Jorge Olivares Funes^{1}
, Elvis Valero^{2}

^{1}University of Antofagasta, Department of Mathematics, Chile

^{2}Universidad de Tarapacá, matemáticas, Chile

In this paper, we will show the solutions of certain exact differential equations that are obtained through the interactive GeoGebra software. GeoGebra software has been of great motivational support in the processes of teaching and modeling in mathematics in various universities and colleges, especially in the engineering careers of the University of Antofagasta in the courses of differential equations and calculation of several variables.

Jaykov Foukzon^{1}

^{1} Israel Institute of Technology , Center for Mathematical Sciences, Technion - Israel Institute of Technology. , Israel

Novel explanation of the Active Galactic Nuclei.The Power Source of Quasars as a result of vacuum polarization by the gravitational singularities on BHs horizon. J.Foukzon¹, E.Menkova² A.Potapov³ ¹Department of mathematics, Israel Institute of Technology, Haifa, Israel ²All-Russian Research Institute for Optical and Physical Measurements, Moscow,Russia ³Kotel'nikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences, Moscow, Russia In this paper we argue that the current paradigm for AGN and quasars essentially incomplete and rivision is needed. Remind that the current paradigm for AGN and quasars is that their radio emission is explained by synchrotron radiation from relativistic electrons that are Doppler boosted through bulk motion. In this model, the intrinsic brightness temperatures cannot exceed 10¹¹ to 10¹² K. Typical Doppler boosting is expected to be able to raise this temperature by a factor of 10.The observed brightness temperature of the most compact structures in BL Lac, constrained by baselines longer than 5.3Gλ, must indeed exceed 2×10¹³K and can reach as high as ~ 3×10¹⁴K. As well known, these visibilities correspond to the structural scales of 30-40 μas oriented along position angles of 25°-30°.These values are indeed close to the width of the inner jet and the normal to its direction.The observed, T_{b,obs}, and intrinsic, $$T_{b,int}$$, brightness temperatures are related by Eq.1: $$T_{b,obs}=δ(1+z)⁻¹T_{b,int}$$ with δ=7.2.The estimeted by Eq.1 a lower limit of the intrinsic brightness temperature in the core component of our Radio Astron observations of $$T_{b,int}>2.910¹² K$$. It is commonly considered that inverse Compton losses limit the intrinsic brightness temperature for incoherent synchrotron sources, such as AGN, to about 10¹²K [1].In case of a strong flare, the "Compton catastrophe" is calculated to take about one day to drive the brightness temperature below 10¹²K [1]. The estimated lower limit for the intrinsic brightness temperature of the core in the Radio Astron image of $$T_{b,int}>2.910¹²K$$ is therefore more than an order of magnitude larger than the equipartition brightness temperature limit established in [1] and at least several times larger than the limit established by inverse Compton cooling. Remark 1.Note that if the estimate of the maximum brightness temperature given in [1], is closer to actual values, it would imply $$T_{b;int}=5×10¹³K$$. This is difficult to reconcile with current incoherent synchrotron emission models from relativistic electrons, requiring alternative models such as emission from relativistic protons. Remark 2. However the proton, as we know, is 1836 times heavier than an electron and absolutely huge energy is required to accelerated it to sublight speed. We argue that these alternative models such as emission from relativistic protons can be suported by semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved distributional spacetimes presented by rotating gravitational singularities [2]. [1] J. L. Gómez, A. P. Lobanov, G. Bruni, Y. Y. Kovalev,Probing the innermost regions of AGN jets and their magnetic fields with Radio Astron.I.Imaging BL Lacertae at 21 microarcsecond resolution. Astrophysical journal 817 (2016) 96, DOI:10.3847/0004-637X/817/2/96 arXiv:1512.04690 [astro-ph.HE] [2] J.Foukzon,E.Menkova,A.Potapov, Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance.110 pp. Published November 14, 2019 ISBN-13 (15) 978-93-89562-22-4 https://doi.org/10.9734/bpi/mono/978-93-89562-22-4

SANTANU CHATTERJEE^{1}
, Sanjoy Mukherjee^{2}

^{1}RGM International Pvt. Ltd., Civil Department, India

^{2}Vikram Solar Limited, Innovation, India

String theory provides an encouraging way to unify all force fields in our universe into a single framework. Different vibrational patterns of a single string resemble different particles. Bosonic strings require 26 spatial dimensions in order to produce particles in a similar fashion as 10 spatial dimensions are required to produce Fermions states. These extra dimensions (beyond 3 spatial & 1-time dimension) are compactified into very small scale and thus in today’s scale of probing energy it is not possible to detect them experimentally. This idea of extra dimensions, hidden from our perception is tempting & worth in-depth theoretical work. Certain topological features and its application on String theory will also be discussed in this paper. We will try to fathom these small plank scale compactified dimensions & will try to throw some light on the various topological aspects of quantum geometry offered by this remarkable theory.

SANTANU CHATTERJEE^{1}
, Sanjoy Mukherjee^{2}

^{1}RGM International Pvt. Ltd., Civil Department, India

^{2}Vikram Solar Limited, Innovation, India

In today’s era of modern medicines, mathematical modeling plays an important role in analyzing difficult aspects of cell divisions and proliferation to speculate the result of the actual experiment. In these concepts of modeling, differential equation as a technique for determining relation between a function and its derivatives played a very important role. If we know a function and few of its derivatives at a particular point then that information along with differential equation can plot the function over its entire domain [1]. By choosing suitable parameters & boundary conditions different types of differential equations can be formed to represent actual biological process and up-to the complexity levels as we may desire. Thus, mathematical modeling of metastatic cancer & growth of cancerous cell has become an active area of research. In normal cells, hundreds of genes intricately control the process of cell division. Cells become cancerous after mutations accumulate in the various genes that control cell proliferation. In other words, they no longer respond to most of the signals that control cellular growth and death. Mutations in genes can cause cancer by accelerating cell division rates or inhibiting normal controls on the system, such as cell cycle arrest or programmed cell death [2]. In this paper will investigate spherical tumor considering three types of cell formation or layer of cell inside it, namely, proliferating cell layer, quiescent cell layer & necrotic cell layer and we will study their properties & dynamics of expansion or contraction with respect to time. Instead of oxygen (as in; Greenspan, H. (1972)), we will model our case with flow of nutrients that are required for a living cell to be alive and proliferate. Through a simple linear 1st order differential equation, different stages of tumor growth and invasion of tumor cells to surrounding tissue in the form of micro metastatic cells will also be investigated. Our approach will be to modify the 1st order differential equation for standard population dynamics (growth & death type) with considerable changes in the parameters to catch up all three earlier mentioned phases or layers of a growing tumor.

Jorge Olivares Funes^{1}
, Pablo Martin^{2}
, Elvis Valero^{3}

^{1}University of Antofagasta, Department of Mathematics, Chile

^{2}University of Antofagasta, physics department, Chile

^{3}Universidad de Tarapacá, matemáticas, Chile

The fractional derivative of Caputo, has huge and important applications in various areas of science and engineering. In this case, through the definition of the Caputo derivative and the Laplace transform and its inverse, we propose to show the solutions that can be obtained for each specific value of alpha of the following fractional differential equations $$\frac{d^\alpha y}{{dx}^\alpha}= I_3/4(x)$$ , $$\frac{d^\alpha y}{{dx}^\alpha}= I_-3/4(x)$$ , with $$m-1<α\le m$$ , $$m\ \in N $$. Where the non-homogeneous parts I_3 / 4 (x) and I_-3/4 (x) are the modified Bessel functions of the first species.

MARÍA ROJAS^{1}
, Jorge Olivares Funes^{2}
, Daniza Rojas^{3}

^{1}UNIVERSIDAD DE ANTOFAGASTA, MATEMÁTICAS, Chile

^{2}University of Antofagasta, Department of Mathematics, Chile

^{3}University of Antofagasta, Department of Mathematics, Chile

GeoGebra and Wolfram Αlpha are educational software used in many areas of science, engineering and mathematics. In this work we will use these programs to see the homogeneous reducible differential equations along with their solutions. The present material that will be presented corresponds to the one designed for the course of differential equations for engineering and pedagogy careers in mathematics at the University of Antofagasta-Chile.

Fedor Grigoriev^{1}

^{1}M.V. Lomonosov MSU, Russian Federation, RCC, Russian Federation

Glancing angle deposition (GLAD) is one of the technique for the fabrication of the anisotropic thin films with high porosity and low refractive index. In this technique the incoming flux of the deposited atoms is directed almost parallel to the substrate surface. Due to density fluctuation in the deposited atoms flow and shadow effect, it results in formation of the different separate nanostructures - tree-like column, slanted column, chevron-like structures and so on – on the substrate. GLAD films are widely used in the optical coating due to low reflectance and anisotropy properties. The structural and optical properties of GLAD-films essentially depend on their fabrication conditions. Experimental investigation of these dependencies is still challenge for the existing experimental techniques. On the other hand due to the progress in high performance computing, now it is possible to study thin films deposition process using the atomistic simulation. In the present work the anisotropy of the GLAD SiO2 films is investigated using the classical atomistic simulation and anisotropic Bruggeman effective medium theory. The depolarizing factors defining the difference of the refractive index components are calculated based on the geometry parameters of the pores between the slanted columns forming the large-scale GLAD structure. Averaged shape parameters of these ellipsoids are defined using the density gradient tensor. It is revealed that the values of difference of main components of refractive index tensor are equal 0,03 and 0,05 for deposition angles 60 grad and 80 grad, if the free volume fraction is calculated using the dependence of film density on the deposition angle. The obtained values coincide with experimental results for silicon dioxide films. Simulation is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University.

The work is supported by the Russian Science Foundation (grant number 19-11-00053).

Jorge Olivares Funes^{1}
, Pablo Martin^{2}
, Elvis Valero^{3}

^{1}University of Antofagasta, Department of Mathematics, Chile

^{2}University of Antofagasta, physics department, Chile

^{3}Universidad de Tarapacá, matemáticas, Chile

Consider $$A∂z/∂y+B∂z/∂x=0$$, Where A, B 𝜖 R. We will solve these types of linear partial differential equations using the GeoGebra Graphical Calculator and their solutions will be seen by GeoGebra 3D. The use of GeoGebra in the study of PDE is a subject that is still being studied and developed with great potential.

Andrei Melekhin^{1}

^{1}National Research Moscow State University of Civil Engineering , heat, gas supply and ventilation, Russian Federation

The use of renewable energy sources to generate electricity is a hot topic. The author has developed a mathematical model of the aerodynamic process of a wind power plant with the solution of a multi-criteria optimization problem. The optimal range of controlled parameters affecting the aerodynamic process with a minimum amount of blown surface and maximum electrical energy production is determined. Regularities of aerodynamic process are established. The convergence of the results of the study in the calculation on the basis of theoretical dependencies and the solution of the mathematical model is determined. To find the optimal controlled parameters of the aerodynamic installation, a complex research method developed by the author is applied, based on multi-criteria optimization of parameters with the introduction of empirically obtained data.The preliminary procedure of IOSO NM 3.8 consists in the formation of an initial plan of the experiment, which can be implemented both in a passive way (using information about various parameters, optimization criteria and constraints obtained earlier) and in an active way, when too much is generated in the initial search area in accordance with a given distribution law. For each vector of variable parameters, the values of optimization criteria and constraints are determined by direct reference to the mathematical model of the object under study. The number of points that make up the initial plan of the experiment depends on the dimension of the problem and the chosen approximation functions. The solution results in a Pareto (optimal) set of solutions.

Mojtaba Ashhadi^{1}

^{1}University of Sistan and Baluchestan, physics, Iran (Islamic Republic of)

The spin-dependent transport properties through a single aromatic molecule sandwiched between two ferromagnetic (FM) electrodes is investigated theoretically. The transmission probability, current–voltage characteristic and tunnel magnetoresistance (TMR) are analyzed by the tight-binding Hamiltonian model and the nonequilibrium Green’s function technique. It is shown that all these characteristics are sensitive to the quantum interference effects originated from the molecule-to-electrode coupling. The spin-dependent transport properties are characterized by several significant factors. One of the important factors in the spin-dependent transport properties of single molecules is the effect of quantum interference that has recently attracted much attention in recent years [1-2]. The effect of quantum interference associated with the molecule-to-electrode interface geometry. In other words, such effects occur when the electronic wavefunctions propagating along the various pathways through ferromagnetic junction. [1] Guimarães M H D, Zomer P J, Vera-Marun I J, van Wees B J 2014 Nano Lett. 14 2952 [2] Stadler R 2009 Phys. Rev. B 80 125401

Jorge Olivares Funes^{1}
, Elvis Valero^{2}

^{1}University of Antofagasta, Department of Mathematics, Chile

^{2}Universidad de Tarapacá, matemáticas, Chile

Let's consider the differential equations of the shape $$ -\ \frac{d}{\ dx}(p (x)\frac{dy}{dx\ })\ + q(x)y = f(x)$$, $$ y(0)=y(a)=0, a>0.$$ Using GeoGebra software and the numerical methods of finite elements and finite differences, We will display the various numerical approximations they get for each value of "a" along with their absolute and relative error in various applets and examples.

Duong Ngoc Hai^{1}
, Nguyen Quang Thai^{2}

^{1}Vietnam Academy of Science and Technology (VAST), Graduate University of Science and Technology, Viet Nam

^{2}Vietnam Academy of Science and Technology (VAST), Institute of Mechanics, Vietnam

While an object at ambient temperature moving within a fluid, if the relative speed between the object and fluid is large enough, due to evaporation the natural vapour cavities can be appeared on the object’s surface. The mixture flows of such fluids and cavities are called as the natural cavitating flows or cavitating flows. The mathematical modelling for this kind of flows are usually complex because of the transient of laminar to turbulent region in flow near body wall, and moreover the existence of phase transition and vapour cavity with changed shapes. In this case, the mathematical models will contain pairs of models, such as flow pattern model for turbulent transient flows and cavitation model for cavitating flows to achieve the correct calculation results. In this paper, the typical mathematical models of cavitating flow around a slider body in water based on the combining of two possible turbulent flow models (LES – Large Eddy Simulation and RAS – Reynolds Averaged Simulation) and, to a pair, three cavitation models (Kunz, Schnerr-Sauer and Merkle models) are presented. Based on those the numerical simulations for cavitating flow around the two different shape bodies (hemisphere head body and sphere shape bodies) at same flow condition (cavitation number σ = 0.2) are performed by using each above mathematical model. The comparisons of numerical results with published experimental measurement results are performed to evaluate the effect of body shapes (existence of the cylinder body) and the accuracy of numerical results. The paper results might be helpful for investigation of cavitation phenomena

This work was supported partly by Grant of the NCVCC42.02/20-20 from the Vietnam Academy of Science and Technology (VAST).

Sergei V Ryzhkov^{1}
, Victor Kuzenov^{2}
, Sanya Dobrynina^{3}
, V. Shumaev^{4}
, Andrey Starostin^{5}

^{1}Bauman Moscow State Technical University, Thermal Physics Department, Russian Federation

^{2}Dukhov VNIAA, , Russian Federation

^{3}BMSTU, , Russian Federation

^{4}BMSTU, Thermal Physics Department, Russian Federation

^{5}BMSTU, , Russian Federation

A method is proposed for the transition from a tetrahedral to a hexagonal irregular computational grid. A variant of the elliptic “regularizer” of the grid is developed, which is based on the “mechanical analogy” and is based on the solution of linear equations of the theory of elasticity. The paper presents the initial results of the reconstruction and “regularization” of the computational grid, as well as the distribution of the “angular” criterion for assessing its quality. The hexagonal “regularized” computational grid is shown, as well as the distribution of the “angular” criterion for assessing its quality for the geometric model of a hypersonic aircraft. From the calculation results it follows that the “regularized” grid fills almost the entire volume of the computational domain, and the criterion for assessing the quality of the hexagonal “regularized” computational grid is more than 0.7. Moreover, to achieve this result, ~20 iterations were required only.

Yuriy Gubarev^{1}
, Shuang Sun^{2}

^{1}Lavrentyev Institute for Hydrodynamics, Laboratory for Fluid and Gas Vortex Motions, Russian Federation

^{2}Novosibirsk National Research State University, Department for Differential Equations, Russian Federation

The Vlasov-Poisson model of boundless collisionless gas of neutral particles in a self-consistent gravitational field continues to be one of the basic models of modern astrophysics. This is due to simplicity, clarity, and obvious effectiveness of the model in describing large-scale processes in the Universe. Despite the fact that this model has been intensively studied for a long time, from the point of view of the mathematical stability theory, it was possible to establish, by and large, only sufficient conditions for the theoretical stability (at semi-infinite time intervals) of a number of dynamic equilibrium states with respect to both small and finite perturbations, but from incomplete unclosed subclasses. In this work, we consider the spatial motions of the boundless collisionless self-gravitating Vlasov-Poisson gas of neutral particles in a three-dimensional Cartesian coordinate system: $$\frac{\partial f}{\partial t} + v_i\frac{\partial f}{\partial x_i} - \frac{\partial \varphi }{\partial x_i}\frac{\partial f}{\partial v_i} = 0, \frac{\partial ^2\varphi }{\partial x^2_i} = 4\pi \int \limits _{\mathbb{R}^3}f({\bf x}, {\bf v}, t)d{\bf v} \ (1)$$ $$i = 1, 2, 3; f = f({\bf x}, {\bf v}, t) \geq 0; f({\bf x}, {\bf v}, 0) = f_0({\bf x}, {\bf v})$$ Here $f$ denotes the distribution function of neutral particles (for reasons of convenience, their masses are assumed to be the same and equal to unity); $t$ is time; ${\bf x} = (x_1, x_2, x_3)$ and ${\bf v} = (v_1, v_2, v_3)$ denote coordinates and velocities of neutral particles; $\varphi ({\bf x}, t)$ is the potential of a self-consistent gravitational field; $f_0({\bf x}, {\bf v})$ denotes the initial data for the function $f$. We suppose that the distribution function $f$ asymptotically approaches zero as $|{\bf v}| \rightarrow \infty $, and this function along with the potential $\varphi $ are periodic in argument ${\bf x}$ or asymptotically approach zero as $|{\bf x}| \rightarrow \infty $ too. Summation is performed on repeating lower index $i$ throughout the work. It is assumed that the mixed problem (1) has the following exact stationary solutions: $$f = f^0({\bf x}, {\bf v}) \geq 0, \varphi = \varphi ^0({\bf x}) \ (2)$$ $$v_i\frac{\partial f^0}{\partial x_i} = \frac{\partial \varphi ^0}{\partial x_i}\frac{\partial f^0}{\partial v_i}, \frac{\partial ^2\varphi ^0}{\partial x^2_i} = 4\pi \int \limits _{\mathbb{R}^3}f^0({\bf x}, {\bf v})d{\bf v}$$ The aim of this work is to prove the absolute linear instability for the spatial states of dynamic equilibrium (2) of the boundless collisionless self-gravitating Vlasov-Poisson gas with respect to small three-dimensional perturbations $f^\prime ({\bf x}, {\bf v}, t)$ and $\varphi ^\prime ({\bf x}, t)$: $$\frac{\partial f^\prime }{\partial t} + v_i\frac{\partial f^\prime }{\partial x_i} - \frac{\partial \varphi ^\prime }{\partial x_i}\frac{\partial f^0}{\partial v_i} - \frac{\partial \varphi ^0}{\partial x_i}\frac{\partial f^\prime }{\partial v_i} = 0 \ (3)$$ $$\frac{\partial ^2\varphi ^\prime }{\partial x^2_i} = 4\pi \int \limits _{\mathbb{R}^3}f^\prime ({\bf x}, {\bf v}, t)d{\bf v}; f^\prime ({\bf x}, {\bf v}, 0) = f_0^\prime ({\bf x}, {\bf v})$$ where $f_0^\prime ({\bf x}, {\bf v})$ denotes the initial data for the function $f^\prime $. In the work, a transition from kinetic equations (1) which describe the motion of the gas under study to an infinite system of relations similar to the equations of isentropic flow of a compressible fluid medium in the “vortex shallow water” and Boussinesq approximations was carried out. In the course of the instability proof, the well-known sufficient Newcomb-Gardner-Rosenbluth condition for stability of dynamic equilibrium states (2) with respect to one incomplete unclosed subclass of small spatial perturbations was conversed. Also, a linear ordinary differential second-order inequality with constant coefficients was obtained. An a priori exponential lower estimate for the growth of small three-dimensional perturbations (3) follows from this inequality when the sufficient conditions for linear practical instability of the considered dynamic equilibrium states found in this work are satisfied. Since the obtained estimate was deduced without any additional restrictions on the equilibrium states under study, then the absolute linear instability of the spatial states (2) of the dynamic equilibrium of the Vlasov-Poisson gas with respect to small three-dimensional perturbations (3) was thereby proved. The results of the work are fully consistent with the classical Earnshaw instability theorem. This theorem states that any equilibrium configuration of point electric charges is unstable if, besides its own Coulomb forces of attraction and repulsion, no other forces act on them. Now the area of applicability for the Earnshaw theorem is expanded from electrostatics to kinetics, namely, to the boundless collisionless self-gravitating Vlasov-Poisson gas of neutral particles. Constructiveness is inherent in the sufficient conditions for linear practical instability established here, which allows them to be used as a testing and control mechanism for conducting physical experiments and performing numerical calculations.

yehuda Roth^{1}

^{1}Oramin academic college, Sceince, Israel

Based on entangled states, quantum computers have the advantage of simultaneously implementing a large number of processes. The coherence of entanglement enables a single operator (logical gate) to be activated simultaneously on all of the states in the superposition Consequently, to implement a quantum computer, a quantum algorithm has to be implemented. In this paper, we propose a different quantum approach that can simultaneously analyze a large amount of data. Although our process allows many processes to work simultaneously, it is not within the conventional frame of quantum computers.

Mikhail Dokukin^{1}

^{1}Bauman Moscow State Technical University, Department of Physics, Russian Federation

Need of a research of processes for the heavy current discharge with the refractory hollow cathode when pumping through it inert working gas is dictated by the requirements of technologies of welding and melting of the chemically active metals in a vacuum. In the offered work the experimental and theoretical research of the magnetooperated hollow cathode for the purpose of determination of the intra cathodic plasma parameters and the influence of the last on the power characteristics of the external discharge is conducted. Good compliance of the settlement quantities of this plasma with the results of experiments on the model device is obtained. Recommendations for improvement of the operational characteristics of the used arc vacuum technological devices are made.

Julio S Espinoza-Ortiz^{1}
, Roberto E. Lagos-Monaco^{2}

^{1}Federal University of Goias-Catalão, Physics, Brazil

^{2}IGCE-Universidade Estadual paulista, Physics, Brazil

In open billiards a particle can escape from the cavity through a leak. This type of systems have received special attention because of their applications to a wide variety of physical phenomena ranging from hydrodynamics to quantum chaos and astronomy. Chaotic leaked billiards are characterized by a so called transient behavior, i.e. by the presence of chaotic motion with a finite life time impossible to be studied just through the analysis of its asymptotic behavior. Under this scenario, we consider the quarter stadium billiard and study the influence of its leaking marginal unstable periodic orbits on the survival trajectories. A rigorous statistical analysis of the survival probability is presented. To pursue this objectives, we set up the classical trajectories' solution in such a way that the system only depends on its partial separability and then from it we pass to construct the Birkhoff map. The possibility of more than one leak into a billiard is also considered.

L. Diógenes T. Câmara^{1}

^{1}IPRJ_UERJ, Dep. Mechanical Eng. and Energy, Brazil

Authors Tedesco, J.C.G. and Câmara, L.D.T. Affiliations Polytecnique Institute of UERJ, Nova Friburgo-RJ, Brazil Abstract Magnetic refrigeration is a new promising technology based on the magneto-caloric effect of solid materials like gadolinium that offers smaller global environmental impact if compared to conventional refrigeration vapor compression processes which utilizes in general the ozone depletion chlorofluorocarbons refrigerants. The rotary refrigerators presents a new challenge in terms of complexity if compared to reciprocating ones which is compensated by refrigeration capacity, steady process operation, performance etc. The modeling and simulation of magneto-caloric refrigerator processes can provide important data in the development and optimization of the experimental units which are in general the only research step carried out by the researchers. A novel full process simulator of a magneto-caloric refrigerator was implemented to simulate the process performance over different conditions of rotating frequency, pump flow rate, room temperature etc. A stepwise modeling approach of the clockwise regenerator movement was implemented which simplifies the phenomena of heat transfers in the regenerators leading to ordinary differential equations which are solved more easily if compared to the partial differential equations in general applied to such complex process. The magneto-caloric bed material utilized in the rotating clockwise wheel was gadolinium with an anticlockwise closed flow loop of water which percolates the six gadolinium porous beds and also the hot and cold heat exchanger. The simulator was able to represent the transient aspects as well as the steady state conditions of the magneto-caloric refrigerator processes in terms of both low time performance and numerical stability. The inversion in heat transfer profiles along the process can be used as a limit in the calculation of the maximum heat transfer absorption in the refrigerator cold exchanger according to the operating conditions assumed.

Mathew Aibinu^{1}

^{1}Durban University of Technology, Institute for Systems science, South Africa

Real life problems are governed by equations which are nonlinear in nature. Nonlinear equations occur in modeling problems, such as minimizing costs in industries and minimizing risks in businesses. A technique which does not involve the assumption of existence of a real constant whose calculation is unclear is used to obtain a strong convergence result for nonlinear equations of (p, η)-strongly monotone type, where η > 0, p > 1. As a consequence of the main result, the solutions of convex minimization and variational inequality problems are obtained. This solution has applications in other fields such as engineering, Physics, Biology, Chemistry, Economics and game theory.

Simona Moldovanu^{1}

^{1}Dunarea de Jos University, Computer Science and Information Technology, Romania

A new approach for the digital watermarking process is proposed to be part of the pre-processing stage of a computer-aided diagnosis system. We propose to embed a denoised image acting as watermark image in the original host image with the final goal of improving the quality of demoscopic images for further image processing operation related to CAD. The proposed algorithm uses Discrete Wavelet Transform (DWT) corroborated with some basic properties of Human Visual System such as Contrast Sensitive Function (CSF) and Noise Visibility Function (NVF) with the goal of correlating the texture properties and noise. This approach hides the watermark (i.e. denoised version of the host image) in high-pass subbands that are focused on image features. The main concern is to evaluate the distortion produced to the host image by watermarking and an objective quality measure function, i.e. Weighted Peak Signal-to-Noise Ratio (WPSNR), is used to evaluate the existing differences between the original and watermarked images. The proposed approach is tested using the available skin lesion images from the digital image archive of the Department of Dermatology of the University Medical Center Groningen. The experiment results show the improved performance of the proposed scheme against a 3 3 median filtering attack in comparison with the a 5 5 median filtering attack.

Pablo Martin^{1}
, Jorge Olivares Funes^{2}
, Adrian Sotomayor^{3}

^{1}University of Antofagasta, Physics department, Chile

^{2}University of Antofagasta, Department of Mathematics, Chile

^{3}University Antofagasta, Mathematic, Chile

In a previous work a method to approximate modified Bessel functions of integer order were presented and applied to the modified Bessel function I1(x) [1]. An improvement to this technique is carry out in the present work. The procedure is, as in previous work , to use simultaneously power series and asymptotic expansions . Starting from both expansions an analytic function is determined as a combination of rational functions and elementary ones [2,3]. This last one in the present work was selected as the hyperbolic function sinh(x), this change produce a lot differences Fist ( where x is the independent variable) , and in previous work was cosh(x)[1]. The parameters of the actual rational function are different to those presented in our previous work . Besides, the important point, is that the efficiency now is better, and the results , using the same number of parameters that in previous one , have now much lower error . In the simplest case of the three parameters , the largest relative error is now 0.007 instead of 0.011 . The evolution of both relative errors can be shown graphically as a function of the independent variable . There is a confluence of the relative errors of both approximations for large values of the independent variable , and this relative error is about 0.004 . Reference [1]. P. Martin, J. Olivares and A. Sotomayor. “Precise analytic approximation for the modified Bessel function I1(x)”. Rev. Mex. Fisica 63, 130-133 (2017). [2]. P. Martin, JJ. Rodriguez-Nunez, JL. Marquez, “Two-dimensional hydrogenlike atoms in the presence of a magnetic field: Quasifractional approximations”. Phys. Rev. B45 , 8359-8362 (1992). [3]. P. Martin, E. Castro, J.L. Paz, “Multi-point quasi-rational approximants for the energy eigenvalues of two-power potentials “. Rev. Mex. Física 58, 301-307 (2012).

Alexey Grushevskii^{1}

^{1}Keldysh Institue of Applied Mathematics - KIAM, Russian Academy of Sciences, Russian Federation

The design of interplanetary trajectories using a series of gravity assists maneuvers begins with the ballistic mission design. It is reasonable to construct the corresponding initial approximation using the patched conics method within the model of the circular restricted three-body problem. Such a construction requires the calculation of the “transfer parameter” Vinf - the asymptotic velocity of the spacecraft relative the target planet, when switching from heliocentric arcs to planetocentric segments and vice versa. In the circular restricted three-body problem model, can be calculated using the Jacobi integral J (or using it's analogue - the Tisserand parameter Ti ) and the basic property of the Jacobi integral for the gravity assists maneuvers within the framework of the circular restricted three body problem: J=3 - Vinf*Vinf . According to this property, the J value does not change during the multiple gravity assists maneuvers that preserve the Jacobi integral constant are performed. This fact is known in astrodynamics but it is classically derived in a rather cumbersome method. In this study, a shorter method for it’s obtaining is proposed. The modifications of the representation of the Jacobi integral in the circular restricted three-body problem for the various configurations of three bodies and the table of transformations of the Jacobi integral and the Tisserand parameter are presented for all cases.

Irina Andreeva^{1}

^{1}Peter the Great St.Petersburg Polytechnic University, Higher Mathematics, Russian Federation

Dynamic systems in applications are useful as mathematical models of those processes and phenomena, where statistical events, or fluctuations, may be disregarded. Dynamic systems may be divided into the two main categories - the systems with continuous time (the flows), and systems with discrete time (the cascades). During the investigations of, first of all, flows, normal autonomous systems of ordinary differential equations are used. The present work is devoted to the original rigorous research of some important family of dynamic systems having reciprocal polynomial right parts, which are the forms of their phase variables. The whole wide family under consideration is being split into numeric subfamilies belonging to different hierarchical levels, and is subjected to the first and the second Poincare transformations, or mappings. As a result, the full qualitative pattern of trajectories is constructed - using the Poincare sphere - in the Poincare disk. A series of new special investigation methods was developed, useful for further investigations of similar dynamic systems’ classes. References. 1. Andreeva,Irina, Andreev, Alexey. Investigation of a Family of Cubic Dynamic Systems. //Vibroengineering Procedia. Vol.15. Dec.2017. Pp.88-93. 2. Andreeva I.A., Andreev A.F. Phase Portraits of One Family of Cubic Systems in a Poincare Circle. I.//Vestnic RAEN. 2017. Vol.17. №4. Pp.8–18. 3. Andreeva I.A., Andreev A.F. Phase Portraits of a Family of Cubic Systems in a Poincare Circle.I I.//Vestnic RAEN. 2018. Vol.18. №4. Pp.11–15. 4. Andreev A.F., Andreeva I.A. Phase Portraits of Some Family of Cubic Dynamic Systems in a Poincare Circle. III.//Vestnic RAEN. 2019. Vol.19. №2. Pp.20–24. 5. Andreev A.F., Andreeva I.A. On a Behavior of Trajectories of a Certain Family of Cubic Dynamic Systems in a Poincare Circle. // IOP Journal of Physics, Conference Series, 2018, 1141. 6. Andreeva I.A., Efimova T.O. Phase Portraits of a Special Class of Dynamic Systems in a Poincare Circle.//IOP Journal of Physics, Conference Series, 2019, 1236.

Prof. Dr. Alexey F. Andreev, St.Petersburg State University, Russia

Oleksandr Denisov^{1}

^{1}Harbin Institute of Technology, Microwave Engineering, China

. Stealth coating is the antidote against of the radar. But it can be disclosed by the microwave radiometer because their job based on the measuring the radio-brightness contrast between Stealth object and the background of the environments (sky is cold, Earth is warm). This short report presents some math model for the simplest calculation of the possible disclosing distance till the Stealth object.

Authors wishing to acknowledge assistance from colleagues in Harbin Institute of Technology.

Valeriy Zakharov^{1}

^{1}Lomonosov Moscow State University, Mathematical Analysis, Russian Federation

Informatics reached extraordinary heights in its development. However, a satisfactory solid theoretical foundation for this science has not been established until now. The reason is the absence of satisfactory general definitions of the notions of information and information system. The distinguished peculiarity is inherent not to Informatics only. In Physics despite of its longer existence the situation with the solid theoretical foundation gets on in the same way: there are no satisfactory general definitions of the notions of energy and energy system. In 1964 the outstanding American physicist Richard Feynman in his famous lectures [Feynman R.F., Leighton R.S., Sands M. The Feynman Lectures on Physics. V. 1: Mainly Mechanics, Radiation, and Heat. − United States of America: Addison Wesley Publishing Company, 1964. − 270 p.] in §1 of Chapter 4 has written: «It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, …». The following conclusion may be done from this apparently not random coincidence: the cause of the described situation is founded not in Informatics itself and not in Physics itself, but in the absence of satisfactory general solid theoretical conception of the world and its being, in which the notions of information and energy might appear in some natural deductive way. Some joint (synthetic) closed in itself idea about the world and its being is expounded in the report, which gives the opportunity to invent the parallel definitions of the energy and the information not exceeding the bounds of the united world. This allows us to introduce some sufficiently general notion of the produciпg (conservatively-dynamic surrounded stream) system, described by the proper system of evolutional equations. In the capacity of important partial cases of such systems the proper notions of the energetic producing system and the informatic producing system are introduced. The explicitly analyzed examples of the heating stove (as the energetic producing system) and the personal computer (as the informatic producing system) expose the applicability of proposed idea to a generalized and formalized description of some wide class of systems really existing.

Dimitrios Vlachos^{1}

^{1}University of Peloponnese, Department of Informatics and Telecommunications, Greece

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