ultimate energy bound of the general solution and the classification of possible asymptotic behaviors, for instance the number of different periodic solutions when $f$ is periodic. It is known that for large periodic forcing terms, subharmonic solutions and chaotic behaviors can appear. In this talk we recall some of the well known results in this direction and more recent advances in the non-chaotic situation both for the single well and the twin well problem corresponding to the case $a<0$. The methodology developped in the last ten years to devise as precise as possible bounds and conditions for the existence of only one (resp. only three) asymptotic regimes has also given interesting results for infinite dimensional « Duffing-like » evolution equations of the form $$u’’ + cu’ + A^2u + g(u) = f(t)$$ where $A$ is a positive self-adjoint operator on an infinite dimensional Hilbert space. The special case $$ g(u) = (b||A^{1/2}u||^2 – a)Au $$ has been recently studied in a joint work of with Marina Ghisi and Massimo Gobbino by improving some of the techniques devised for the twin-well scalar equations, and more realistic models representing small deviations by an external force of some magneto-elastic systems will be the object of future studies, hopefully by using similar tools adapted to this more difficult situation.

Jaykov Foukzon^{1}

^{1}Israel Institute of Technology, Haifa, Israel, math, Israel

In present article the original proposition is a generalization of the Einstein's world tensor $g_{ij}$ by the introduction of pure inertial field tensor $g^{ac}_{ij}$ such that $R_{{\mu}{\nu}{\lambda}}^{{\alpha}}(g^{ac}_{ij})\neq0$.Bimetric theory of gravitational-inertial field is considered for the case when the gravitational-Inertial field is governed by either a perfect magnetic fluid.In a series of papers published during the past decade with respect to M\"ossbauer experiments in a rotating system [71]-[75],it has been experimentally shown that the relative energy shift {\Delta}E/E between the source of resonant radiation (situated at the center of the rotating system) and the resonant absorber (located on the rotor rim) is described by the relationship ${\Delta}E/E=-ku^2/c^2$, where u is the tangential velocity of the absorber, c the velocity of light in vacuum, and k some coefficient, which -- contrary to what had been classically predicted equal 1/2 (see for example [35]) -- turns out to be substantially larger than 1/2. It cannot be stressed enough that the equality $k=1/2$ had been predicted by general theory of relativity (GTR) on account of the special relativistic time dilation effect delineated by the tangential displacement of the rotating absorber, where the "clock hypothesis" by Einstein (i.e., the non-reliance of the time rate of any clock on its acceleration [35]) was straightly adopted. Hence, the revealed inequality $k>1/2$ indicates the presence of some additional energy shift (next to the usual time dilation effect arising from tangential displacement alone) between the emitted and absorbed resonant radiation. By using Bimetric Theory of Gravitational-Inertial Field [76] we obtain $k=0.625$ in a good agreement with experimental result $k=0.69$ [75].

MICHAEL AGARANA^{1}

^{1}COVENANT UNIVERSITY, MATHEMATICS, Nigeria

This paper analysed, mathematically, the mechanical properties of engineering materials that are of uniform composition throughout that cannot be mechanically separated into different materials and having identical values of a property in all directions. The analysis was carried out analytically. Comparisons were made to establish relationships between the mechanical properties of the materials considered. The results obtained are consistent with the ones in the literature. In particular, strength of the various engineering materials vary according to the type of loading.

MICHAEL AGARANA^{1}

^{1}COVENANT UNIVERSITY, MATHEMATICS, Nigeria

Robots are fast becoming a fixture in our lives. Kinematics and dynamics of self-balancing Inverted Pendulum Robot, modelled as an inverted are derived in this paper using Lagrange energy method. The derived equation of motion of the inverted pendulum robot was analysed via Taylor differential transformation. Computer software - Maples was used for the plotting of graphs for the result obtained. The results show that the position and motion of the inverted pendulum robot have significant effect on achieving its self-balance.

MICHAEL AGARANA^{1}

^{1}COVENANT UNIVERSITY, MATHEMATICS, Nigeria

This paper is on the dynamics of inverted pendulum as a harmonic mechanical Oscillator. The pendulum considered is inverted, damped and driven. The motion of the dynamical system was modelled. Computer software - Maple, was used to simulate and analyse the model. Some oscillatory phenomena were observed by the exhibition of interesting dynamics by the damped driven inverted pendulum. Specifically, the lower the fixed value of the angular driven force the higher the angular velocity at various values of the angular displacement.

MICHAEL AGARANA^{1}

^{1}COVENANT UNIVERSITY, MATHEMATICS, Nigeria

Dynamical Systems exhibit their natural frequencies when given an initial disturbance and allowed to vibrate freely. This paper attempts to characterize inverted pendulum in terms of one or more natural frequencies. The Energy method was used to determine the natural frequency. This method yielded the equation of motion of the system, which was solved analytically. A computer software – MAPLE, was used to display the three-dimensional figures of the results. The results revealed that the Energy method is closely related to both Rayleigh’s equation methods. Specifically, the results show that the natural frequency of an inverted pendulum is typically a function of time. It was observed that as time increases the natural frequency of the inverted pendulum system decreases.

ABDALLAH SLAOUI^{1}

^{1}Laboratory Of High Energy Physics, Modeling and Simulation. Faculty of Sciences. Rabat. University mohammed V Morocco, Department of Physics, Morocco

Recently, quantum correlations have been the subject of deep studies, principally due to the general belief that they are fundamental resources for different quantum, information processing tasks. These correlations are diffcult to maintain because a quantum system is usually not closed, but in interaction with its environment. In this in work, we employ the concept of local quantum uncertainty (LQU) and trace quantum discord (TQD) to investigate the environmental effects on quantum correlations of a two qubit system. Two decoherence scenarios are considered. The rst one concerns the coupling of the system with two independent bosonic reservoirs. The second scenario deals with a two level system interacting with a quantized electromagnetic radiation.

George Jandieri^{1}

^{1}Institute of Cybernetics, Georgian Technical University, Stochastic analyses and mathematical simulation, Georgia

Statistical characteristics of scattered electromagnetic waves in the ionospheric plasma are calculated solving stochastic differential equation for the phase fluctuations taking into account boundary condition, diffraction effects and polarization coefficients for both ordinary and extraordinary waves. Variance and correlation function of the phase fluctuations are obtained for arbitrary correlation function of electron density fluctuations. These second order statistical moments allows to estimate the broadening and shift of maximum of the spatial power spectrum of scattered radiation, and also investigate scintillation effects in the F region of the ionosphere using the experimental data. Numerical calculations are carried out for 3 MHz incident wave and 3D anisotropic spectral function of electron density fluctuations characterizing anisotropic plasma irregularities containing both anisotropy factor and orientation angle of elongated plasma irregularities with respect to the external magnetic field. It is shown that displacement and the width of the SPS for the ordinary and extraordinary waves tends to the saturation increasing anisotropy factor. Shift of maximum of the SPS strongly depends on the orientation angle of anisotropic plasma irregularities, particularly, varying angle in the interval displacement of its maximum increases six times. The angle-of-arrivals in the main plane (containing external magnetic field and wave vector of an incident wave) less than in normal direction. Phase scintillation index for small-scale irregularities fast growth in proportion to the orientation angle and reaching maximum slowly decreases inversely proportion to the characteristic linear scale of plasma irregularities. Small ( ) level scintillations is associated with both positive and negative intensity fluctuations, while the large ( ) levels (primarily corresponds to the positive intensity fluctuations. Sinusoidal type oscillations are observed in the intensity spectrum and are attributed to a Fresnel filtering effect for plasma irregularities having characteristic spatial scale less than the Fresnel radius. Oscillation minima satisfy the “standard relationship” 1: : : … Scintillation level allows to calculate the spectral width (first and second moments) computing the power spectrum and scintillation period. The spectral widths (first and second moments) of the power spectrum and scintillation periods are calculated for “frozen-in” drifting elongated plasma irregularities with apparent velocity transverse to the line of the sight path using the experimental data. If “frozen-in” irregularities drift along the X-axis (perpendicularly to the main plane) with the velocity 100 m/sec, spectral width mHz, period is 100 sec and mHz. If elongated plasma irregularities are moving along the Y-axis in the main plane, sec and mHz. Knowledge data of these oscillations allows to solve the reverse problem - calculate the velocity of plasma irregularities in the principal and perpendicular planes, to estimate characteristic spatial scales and the r.m.s. electron density fluctuations for plasma irregularities smaller than the Fresnel radius.

This work has been supported by the International Science and Technology Center (ISTC) under Grant # G-2126 and Shota Rustaveli National Science Foundation under Grant # FR/3/9-190/14.

Gerard Lemaitre^{1}

^{1}Aix Marseille Universite, Laboratoire d'Astrophysique Marseille, France

7eme Conférence Internationale - Moscou – 25-29 Juin 2018 – Université Lomonosov Proposition - Conférence invitée sur «Modélisation mathématique en sciences physiques» Auteur : Gerard R. Lemaitre ENGLISH Title : “ Active optics in astronomy – Modeling of deformable substrates : Freeform surfaces for FIREBall and MESSIER” Abstract : Active optics techniques as currently implemented on large telescopes and astronomical instrumentations provide high imaging quality. For ground-based astronomy, the co-addition of adaptive optics again increases angular resolution up to provide diffraction-limited imaging at least in the infrared. Especially active and adaptive optics marked milestone progress in the detection of exoplanets, black holes, and large scale structure of galaxies. This paper is dedicated to highly deformable active optics that can generate non-axisymmetric aspheric surfaces – or freeform surfaces – by use of a minimum number of actuators : a single uniform load acts over the surface of a vase-form substrate whilst under reaction to its elliptical perimeter ring. Two such instruments are presented, 1) the FIREBall telescope and MOS where the freeform reflective diffraction grating is generated by replication of a deformable master grating, and 2) the MESSIER wide-field low-central-obstruction TMA telescope proposal where the freeform mirror is generated by stress figuring and elastic relaxation. Modeling codes to generate these freeform surfaces by plane super-polishing were applied from preliminary analyses with the optics theory of 3rd-order aberrations and elasticity theory of thin elliptical plates. Cross-optimizations required use of Zemax raytracing code and Nastran FEA elasticity code to determine geometry of the deformable substrates. FRANCAIS Titre : Optique active en astronomie – Surfaces « freeform » et modélisation de substrats déformables Résumé : Les techniques d’optique active actuelles installées sur les grands télescopes et l’instrumentation astronomique fournissent des images de haute qualité. Pour l’astronomie au sol, l’ajout de l’optique adaptative augmente encore la résolution angulaire jusqu’a fournir des images de diffraction au moins en infrarouge. Notamment les optiques active et adaptative ont permis des progrès remarquables sur la détection d’exo-planètes, de trous noirs, et des structures à grande échelle des galaxies. Cet article est dédié à l’optique active de grande déformation pouvant générer des surfaces asphériques non de révolution – dites surfaces freeform – par l’emploi d’un nombre minimum d’actionneurs : une seule charge uniforme agit sur la surface d’un substrat de forme en vase tandis que la réaction est absorbée par la jupe elliptique du contour. Deux tels instruments sur présentés, 1) le télescope et MOS FIREBall dont le réseau de diffraction freeform par réflexion est généré par réplique d’un réseau mère sous déformation, and 2) le projet de telescope TMA a grand champ et faible obstruction centrale MESSIER dont le miroir freeform est généré pas taille plane sous contrainte et relaxation élastique. Les logiciels de modélisation permettant de générer ces surfaces freeform par super-polissage ont été appliqués à partir d’analyses préliminaires de la théorie optique des aberrations du 3eme ordre et de la théorie de l’élasticité des plaques minces elliptiques. Les optimisations croisées avec le logiciel de passage de rayons Zemax et le logiciel d’élasticité par éléments finis Nastran ont permis de déterminer la géométrie des substrats déformables.

Aidar Ilyasov^{1}
, Guzel Bulgakova^{2}
, Timur Kireev^{3}

^{1}Ufa State Aviation Technical University, , Russian Federation

^{2}Ufa State Aviation Technical University, Mathematics, Russia

^{3}Ufa State Aviation Technical University, , Russian Federation

An important engineering task for mature oil fields with high water cut production wells is water shut-off (WSO) in these wells. For successful WSO operations in the intervals of porous and fractured-porous formations it is necessary to create appropriate mathematical models, taking into account rock mechanical properties, elastic properties of water-shutoff agents as well as other technological parameters. In this regard, the problem arises on the mathematical model development for evaluating the strength of water shutoff baffles from cement slurry and synthetic resins in porous and fractured-porous formations in production water-cut wells. Based on the solution of Lame problem for an isotropic ideally elastic body, a model is proposed that predicts the WSO synthetic resin baffle strength in porous and fractured-porous formations in production water-cut wells. The strength diagrams were constructed as a function of WSO material volume, bottomhole pressure, reservoir geomechanical properties, WSO material elastic properties, and formation depth. The results of mathematical modeling are compared with actual field data.

This study was supported by the Russian Foundation for Basic Research (Project 17-41-020226 r_a).

olutayo onanuga^{1}

^{1}Lagos state polytechnic, Ikorodu, Lagos ,Nigeria, PHYSICAL SCIENCE, Nigeria

The focus of the study is to examine thermal radiation and viscous dissipative heat transfer of magnetohydrodynamics (MHD) stagnation point flow past a permeable confined stretching cylinder with non-uniform heat source or sink. The formulated equation governing the flow is non-dimensional. The dimensionless momentum and energy equation are solved using shooting technique coupled with order four of Runge-Kutta integrated scheme which satisfied smoothness conditions at the edge of the boundary layer. The results obtained are presented graphically and discussed for the velocity and temperature distributions to portray the effects of some important embodiment parameters on the flow. The Nusselt number and skin friction were obtained and compared with the previous scholars’ results in other to validate the present research work.

1Onanuga O.K., 2Chendo M.A.C., and 2Erusiafe N.E. 1 Department of Physics, Lagos State Polytechnic, Ikorodu, Lagos State 2Department of Physics, University of Lagos, Akoka, Lagos State

Gottfried Berton^{1}

^{1}Société de Calcul Mathématique, Mathématiques, France

The aim of this study is to compare two methods of interpolation, namely Kriging (a standard algorithm), mainly used in geostatistics, and the Experimental Probabilistic Hypersurface (developed by SCM SA). We study several technical points, such as their ability to take uncertainties into account, to return an uncertainty on the interpolation, the quality of the numerical procedures, etc. The Experimental Probabilistic Hypersurface (EPH) is a minimal information model, which only uses the existing data and makes as less artificial hypothesis on the data as possible. The Kriging on the contrary, relies on an estimation of the variability of the data using a variogram.

Semyon Levitsky^{1}
, Rudolf Bergman^{2}

^{1}Shamoon College of Engineering, Mathematics, Israel

^{2}Shamoon College of Engineering, Mathematics, Israel

The objective of the study is to describe the effect of liquid viscoelasticity on the propagation of a pressure signal, initially generated at the pipe end, in the case of a layered tube. The studied waveguide consists of a thin double-layered cylindrical shell filled with compressible viscoelastic liquid. It is supposed that liquid rheology follows the 3-parameter Oldroyd rheological model. Liquid dynamics in the wave is described within the quasi-one-dimensional approach. It is assumed that the bending stresses in the tube wall can be neglected, as compared with the membrane ones. The solution of the momentum balance equation for the layered shell, found in this approximation, is coupled with a transient solution, describing liquid flow in the wave, by appropriate boundary conditions. The resulting system of equations is solved by the operational method for a standard finite pressure signal, generated at the initial moment of time at the pipe end. The Laplace image of the pressure disturbance in the wave is inverted numerically. The obtained data indicate that combin