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.

Manuel Fiolhais^{1}

^{1}University of Coimbra, Department of Physics, Portugal

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)

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

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

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

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.

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.

Gordana Jovanovic Dolecek^{1}

^{1}Institute INAOE, Electronics, Mexico

Decreasing the sample rate by an integer is called decimation. Decimation find applications in audio signal processing, Analog/Digital (A/D) converters, Digital receivers, among others. The signal must be filtered by a digital filter, before decimation, to prevent aliasing which may deteriorate decimated signal. This filter is called anti-aliasing or decimation filter. The simplest decimation filter is a comb filter which has all coefficients equal to unity, and consequently does not require multipliers for its implementation. This filter is usually used in the first stage of decimation. The aliasing occur in the bands around comb zeros, called folding bands. In that way comb filter naturally provides the aliasing rejection in folding bands. However, this attenuation is not enough in many applications, and must be increased. Many methods were proposed to improve aliasing rejection in comb folding bands. The principal goal is keeping comb low complexity, while improving its aliasing rejection. In this paper we propose novel method based on filters with stepped triangular impulse responses (STIR). It is shown that STIR filters are multiplierless filters with three parameters. Using mathematical methods, it is elaborated how to choose the STIR filter parameters to get an improved comb aliasing rejection. The possible filter structures are also analyzed. The method is compared with other methods proposed in literature for comb aliasing rejection improvement,

Conacyt Mexico

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.

Guzel Bulgakova^{1}

^{1}Ufa State Aviation Technical University, Mathematics, 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 magneto hydrodynamics (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 email: 1 onanuga.olutayo@gmail.com

Anton Belyakov^{1}

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

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. It turns out that comparison of approximate stability boundaries of fourth order with exact stability domains are very close to each other, showing effectiveness of the method. Addition of small damping shifts the stability boundaries upwards, thus resulting to both stabilization and destabilization effects.

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 combination of elastic properties of the shell’s layers may influence essentially the manifestation of liquid viscoelasticity through a characteristic time of the pressure transient, which is highly sensitive to effective parameters of the pipe wall. The general conclusion, following from the data is that increase in the liquid relaxation time yields a marked decrease in the signal attenuation along the pipe.

The study was supported by the SCE (Research Authority Grant No. 03/Y15/T2/D3/Yr3).

Jorge Olivares^{1}

^{1}Universidad de Antofagasta, Matemáticas, Chile

(Jorge Olivares, Luis Cortes, Daniza Rojas, Dalia Escalier and María Drina Rojas) GeoGebra is one of the software that is used in the teaching of mathematics and differential equations. By using GeoGebra, which is a dynamic and educational software, one can acquire the expected learning, especially in linear first order differential equations, which is a lesson in which intense learning difficulties are experienced.In this paper, interactive animated applications have been made for students to learn through technological resources. References [1] Oldknow, A. & Tetlow, L. (2008). “Using dynamic geometry software to encourage 3D visualization and modeling” . The Electronic Journal ofMathematics and Technology, 2(1), 54-61. [2 ] Carter, J., & Ferrucci, B. (2009). “Using GeoGebra to Enhance Prospective Elementary School Teachers' Understanding of Geometry”. The Electronic Journal of Mathematics and Technology, 3(2) , 149-165.

Vyankat Pawar^{1}

^{1}Sunderrao Solanke Mahavidyalaya, Department of Physics, India

The complex permitting spectra and thermodynamic properties of 1,2-dichloroethane (DE) and n,n-dimethylformamide (NMF) has been obtained at 15°C temperature in the frequency range of 10 MHz to 30 GHz by using time domain reflectometry technique for 11 different concentrations of the system. A complex spectrum gives information regarding dielectric dispersion (ɛʹ) and dielectric absorption (ɛʺ), which indicates the purity of the liquid mixture. In binary mixture of polar liquids mixed together, there is a change in the energy of the system. This change in energy can be interpreted with thermodynamic parameters such as free energy of activation (G), molar enthalpy of activation (H) and molar entropy of activation (S).

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.

Pablo Martin^{1}

^{1}Universidad de Antofagasta, departamento de física, Chile

P. Martin, J. Olivares and L. Cortes, Universidad de Antofagasta, Chile. Fractional derivatives have become very important in several areas of Engineering as hysteresis cycle, etc. In this paper the solutions of linear nonhomogeneous differential equations of order α are considered . Simple initial conditions are taken first are more complicated forms are also analyzed later. Fractional derivatives of order α are studied for α=1/4,1/3,1/2,2/3 ,3/4 and 1. With simple conditions the solutions are combinations of hypergeometric functions of the type: 〖_1^ 〗F_2(a,b,c;t^3/9) and 〖_2^ 〗F_3(a_1,a_2;〖b_1,b_2,c;t〗^3/9). Caputo definition for fractional derivatives are used . The solutions are found using the Laplace transform first and the inverse transform later . The analysis using α as a rational number m/p do not lead to a general solution.

fractional derivatives , Airy, hypergeometric , electric engineering

Svetlana Absalyamova^{1}
, Natalia Ivanova^{2}
, Chulpan Mukhametgalieva^{3}
, Asiya Khusnullova^{4}

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

^{2}Kazan Federal University, , Russian Federation

^{3}Kazan Federal University, , Russian Federation

^{4}Kazan Cooperation Institute, , Russian Federation

Currently, the global economy is experiencing a global innovation gap, representing a set of significant disparities in the innovative development of individual countries. The global innovation gap is a consequence of a number of objective phenomena and processes taking place in different countries and regions. The study of the nature of the innovation gap implies the study of specific problems in the innovative development of countries, as well as the search for ways to solve them. This article examines the relationship between the global innovation index and the positions of national universities in international rankings. We have chosen a group of countries (developed, developing and transition economies) and their universities which ranked best in international rankings. The sample consisted of 24 countries, the university's rating was chosen as the dependent variable, and the country's rating on the global innovation index was selected as an independent variable. After the calculations, the linear pair correlation coefficient was set equal to 0.913, which indicates a high level of coupling between the variables. The coefficient of determination indicates that in 83.35% of cases the changes in x lead to a change in y. In other words, the accuracy of selecting the regression equation is high. The remaining 16.65% change in Y is due to factors not taken into account in the model (as well as by specification errors). The conducted studies confirmed the connection between the quality of education, the research sector of universities and the level of innovative development of countries. This made it possible to propose a set of measures to strengthen the positions of national universities in international rankings and increase their contribution to the transition of the Russian economy to an innovative development path.

Guo Liang^{1}
, Zhanmei REN^{2}
, QI GUO^{3}

^{1}Shangqiu Normal University, School of Physics and Electrical Information, China

^{2}South China Normal University, School of Information and Optoelectronic Science and Engineerin, China

^{3}South China Normal University, School of Information and Optoelectronic Science and Engineerin, China

Analytically discussed is the nonlinear propagations of spiraling elliptic beams in nonlocal nonlinear media with linear anisotropy by using an approach of two dimensional asynchronous fractional Fourier transform. The spiraling elliptic beams exhibit a kind of molecule-like libration due to the combined effects of the linear anisotropy and the orbital angular momentum. Depending on the anisotropy parameter of the media, the molecule-like libration mode has two different kinds of dynamical behaviors. When the anisotropy parameter of the media is a rational number, the spiraling elliptic beams evolve as the breathers and can recover their initial shapes after one propagation period. However, when the anisotropy parameter of the media is an irrational number, all the related parameters of spiraling elliptic beams such as the optical intensity, the beam width, and the angular velocity cannot evolve in a periodic manner, and no spiraling elliptic breathers exist in this case.

Guo Liang^{1}
, QI GUO^{2}

^{1}Shangqiu Normal University, School of Physics and Electrical Information, China

^{2}South China Normal University, School of Information and Optoelectronic Science and Engineerin, China

It is demonstrated that the orbital angular momentum (OAM) carried by the elliptic beam without the phase-singularity can induce the anisotropic diffraction (AD). The quantitative relation between the OAM and its induced AD is analytically obtained by a comparison of two different kinds of (1+2)-dimensional beam propagations: the linear propagations of the elliptic beam without the OAM in an anisotropic medium and that with the OAM in an isotropic one. In the former case, the optical beam evolves as the fundamental mode of the eigenmodes when its ellipticity is the square root of the anisotropic parameter defined in the paper; while in the latter case, the fundamental mode exists only when the OAM carried by the optical beam equals a specific one called a critical OAM. The OAM always enhances the beam-expanding in the major-axis direction and weakens that in the minor-axis direction no matter the sign of the OAM, and the larger the OAM, the stronger the AD induced by it. Besides, the OAM can also make the elliptic beam rotate, and the absolute value of the rotation angle is no larger than $\pi/2$ during the propagation.

This research was supported by the National Natural Science Foundation of China, Grant Nos.~11474109 and 11604199.

Spyridon Vossos^{1}
, Elias Vossos^{2}

^{1}NKUA, Chemistry, Greece

^{2}NKUA, Physics, Greece

“Any space of constant curvature is conformally flat; that is, has a metric that can be expressed as a multiple of the Euclidean metric” [1]. At the same way, Generalized Special Relativity (SR) relates the frames of Relativistic Inertial observers (RIOs) where the spacetime has the metric gI=diag(gI00, gI11, gI11, gI11). The parameter ωI (where ωI^2=gI11/gI00) is contained in the matrix (ΛI) of the Euclidean Closed Linear Transformation of complex spacetime (LSTT) [2]. If ωI is an imaginary number, the elements of ΛI are complex numbers. So, the corresponding spacetime is necessarily complex and there exists real Universal Speed UI (where UI^2=-c^2/ωI^2). In addition, the complex Cartesian Coordinates (CCs) of the theory may be turned to the corresponding real CCs, in order to be perceived by human senses. The new real transformation is not closed (the corresponding matrices ΛIR do not form a group) and the successive transformations produce Generalized Thomas Rotation. The specific value ωI=±i gives Vossos transformation (VT) endowed with Lorentz metric (for gI11=1) of complex spacetime and invariant speed of light in vacuum (UI=c), which produce the theory of Euclidean Complex Relativistic Mechanics (ECRMs) [3]. The corresponding real matrix (ΛIR) is the matrix of Lorentz Boost (ΛIR=ΛL). If ωI is a real number (ωI≠0) the elements of the ΛI are real numbers. So, the corresponding spacetime is real, but there exist imaginary UI. The specific value ωI=0 gives Galilean Transformation (GT) with invariant time, in which any other closed LSTT is reduced, if one RIO has small velocity wrt another RIO. Thus we unify Theories of Physics (TPs) such as Newtonian Physics (NPs) keeping the formalism of Einstein Relativity Theory (ERT). The generalized definition of Proper Time (τ) gives us the possibility to compute four-velocity, four-momentum, Relativistic Doppler Shift etc, building the whole structure of the Generalized SR and General Relativity (GR). For instance the Generalized Relativistic Energy of Rest Mass (m) is E=-m c^2/ωI^2. In case of NPs, the annihilation energy becomes infinite. Thus the Lomonosov-Lavoisier Law becomes clear theorem of NPs. In case that we relate accelerated observers with variable metric of spacetime, we have the case of Generalized GR. For being that clear, we produce the 1st Generalized Schwarzschild metric (1GSM) and 2nd Generalized Schwarzschild metric (2GSM), which are in accordance with any SR based on Generalized metrics gI and Einstein field equations [4]. We combine 2GSM with Modified Newtonian Dynamics (MOND) [5-9], achieving MOND relativization and we then pass to the case of flat spacetime with Lorentz metric (Minkowski space), extending MOND methods to ERT. We use Simple and Standard Interpolating Function (μ) to the Lorentzian-Einsteinian 2GSM for the explanation of the Rotation Curves in Galaxy NGC 3198 as well as the Solar system, eliminating Dark Matter [10-14]. Generally, our approach, in non rotating black hole, planetary and star system scale, coincides to the original Schwarzschild metric, while in galactic scale, it gives MONDian results. In universal scale, the gravitational field strength becomes negative producing slight antigravity. Moreover, we compute the corresponding Lagrangian, geodesics, equations of motion, precession of planets’ orbits etc, resulting formulas which are referred to any TPs. The used theory is open to any other modification as well as the Generalization can be extended to other metrics of spacetime such as Kerr metric, producing Generalized field strength of rotating mass systems. Besides, the theory gives us the possibility of using mathematical objects (such as μ) which have been produced by one specific TPs (such as NPs) to be suitably transformed and then effectively be used in another TPs (such as ERT).

Hossein Ghaffarnejad^{1}

^{1}Semnan University, Physics, Iran (Islamic Republic of)

We solve Wheeler-De Witt metric probability wave equation for the Schwarzschild de Sitter black hole by applying the perturbation method. We obtained leading order solution which reduces to maximal probability condition of the metric as quantized spherically symmetric slices on $`t=Constant`$ hyper-surfaces with radiuses $\frac{r_n}{2M}=e^{\frac{2\pi n}{\sqrt{|12a+1|}}};$ $n=0,\pm1,\pm2,\cdots,$ where `$2M$` is Schwarzschild radius of the black hole and $a<0$ is a suitable dimensionless integral constant. This result satisfies Penrose hypotheses for cosmic censorship where a causal singularity may be covered by a horizon surface.

Songmin Shang^{1}
, Ka Lok Chiu ^{2}

^{1}The Hong Kong Polytechnics University, Institute of Textiles and Clothing, Hong Kong

^{2}The Hong Kong Polytechnics University, Institute of Textiles and Clothing, Hong Kong

In this project, a cyclodextrin-based adsorbent was synthesized for the removal of ibuprofen (IB) from pharmaceutical sewage. Our adsorbent was prepared via solution blending of (2-hydroxypropyl)-β-cyclodextrin (HPBCD) and poly(vinyl alcohol) (PVA), followed by glutaraldehyde treatment. The obtained adsorbent is in the form of transparent film and is very easy to handle. Its FTIR and thermal analysis results show that it is well crosslinked, preventing it from dissolving in sewage. Our cyclodextrin-based adsorbent has good IB adsorption ability and can adsorb up to 2.56 mg/g IB. It has better IB adsorption ability than other absorbents including mesoporous silica, volcanic soil, and zeolite. Its IB adsorption capacity increases with the HPBCD content and IB concentration. It can remove IB effectively in acidic condition but it has a poor IB adsorption ability at high pH value. Its adsorption kinetics can be well described by the pseudo-second-order kinetic model, as shown in Figure 1. More than 90% of adsorbed IB molecules desorb within 1 hour by simply soaking the consumed sample in 5% ethanol-water solution. As shown in Figure 2, IB molecules are entrapped by HPBCD inside our eco-adsorbent via host–guest interaction [1]. This provides strong attraction between IB and our eco-adsorbent for IB adsorption. Meanwhile, this interaction is reversible so the desorption of adsorbed IB molecules from the eco-adsorbent is allowed.

Jaykov Foukzon^{1}

^{1}Israel Institute of Technology, Haifa, Israel, math, Israel

It seemed natural to identify gravitation singularities with singular values of the metric or curvature components and their scalar combinations [1]. However, under formal and mathematically abnormal calculation which known from a very old physics handbooks [2]-[3], such a notion depends on choosing a reference frame and includes fictious singularities which being real for some observers are absent for others.In a nutshell, there is a widespread belief that there exist true physical singularities and unphysical,i.e.,coordinate singularities. We try to base our approach to the problem of the gravitation singularities on the fact that a gravitation singularity leads by a natural way, directly to a singularity of a space-time structure in sense of Colombeau distributional geometry [4]-[6].We aim to describe gravitation singularities using mathematically rigorous approach via Colombeau nonlinear distributional space-time structures with distributional Levi-Civit‘a connection.We pointed out that some important physical singularities which many years mistakenly considered as coordinate singularities. [7]-[8] [1] D. Ivanenko and G. Sardanashvily,Foliation analysis of gravitation singularities, Physics Letters A Volume 91, Issue 7, 27 September 1982, Pages 341-344. https://doi.org/10.1016/0375-9601(82)90428-5 [2] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 7th ed. (Nauka, Moscow, 1988; Pergamon, Oxford,1975). [3] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, New York, 1973; Mir, Moscow, 1977). [4] M, Kunzinger, R. Steinbauer,Foundations of a Nonlinear Distributional Geometry, Acta Applicandae Mathematica,April 2002, Volume 71, Issue 2, pp 179–206. [5] J.A. Vickers,Distributional geometry in general relativity,Journal of Geometry and Physics 62 (2012) 692–705. [6] R. Steinbauer, Nonlinear distributional geometry and general relativity, https://arxiv.org/abs/math-ph/0104041v1 [7] J. Foukzon,Distributional Schwarzschild Geometry from Non Smooth Regularization Via Horizon,British Journal of Mathematics & Computer Science, ISSN: 2231-0851, 11(1): 1-28, 2015, Article no.BJMCS.16961,DOI : 10.9734/BJMCS/2015/16961 [8] J. Foukzon, A. Potapov and E. Menkova,Distributional SAdS BH-Spacetime Induced Vacuum Dominance,British Journal of Mathematics & Computer Science 13(6):1-54, 2016, Article no.BJMCS.19235,DOI:10.9734/BJMCS/2016/19235

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

^{1}Universidade Federal de Goias, Physics Department, Brazil

^{2}Universidade Estadual Paulista (UNESP), Departamento de Física IGCE, Brazil

We consider a special relativity scenario in order to study the dynamical behaviour of a particle flying under the influence of a gravitational field, colliding successive times against a rigid vibrating surface, via a restitution coefficient. We define two re-scaled dimensionless dynamical variables, namely: the relative particle velocity {$W$} with respect to the surface's velocity, and the real parameter {$\tau$} accounting for the temporal evolution of the system. In order to analyze the system's nonlinear dynamical behaviour, we construct the mapping at the particle-surface contact point, described for the {$k'th$} collision, by the couple of variables {$\left(\tau_{k}\,W_{k}\right)$}. Then, from the dynamical mapping, we compute the fixed point trajectory and analyze its stability. We find the dynamical behaviour of the fixed point trajectory to be stable or unstable, according to the values of the re-scaled vibrating surface amplitude, the restitution coefficient and the auxiliary variable {$\beta\in\left(0,1\right)$}, sweeping from the non relativistic to the ultra relativistic regimes. Other important dynamical aspects such as the phase space volume and the one cycle vibrating surface (decomposed into absorbing and transmitting regions) are also discussed. Furthermore, the model rescues well known results in the non relativistic limit.

The authors would like to thank the support of the Goiás Research Foundation - FAPEG.

Jorge Olivares^{1}

^{1}Universidad de Antofagasta, Matemáticas, Chile

With the arrival of technology in our time, several investigations have been carried out and several mathematical softwares and programs have been developed using computers in the mathematical teaching . The aim of this work is to solve fractional differential equations $(df^α (x))/(dx^α )=Ax+B$ , (A, B constants ) using the mathematical software geogebra.The initial condition is $f(0)=0 , 0< α<1$. Caputo definition for the fractional derivatives is considered.

Gauranga Samanta^{1}
, Parth Shah^{2}

^{1}Birla Institute of Technology and Science Pilani K K Birla Goa, Mathematics, CC-115, India

^{2}BITS Pilani K K Birla Goa Campus, Mathematics, India

In this talk, the dynamical systems analysis approach is applied; to study the behavior of the cosmological models in general theory of relativity and modified theory gravity will be discussed. The cosmological models are buildup with a set of complicated nonlinear differential equations. It is a challenging problem to find an exact solution of a system of nonlinear differential equations in cosmology. Therefore, in this talk the dynamical systems analysis techniques will be applied to study the qualitative behavior of the cosmological models through the system of nonlinear differential equations, which are involved in dark energy models from different aspects. One of the main objectives of this talk is to bridge the gap between dynamical systems analysis, an area of applied mathematics, and cosmology, an area of astrophysics and astronomy that studies the universe as a whole.

PRADYUMN SAHOO^{1}

^{1}Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Mathematics, India

Wormholes are tunnels connecting different regions in space-time. They were obtained originally as a solution for Einstein's General Relativity theory and according to this theory they need to be filled by an exotic kind of anisotropic matter. In the present sense, by ``exotic matter'' we mean matter that does not satisfy the energy conditions. In this article we propose the modelling of wormholes within an alternative gravity theory that proposes an extra material (rather than geometrical) term in its gravitational action. Our solutions are obtained from well-known particular cases of the wormhole metric potentials, named redshift and shape functions, and yield the wormholes to be filled by a phantom fluid, that is, a fluid with equation of state parameter $\omega<-1$. In possession of the solutions for the wormhole material content, we also apply the energy conditions to them. The features of those are carefully discussed.

Zainab Nayyar^{1}

^{1}Bahria University, Software Engineering, Pakistan

Few years back supercomputers were used for large computational work and they were used only for specific purposes, but after the completion of particular task one could not use them for any other general purpose and not more than one task could run on supercomputers at a time. After the invention of desktops it becomes easy to run more than one task but large computational work cannot be handled by them due to the limitation of resources. Due to these limitations, grid technology has emerged to a large scale to cater the jobs of supercomputers but without the resource limitation. This paper shows the implementation and working of a computational grid, by submitting jobs we can monitor the utilization of resources in grid environment.

The authors would like to thank the support of the UET Taxila.

Elena Solovyeva^{1}

^{1}St. Petersburg Electrotechnical University, Theoretical Electrical Engineering Department , Russian Federation

In the cases when the mathematical model of a device is complicated, or it can not be constructed because of the lack of sufficient information about the researched object, an approach based on the description of a unique relationship between the sets of input and output signals is used. Such situations often arise on modelling non-linear devices with high accuracy. The point is to approximate a non-linear operator establishing the unique input-output correspondence by mathematical constructions (multidimensional polynomials, regression models, neural networks). Recurrent neural networks possessing the properties of dynamics and nonlinearity are considered as mathematical models within the input-output approach. Among the different types of neural networks and multidimensional polynomials, it is a cellular neural network, that provides the highest accuracy of modelling a non-linear filter used to restore images distorted by non-Gaussian noise, for instance, by impulse one. It results from the comparison of root-mean-square errors obtained for the different non-linear filters models and for the different probability densities of impulse noise in images. It is shown, that there is possibility to rise the filtration accuracy by applying cascade filters with the different types of the included units models.

Yehuda Roth^{1}

^{1}Oranim College, Science, Israel

We identify a living system as one that, in order to survive, continual measurements on its surroundings (and on itself). We also define a conceptual observer that reads and interprets the output of the measurements. In a different discipline of science, quantum measurements are also associated with an observer who interprets reality through the collapse procedure. However, this time, the interpretation has nothing to do with survival skills. Nevertheless, since both disciplines belong to the science of Nature, we suspect they obey similar rules. Indeed, we demonstrate that both measurements follow one fundamental principle: they both correspond with re-coherent processes. Applying a quantum-like model, such as that of a spin glass neural network, we describe brain activity with the Hilbert space concept in what we refer to as quantum-like measurements.

Zainab Nayyar^{1}

^{1}Bahria University, Software Engineering, 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. This study addresses the reasons of trust deficiency among different organizations while availing the cloud services. For this purpose a mixed method approach is used in which qualitative data is gathered through semi structured interviews which are conducted from the cloud service providers to find out what difficulties they are facing while providing services to different organizations, whereas quantitative data is also gathered through questionnaires from those organizations which are availing cloud services. Quantitative data is analyzed through SPSS and qualitative data is analyzed through Nvivo. Results gathered after analysis provided remarkable measures/solutions that can be applied to increase trust level on cloud services. The study is very useful especially for the technological development of information security organizations that are providing cloud services to different organizations. It can not only give an economical increase to information security organizations but also stabilize the market position of the organization in information security domain.

The authors would like to thank the support of the Bahria University Islamabad.

Mahmoud El-Borai^{1}

^{1}Alexandria University Faculty of Science Alexandria Egypt, Mathematics and Computer Sciences, Egypt

Problems of the form: $$\partial ^α u_i (x,t))/(\partial t^α )=c L_i u_i (x,t)+c f_i (H(t),u)+ c^2 F_i (S(t),x,u),t\lt 0 ,0\lt α\leq 1 \\ u_i (x,0)=[φ] _i (x), i = 1,……,n$$ Are considered, where $L_1,L_2,...,L_n$ are elliptic partial differential operators of higher orders and with coefficients depending on $x={x_1,….,x_m } ,\infty \lt x_j \lt \infty, j=1,2,…,m$. It is supposed that H(t) ia an nxN matrix with elements that are statistically stationary processes with the property of ergodicity of the mean. It is supposed also that S( t) is a matrix of order nxN with stochastic processes elements. The existence and asymptotic behavior of solutions as c tends to zero are studied.

Irina Andreeva^{1}
, Alexey Andreev^{2}

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

^{2}St. Petersburg State University, Differential Equations, Russian Federation

We present the results of the investigation of a family of dynamic systems on a real plane containing reciprocal cubic and square polynomials in their right parts. A Poincare method of serial mappings have been used. All possible for the systems under consideration topologically different types of phase portraits in a Poincare circle have been described and constructed. The total amount of such portraits appears to be over 200. Coefficient criteria of their realization have been outlined.

Jaykov Foukzon^{1}

^{1}Israel Institute of Technology, Haifa, Israel, math, Israel

The vacuum energy density of free scalar quantum field Ф in a Rindler distributional spacetime with distributional Levi-Civita connection is considered. It has been widely believed that, except in very extreme situations, the inuence of acceleration on quantum elds should amount to just small, sub-dominant contributions. Here we argue that this belief is wrong by showing that in a Rindler distributional background spacetime with distributional Levi-Civita connection the vacuum energy of free quantum elds is forced, by the very same background distributional spacetime such a Rindler distributional background spacetime, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on a Rindler distributional space-time with distributional Levi-Civita connection. In particular we obtain that the vacuum uctuations <Ф^2> has a singular behavior on a Rindler horizon. Therefore suciently strongly accelerated observer burns up near the Rindler horizon. Thus Polchinski's account doesn't violation of the Einstein equivalence principle.

Joao Fialho^{1}

^{1}British University of Vietnam, Mathematics, Viet Nam

The mathematical modelling of idiopathic scoliosis has been studied throughout the years. The models presented on those papers are based on the orthotic stabilization of the idiopathic scoliosis, which are based on a transversal force being applied to the human spine on a continuous form. When considering robotic traction devices, the existent models cannot be used, as the type of forces applied are no longer transversal nor applied in a continuous manner. In robotic devices, vertical traction is applied and in addition parameters such as magnitude, direction and angle of the force applied are required and and essential if the best therapy plan is to be administered. In this study we propose a mathematical model to the idiopathic scoliosis, using robotic traction devices, and with the parameters obtained from the mathematical modeling, set up a case-by-case individualized therapy plan, for each patient. To the best of our knowledge, modelling involving these assumptions was never investigated before, neither was the usage of modeling to establish viable and effective bounds for all he possible parameters in a robotic traction device.

Tom Burr^{1}

^{1}International Atomic Energy Agency, Safeguards/Information Management, Austria

Approximate Bayesian Computation (ABC) is a candidate approach when an explicit likelihood for the data is not available, but there is a model that outputs predicted observables for specified input parameters. This paper briefly reviews ABC and then describes a few measurement methods used in nuclear safeguards, some of which have, and others of which do not have corresponding likelihoods. A key aspect of metrology is uncertainty quantification (UQ), whether approached from physical first principles (“bottom-up”) or empirically by comparing measurements from different methods and/or laboratories (“top-down”). Although ABC is not yet commonly used in metrology, these example measurement methods in nuclear safeguards illustrate advantages in using ABC compared to current bottom-up approaches. ABC is also shown to be useful in top-down UQ approaches. As a diagnostic, in bottom-up and top-down applications of ABC, the actual coverages of probability intervals are compared to the true coverages. If an ABC-based interval for a parameter is constructed to contain approximately 95% of the true parameter values, then it is important to verify that the actual coverage is close to 95%. It is shown that one advantage of ABC is its apparent robustness to misspecifying the likelihood while still maintaining good agreement between nominal and actual probability interval coverage compared to other Bayesian approaches.

T. Mnatsakanov^{1}

^{1}Moscow Power Engineering Institute, , Russian Federation

Analytical expression describing the dependence of the p+-n junction leakage current on the doping level of the highly doped p+- type region is derived taking into account a whole set of nonlinear physical effects: electron-hole scattering, Auger recombination, band-gap narrowing, and nonlinear dependences of charge carrier lifetime and mobility on the doping level. It is shown that the dependence of the leakage current has a minimum at which the injection ability of the p+- type emitter is at a maximum. The dependence of the extremum position on the main parameters of the heavily doped p+- layer is analyzed. The data obtained make it possible to optimize the structure of high power silicon devices and to facilitate the adequacy of numerical calculations.

T. Mnatsakanov^{1}

^{1}Moscow Power Engineering Institute, , Russian Federation

Fast and ultrafast switch-on processes in p-n diodes at very high current densities j and very short current rise time t0 have been investigated. It is demonstrated that even at relatively modest values of the rate of current density rise dj/dt, the switch-on transient fundamentally differs from that in the "classical" quasineutral mode. The switch-on processes occur in modes with broken neutrality. The base resistivity is modulated by fast hole and electron waves that propagate toward each other at the maximum possible (saturated) velocity. The conditions for the appearance of fast waves modes have been established. Analytical expressions for the initial stage of these waves propagation have been derived. Adequate computer simulation confirms the analytical results.