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Natela Zirakashvili, Analytical Solution of Interior Boundary Value Problems of Elasticity for the Domain Bounded by the Parabola, Bulletin of TICMI Vol. 20, No. 1, 2016, 3–24. , Tbilisi University Press, 2016
Exact solution of two dimensional problems of elasticity are constructed in the parabolic coordinates in domain bounded by coordinate lines of the parabolic coordinate system. Here we represent internal boundary value problems of elastic equilibrium of the homogeneous isotropic body bounded by coordinate lines of the parabolic coordinate system, when on parabolic border normal or tangential stresses are given. Exact solutions are obtained using the method of separation of variables. Using the MATLAB software numerical results and constructed graphs of the mentioned boundary value problems are obtained.
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Natela Zirakashvili, Lamara Bitsadze, Explicit Solutions of the Boundary Value Problems for an Ellipse with Double Porosity, Advances in Mathematical Physics Volume 2016, Article ID 1810795, 11 pages http://dx.doi.org/10.1155/2016/1810795, Hindawi Publishing Corporation, 2016
The basic two-dimensional boundary value problems of the fully coupled linear equilibrium theory of elasticity for solids with double porosity structure are reduced to the solvability of two types of a problem. The first one is the BVPs for the equations of classical elasticity of isotropic bodies, and the other is the BVPs for the equations of pore and fissure fluid pressures. The solutions of these equations are presented by means of elementary (harmonic, metaharmonic, and biharmonic) functions. On the basis of the gained results, we constructed an explicit solution of some basic BVPs for an ellipse in the form of absolutely uniformly convergent series.
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George Jaiani, On micropolar elastic cusped prismatic shells, Transactions of A. Razmadze Mathematical Institute, 170, 376-384, Tbilisi University Press, 2016
A huge literature is devoted to the study of cusped prismatic shells on the basis of the classical theory of elasticity. It was stimulated by the works of I. Vekua. I. Vekua considered very important to carry out investigations of boundary value and initial
boundary value problems for such bodies, since they are connected with egenerate partial differential equations and, therefore, are not classical, in general. The present paper is devoted to cusped prismatic shells on the basis of the theory of micropolar elasticity. Namely, on the basis of the N = 0 approximation of hierarchical models for micropolar elastic cusped prismatic shells constructed by the I. Vekua dimension reduction method.
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George Jaiani, Vekua type hierarchical models for prismatic shells with mixed conditions on face surfaces, Composite Structures,152, 226-238, Elsevier, 2016
I. Vekua constructed hierarchical models for elastic prismatic shells, in particular, plates of variable thickness, when on the face surfaces either stresses or displacements are known. In the present paper other hierarchical models for cusped, in general, elastic isotropic and anisotropic prismatic shells are constructed and analyzed, namely, when on the face surfaces (i) a normal to the projection of the prismatic shell component of a stress vector and parallel to the projection of the prismatic shell components of a displacement vector, (ii) a normal to the projection of the prismatic shell component of the displacement vector and parallel to the projection of the prismatic shell components of the stress vector are prescribed. We construct also hierarchical models, when other mixed conditions are given on face surfaces. In the zero approximations of the models under consideration peculiarities (depending on sharpening geometry of the cusped edge) of correct setting boundary conditions at edges are investigated. In concrete cases some boundary value problems are solved in an explicit form. As an example of application of the constructed Vekua-type models to composite structures, an unidirectional lamina with fibers parallel to $x_2$-axis under shear strain is considered. Tension–compression is treated as well.
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Lamara Bitsadze, George Jaiani, On basic problems for elastic prismatic shells with microtemperatures , ZAMM - Z. Angew. Math. Mech., 96 (9), 1082–1088, Wiley, 2016
In the present paper on the basis of the linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures the zeroth order approximation of hierarchical models of elastic prismatic shells with microtemperatures in the case of constant thickness (but, in general, with bent face surfaces) is considered. The existence and uniqueness of solutions of basic boundary value problems when the projections of the bodies under consideration are bounded and unbounded domains with closed contours are established. The ways of solving boundary value problems in explicit forms and of their numerical solution are indicated.
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George Jaiani, A Model of Layered Prismatic Shells, Continuum Mechanics and Thermodynamics, 28, 765-784, Springer, 2016
The present paper is devoted to a model for elastic layered prismatic shells which is constructed by means of a suggested in the paper approach which essentially differs from the known approaches for constructing models of laminated structures. Using Vekua’s dimension reduction method after appropriate modifications, hierarchical models for elastic layered prismatic shells are constructed. We get coupled governing systems for the whole structure in the projection of the structure. The advantage of this model consists in the fact that we solve boundary value problems separately for each ply. In addition, beginning with the second ply, we use a solution of a boundary value problem of the preceding ply. We indicate ways of investigating boundary value problems for the governing systems. For the sake of simplicity, we consider the case of two plies, in the zeroth approximation. However, we also make remarks concerning the cases when either the number of plies is more than two or higher-order approximations (hierarchical models) should be applied. As an example, we consider a special case of deformation and solve the corresponding boundary value problem in the explicit form.
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Roman Janjgava, About the plane theory of poroelasticity for the binary mixture with double porosity, Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, 30, 2016, 31-34, Tbilisi University Press , 2016
We consider two-dimensional differential equations of the theory of binary mixtures in case of double porosity. The general solution of this system is represented by five analytic functions of a complex variable and solution of the Helmholtz equation. The general representation of the solution gives the opportunity to construct the analytical solutions of static boundary value problems.
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Roman Janjgava, Miranda Narmania, Some Statically Definable Problems For Cylindrical Shells, AMIM (Applied Mathematics, Informatics and Mechanics) 21(2), 65-73, Tbilisi University Press , 2016
In this paper we consider some statically definable problems for a cylindrical shell with constant thickness. The expand of middle surface of the shell on the plane is a rectangle. The shell is so thin that Hooke’s law does not apply. It means that, the body of the transverse stress field is assigned beforehand and for the tangential stress
components the system of equations is obtained. This system of equations is based on the physical boundary conditions and the problems are solved analytically.
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Roman Janjgava, About one method of construction of approximate solutions of some boundary value problems, Seminar of I. Vekua Institute of Applied Mathematics, Reports, 42,10-26;, Tbilisi University Press , 2016
A simple algorithm for construction of the approximate solution of some classical and nonlocal boundary value problems of the mathematical physics is considered. The efficiency of the offered algorithm for construction of the approximate solutions of problems is shown on the examples of two-dimensional classical and nonlocal boundary value problems of the theory of elasticity and for two-dimensional equations of Laplace and Helmholtz.
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Roman Janjgava, Miranda Narmania, One Effect for Bodies with Double Porosity in the case of Plane Deformation, Bulletin of TICMI, 20(1), 32-47, Tbilisi University Press , 2016
We consider the basic two-dimensional differential equations of static equilibrium poroelastic materials with double porosity. We construct the general solution of this system of equations by means of three analytic functions of a complex variable and solution of the Helmholtz equation. On the basis of the constructed general solution we have defined the
effect caused by pressures in a porous medium which is similar to temperature effect of Muskhelishvili.
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Roman Janjgava, Elastic Equilibrium of Porous Cosserat Media with Double Porosity, Advances in Mathematical Physics, Advances in Mathematical Physics, Article ID 4792148, 9p, Hindawi Publishing Corporation, 2016
The static equilibrium of porous elastic materials with double porosity is considered in the case of an elastic Cosserat medium. The corresponding three-dimensional system of differential equations is derived. Detailed consideration is given to the case of plane deformation. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of three analytic functions of a complex variable and two solutions of Helmholtz equations. The constructed general solution enables one to solve analytically a sufficiently wide class of plane boundary value problems of the elastic equilibrium of porous Cosserat media with double porosity. A concrete boundary value problem for a concentric ring is solved.
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Roman Janjgava, The Approximate Solution of Some Plane Boundary Value Problems of the Moment Theory of Elasticity, Advances in Mathematical Physics, Hindawi Publishing Corporation, Article ID 3845362, 12 p,, Hindawi Publishing Corporation, 2016
We consider a two-dimensional system of differential equations of the moment theory of elasticity. The general solution of this system is represented by two arbitrary harmonic functions and solution of the Helmholtz equation. Based on the general solution, an algorithm of constructing approximate solutions of boundary value problems is developed. Using the proposed method, the approximate solutions of some problems on stress concentration on the contours of holes are constructed. The values of stress concentration coefficients obtained in the case of moment elasticity and for the classical elastic medium are compared. In the final part of the paper, we construct the approximate solution of a nonlocal problem whose exact solution is already known and compare our approximate solution with the exact one. Supposedly, the proposed method makes it possible to construct approximate solutions of quite a wide class of boundary value problems.
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Miranda Narmania, The solution of some two-dimensional problems of thermoelasticity taking into account the microtemperature, Journal of Thermal Stresses, 39(1), 57-64, 2016, Taylor&Francis, Philadelphia,, 2016
In this work we consider the two-dimensional system of the differential equations describing the plane statical thermoelastic balance of homogenous isotropic elastic bodies, the microelements of which have microtemperature in addition to the classical displacement and a temperature field. It is construed the general solution of this system of the equations by means of analytic functions of complex variable and solutions of the equation of Helmholtz. The general representation of the solution obtained gives the opportunity to construct the analytical solutions of a number of plane boundary value problems of microthermoelasticity. As an example we consider the boundary value problem for a concentric circular ring.
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Roman Janjgava, Nuri Khomasuridze, Solution of some boundary value thermoelasticity problems for a rectangular parallelepiped taking into account microthermal effects, Meccacica, An International Journal of Theoretical and Applied Mechanics, 51(1), 211–221, Springer, 2016
A three-dimensional system of differential equations is considered that describes a thermoelastic equilibrium of homogeneous isotropic elastic materials, microelements of which, in addition to classical displacements and thermal fields, are also characterized by microtemperatures. In the Cartesian system of coordinates the general solution of this system of equations is constructed using harmonic and metaharmonic functions. Some boundary value micro-thermoelasticity problems are stated for the rectangular parallelepiped. An analytical solution of this class of boundary value problems is constructed using the above-mentioned general solution. When the coefficients characterizing microthermal effects are zero, the obtained solutions lead to the solutions of corresponding classical boundary value thermoelasticity problems, the majority of which have been solved for the first time. It should be noted that the aim of the given work is to construct an effective (analytical) solution for a class of boundary vale problems rather than to investigate the validity or applicability of the involved theory.
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Roman Janjgava, Miranda Narmania, Some Two-dimensional Thermoelasticity Boundary Value Problems for Cosserat Continuum with Microtemperature, Proceedings of I. Vekua Institute of Applied Mathematics, 66,19-23, Tbilisi University Press , 2016
In the present paper we consider the two-dimensional system of differential equations describing plane thermoelastic equilibrium for elastic bodies of Cosserat with microtemperature. The general solution of this system of equations is constructed using analytical functions of a complex variable and solutions of the Helmholtz equation.
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Dazmir Shulaia, P Ghurtskaia, General representation of solutions of the equation of penetration and diffusion of X-rays in plane geometry, Journal of Mathematical Sciences 218 (6), 829-833, Springer Science and Business Media Deutschland GmbH, 2016
In this paper, we present a general procedure for solving of homogeneous equations that describe penetration and diffusion of X-rays in plane geometry. Starting from Van Kampen’s and Case’s observation that it suffices that “solutions” be distributions, elementary solutions of a homogeneous equation are found. We also prove that general solutions can be obtained by superposition of elementary solutions.
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Dazmir Shulaia, Mariam Avalishvili, A spectral representation of the linear multivelocity transport problem, Georgian Mathematical Journal 23 (3), 329-341, De Gruyter, 2016
The transformation of the original characteristic equation of the multivelocity linear transport theory was carried out by expanding the scattering function for the problem to be solved as a spectral integral over a complete set of eigenfunctions for the previously solved transport problem. The obtained equation represents a singular integral equation containing a spectral integral over the spectrum of the solved problem, whose kernel depends on the difference between the scattering of the problem to be solved and that of the already solved problem. We consider also the examples illustrating the validity of such a transformation. M. Kanal and J. Davies made a similar transformation of the characteristic equation of the one-velocity transport theory.
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Dazmir Shulaia, Giorgi Makatsaria, Green’s Function for the Light Scattering Equations, Bulletin of TICMI 20 (1), 25-31, Tbilisi State University, 2016
The aim of this paper is to construct Green’s function in an infinite medium for
the light scattering equation. To this end the method of spectral resolution of the solutions
by the eigenfunctions of the corresponding characteristic equation is used.
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Temur Jangveladze, Mikheil Gagoshidze , Hopf Bifurcation and its Computer Simulation for One-Dimensional Maxwell's Model, Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, Vol. 30, 27-30, Tbilisi University Press, 2016
One-dimensional system of nonlinear partial differential equations based on Maxwell’s model is considered. The initial-boundary value problem with mixed type boundary conditions is discussed. It is proved that in some cases of nonlinearity there exists critical value $\psi_{c}$ of the boundary data, such that for $0< \psi < \psi_{c}$ the steady state solution of the studied problem is linearly stable, while for $\psi> \psi_{c}$ is unstable. It is shown that when $\psi$ passes through $\psi_{c}$ then the Hopf type bifurcation may take place. The finite difference scheme is constructed. Numerical experiments agree with theoretical investigations.
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Nino Khatiashvili, On the Stokes Nonlinear Waves in 2D, Recent Advances in Mathematics and Computational Science (ed. Imre J. Rudas),2016, 81-84, WSEAS Press, 2016
The Stokes nonlinear waves associated with the nonlinear problem of a free boundary with peaks in incompressible heavy fluid are studied in 2D. In the early works of the author the problem was reduced to the nonlinear integral equation with the weakly singular kernel. The approximate solution of this equation is obtained. The profile of the free boundary is plotted by means of Maple-12.
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Tea Shavadze, Variation Formulas of Solution for One Class of Controlled Functional Differential Equation with Several Delays and the Continuous Initial Condition, International Workshop QUALITDE – 2016, December 24 – 26, 2016, Tbilisi, Georgia, A. RAZMADZE MATHEMATICAL INSTITUTE of I. Javakhishvili Tbilisi State University, 2016
http://www.rmi.ge/eng/QUALITDE-2016/Shavadze_workshop_2016.pdf
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Teimurazi Davitashvili, Givi Gubelidze, Meri Sharikadze, Mathematical modeling of natural gas leak detection and localization in the branched pipelines, Journal Applied Mathematics, Informatics and Mechanics, 21, N 1, 74-88, Tbilisi University Press, 2016
In this article for detection of gas accidental escape localization in the branched gas pipelines two mathematical models are suggested. The first model is indented for leak detection and localization in the horizontal branched pipeline and second one for an inclined section of the main gas pipeline. The algorithm of leak localization in the branched pipeline is not demand on knowledge of corresponding initial hydraulic parameters at entrance and ending points of each sections of pipeline. For detection of the damaged section and then leak localization in this section special functions and equations are constructed. Some results of calculations for horizontal pipelines
having two, four and five sections are presented. Also a method and formula for the leak localization in the inclined section of the main gas pipeline are suggested. Some results of numerical calculations for the inclined pipeline are presented too
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David Gordeziani, Teimurazi Davitashvili, Tinatin Davitashvili, Meri Sharikadze, Mathematical Modeling of Filtration Problem for the Multilayer Liquid-Permeable Horizons, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics,30,19-22, Tbilisi University Press, 2016
The present work is devoted to the analysis of some mathematical models describing a movement of subsoil waters into the soil having the non-homogeneous multilayer structure in the vertical direction. Namely the corresponding systems of two-dimensional differential
equations in stationary and non-stationary cases are considered. For the first one the problem with classical and non-classical boundary conditions is stated. For numerical solution of the problem with nonlocal boundary conditions the iteration process is constructed, which allows one to reduce the solution of the initial problem to the solution of a sequence of classical Dirichlet problems. Some results of numerical calculations for the soil having two-layer structure are
presented
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Natalia Chinchaladze, On One Problem of a Cusped Elastic Prismatic Shells in Case of the Third Model of Vekua’s Hierarchical Model, HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS, Vol. 45 (6), 1665-1673, Hacettepe University, 2016
In the present paper hierarchical model for cusped, in general, elastic prismatic shells is considered, when on the face surfaces a normal to the projection of the prismatic shell component of a traction vector and parallel to the projection of the prismatic shell components of a displacement vector are known.
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Archil Papukashvili, Approximate Solution of Boundary Value Problem for the Ordinary Second-Order Differential Equation with Variable Coefficients by Means of Operator Interpolation Method , Bulletin of the Georgian National Academy of Sciences, vol. 10, no. 3, 2016. p. 7-16. , Georgian Academy Press, 2016
New computing algorithms for approximate solution of the two-point boundary value problem with variable coefficients are described in the paper. Green function of the given boundary value problem considered as a non-linear operator with respect to the variable coefficient is a approximated by means of operator interpolation polynomial of the Newton type. For approximation of the inverse operator two different types of formulae are constructed. Conventionally these formulas can be called direct and modified formulas. Consequently, for approximate solution of the two-point boundary value problem with variable coefficients direct and modified interpolation operator methods are used. Description of the algorithms for approximate solution are provided and the computation results of the test problems are given in tables.
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Teimurazi Davitashvili, Mathematical Simulation of Possible Detrimental Event Development above the Territory of Georgia, Proceedings of Azerbaijan State Marin Academy, 2, 267-273, Azerbaijan State Marin Academy, 2016
In the present work the problem of possible contamination of the Georgian territory by radioactive products, in case of accident at Armenian Nuclear Power Plant, is studied. Radioactive substances transportation, diffusion and fallout in the main towns of Georgia are investigated by mathematical modelling. The mathematical model has taken into account compound orography of Caucasus. Some results of numerical calculations are presented.
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Teimurazi Davitashvili, Nato Kutaladze, Ramaz Kvatadze, Giorgi Mikuchadze, Zurab Modebadze, Showers Prediction by WRF Model above Complex Terrain, Proceedings of the 39th International Convection MIPRO 2016/DC VIS, Opatija, Croatia, 236-241, Croatian Association MIPRO, Zagreb University, 2016
Regional climate formation above the territory of complex terrains is conditioned dominance due to of joint action of large-scale synoptic and local atmospheric processes which is basically stipulated by complex topography structure of the terrain. The territory of Caucasus and especially territory of Georgia are good examples for that. Indeed, about 85% of the total land area of Georgia is mountain ranges with compound topographic sections which play an impotent role for spatial-temporal distribution of meteorological fields. Therefore the territory of Georgia represents our interest. As known the global weather prediction models can well characterize the large scale atmospheric systems, but not enough the mesoscale processes which are associated with regional complex terrain and land cover. With the purpose of modelling these smaller scale atmospheric phenomena and its characterizing features it is necessary to take into consideration the main features of the local complex terrain, its heterogeneous land surfaces and at the same time influence of large scale atmosphere processes on the local scale processes. The Weather Research and Forecasting (WRF) model version 3.7 represents a good opportunity for studding regional and mesoscale atmospheric processes such are: Regional Climate, Extreme Precipitations, Hails, Sensitivity of WRF to physics options, influence of orography on mesoscale atmosphere processes e.c. In this study, WRF is using for prediction heavy showers and hails for different set of physical options in the regions characterized with the complex topography on the territory of Georgia.
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Teimurazi Davitashvili, Nato Kutaladze, Ramaz Kvatadze, Giorgi Mikuchadze, Inga Samkharadze, Precipitations Prediction by Different Physics of WRF Model, International Journal of Environmental Science, 1, 294-299, Iaras Press, 2016
In this article we have configured the nested grid WRF v.3.6 model for the Caucasus region. Computations were performed using Grid system GE-01-GRENA with working nodes (16 cores+, 32GB RAM on each). Two particulate cases of unexpected heavy showers were studied. Simulations were performed by two set of domains with horizontal grid-point resolutions of 6.6 km and 2.2 km. The ability of the WRF model in prediction precipitations with different microphysics and convective scheme components taking into consideration complex terrain of the Georgian territory was tested. Some results of the numerical calculations performed by WRF model are presented.
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David Natroshvili, Tengiz Buchukuri, Otar Chkadua, Mathematical Problems of Generalized Thermo-Electro-Magneto-Elasticity Theory, Memoirs on Differential Equations and Mathematical Physics, Tbilisi State University, 2016
The monograph is dedicated to the theoretical investigation of basic, mixed, and crack type three-dimensional initial-boundary value problems of the generalized thermo-electro-magnetoelasticity
theory associated with Green–Lindsay’s model. The essential feature of the generalized model under consideration is that heat propagation has a finite speed. We investigate the uniqueness of solutions to the dynamical initial-boundary value problems and analyse the corresponding boundary value problems of pseudo-oscillations which are obtained form the dynamical problems by the Laplace
transform. The solvability of the boundary value problems under consideration are analyzed by the potential method in appropriate Sobolev–Slobodetskii (W^s_ p ), Bessel potential (H^s_p), and Besov (B^s_{p;q}) spaces. The smoothness properties and singularities of thermo-mechanical and electro-magnetic fields are investigated near the crack edges and the curves where the different types of boundary conditions collide.
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Giorgi Geladze, Nodar Begalishvili, Nino Begalishvili, Numerical simulation of clouds ansamble and Foehns, Transactions of the Institute of Hydrometeorology at the Georgian Technical University V.123, pp. 50-55, Institute of Hydrometeorology, 2016
The ensemble of humidity processes (fogs, layered clouds) has been simulated on the basis of the numerical model of a non-stationary mesoscale boundary layer of atmosphere (MBLA) developed by us. In this work the accent becomes on interaction and interconversion of humidity processes in the above-stated ensemble. Genesis of Foehns is in detail investigated. They are classified on dryadiabatic, mostadiabatic and most-dryadiabatic Foehns. It is stated a problem about numerical modelling of Foehns in frame of a flat, two-dimensional mesoscale boundary layer. The problem is at a stage of numerical realisation. The first encouraging results are received
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Archil Papukashvili, Beqa Tezelishvili, Zurab Vashakidze, The numerical solution of a two-point boundary value problem with a non-constant coefficient by means of operator interpolation method, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics. Volume 30, 2016. p. 82-85., Tbilisi University Press , 2016
The new numerical algorithms for a two-point boundary value problem with a non-constant coefficient are proposed. The Green function of the given problem is represented as a nonlinear operator with respect to the coefficient. This operator is approximated by an operator
interpolation polynomial of the Newton type. For the inverse operators approximate formulas of different types are derived. The numerical algorithms and results of calculation of test problems are given.
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Temur Jangveladze, Unique Solvability and Additive Averaged Rothe's Type Scheme for One Nonlinear Multi-Dimensional Integro-Differential Parabolic Problem, International Workshop on the Qualitative Theory of Differential Equations – QUALITDE-2016, p.103-106, Georgian Technical University Press, 2016
The paper is devoted to the existence and uniqueness of a solution of the initial-boundary problem for one nonlinear multi-dimensional integro-differential equation of parabolic type. Construction and study of the additive averaged Rothe’s type scheme is also given. The studied equation is based on well-known Maxwell’s system arising in mathematical simulation of electromagnetic field penetration into a substance.
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Tamaz Tadumadze, Phridon Dvalishvili, Continuous dependence of the minimum of functional on perturbations in optimal control problems with distributed and concentrated delays, Differential and Difference Equations with Applications, ICDDEA, Amadora, Portugal, May 2015, Selected Contributions, Springer Proceedings in Mathematics & Statistics, 164 ,339-348, Springer International Publishing Switzerland, 2016
Continuity of the minimum of a general functional is proved with respect to perturbations of the initial data and right-hand side of the equation with variable distributed and concentrated delays. Under the initial data, we understand the collection of initial moment, of variable delays, and initial function. Perturbations of the right-hand side of the equation are small in the integral sense.
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Nika Gorgodze, Tamaz Tadumadze, Ia Ramishvili, Continuous dependence of solution of a neutral functional differential equation on the right-hand side and initial data considering perturbations of variable delays, Georgian Math. J., 23 (4) , 519-535, De Gruyter, 2016
Theorems on the continuous dependence of the solution on perturbations of the initial data and the right-hand side of equation are proved. Under initial data we understand the collection of initial moment, of delay function and initial function. Perturbations of the right-hand side of equation are small in the integral sense.
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Bakur Gulua, On a boundary value problem for the nonlinear non-shallow spherical shell, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, Volume 30, 23-26, Tbilisi University Press, 2016
In this work we consider the geometrically nonlinear and non-shallow spherical shells for I.N. Vekua N=1 approximation. Concrete problem using complex variable functions and the method of the small parameter has been solved.
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Tengiz Meunargia, The problem of existence the neutral surface for the elastic shell, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, Volume 30, 74-77, Tbilisi University Press, 2016
I. Vekua obtained the conditions for the existence of the neutral surface of a shell, when the neutral surface is the middle surface. In this paper the neutral surface is considered as any equidistant surfaces of the shell.
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Temur Jangveladze, Zurab Kiguradze, Difference Scheme for One System of Nonlinear Parabolic Integro-Differential Equations, Applied Mathematics, Informatics, and Mechanics, V.21, N1, p.104-120, Tbilisi University Press, 2016
Nonlinear parabolic integro-differential model which is based on Maxwell system is considered. Large time behavior of solutions of the initial-boundary value problem with mixed boundary condition is given. Finite difference scheme is investigated. Wider class of nonlinearity is studied than one has been investigated before.
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Giorgi Akhalaia, Nino Manjavidze, On one boundary value problem of the generalized analytic vectors, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, Volume 30, 3-6, Tbilisi University Press, 2016
The Dirichlet-type problem for one quasi-linear elliptic system is investigated.
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Tamaz Kaladze, Oleg Kharshiladze, Generation of electrostatic drift zonal flows under the action of mean sheared flows, Physics of Plasmas, v.23, No. 12, 122306, American Institute of Physics, 2016
Generation of large-scale zonal flows by the small-scale electrostatic drift wave turbulence in the magnetized plasma under the action of mean poloidal sheared flow is considered. Attention to large-scale (compared to the ion Larmor radius) drift structures is paid. To this end, the generalized Hasegawa-Mima equation containing both vector and scalar nonlinearities is derived, and the appropriate eigenvalue problem is solved numerically. Destabilizing role of the small amplitude mean shear flow and spatial inhomogeneity of electron temperature is shown.
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Mariam Beriashvili, Measurability properties of certain paradoxical subsets of the real line, Georgian Math. J. 2016; 23 (1):25–32, DE GRUYTER, 2016
The paper deals with the measurability properties of some classical subsets of the real line ℝ having
an extra-ordinary descriptive structure: Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpiński
sets. In particular, it is shown that there exists a translation invariant measure ???? on ℝ extending the Lebesgue
measure and such that all Sierpiński sets are measurable with respect to ????
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Roman Koplatadze, Almost linear functional differential equations with properties A and B, Translations of. Razmadze Math. Inst. 170 , no. 2, 215—242., Elsevier, 2016
For the general functional differential equation, the sufficient conditions n –th order to have Property A (Property B) are established. As particular was, we consider almost linear ordinary differential equation deviating argument. The sufficient conditions are obtained for the solutions to be oscillatory. These criteria cover the wee-known results for the linear differential equations.
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Gogi Pantsulaia, Applications of Measure Theory to Statistics, Springer International Publishing, Springer International Publishing, 2016
Helps the reader to understand some conjectures arising in the criticism of null hypothesis significance testing (NHST)
Includes a special chapter that helps the reader to calculate infinite-dimensional Riemann integrals over infinite-dimensional rectangles in R8
Considers how to construct objective consistent estimates of an unknown parameter in a Polish group
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Gogi Pantsulaia, Zurab Kvatadze, L Labadze, On a consistent estimator of a useful signal in Ornstein-Uhlenbeck model in , arXiv preprint arXiv:1612.03904, Arxiv, 2016
It is considered a transmittion process of a useful signal in Ornstein-Uhlenbeck model in C[−l,l[ defined by the stochastic differential equation
dΨ(t,x,ω)=∑n=02mAn∂n∂xnΨ(t,x,ω)dt+σdW(t,ω)
with initial condition
Ψ(0,x,ω)=Ψ0(x)∈FD(0)[−l,l[,
where m≥1, (An)0≤n≤2m∈R+×R2m−1, ((t,x,ω)∈[0,+∞[×[−l,l[×Ω), σ∈R+, C[−l,l[ is Banach space of all real-valued bounded continuous functions on [−l,l[, FD(0)[−l,l[⊂C[−l,l[ is class of all real-valued bounded continuous functions on [−l,l[ whose Fourier series converges to himself everywhere on [−l,l[, (W(t,ω))t≥0 is a Wiener process and Ψ0(x) is a useful signal.
By use a sequence of transformed signals (Zk)k∈N=(Ψ(t0,x,ωk))k∈N at moment t0>0, consistent and infinite-sample consistent estimations of the useful signal Ψ0 is constructed under assumption that parameters (An)0≤n≤2m and σ are known. Animation and simulation of the Ornstein-Uhlenbeck process in C[−l,l[ and an estimation of a useful signal are also presented.
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Gogi Pantsulaia, Tengiz Kiria, Calculation of Lebesgue integrals by using uniformly distributed sequences, Transactions of A. Razmadze Mathematical Institute 170 (3), 402-409, A. Razmadze Mathematical Institute, 2016
A certain modified version of Kolmogorov’s strong law of large numbers is used for an extension of the result of C. Baxa and J. Schoißengeier (2002) to a maximal set of uniformly distributed sequences in (0,1) which strictly contains the set of all sequences having the form ({αn})n∈N for some irrational number α and having the full ℓ1∞-measure, where ℓ1∞ denotes the infinite power of the linear Lebesgue measure ℓ1 in (0,1).
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L Labadze, Gimzer Saatashvili, Gogi Pantsulaia, Infinite-sample consistent estimations of parameters of the Wiener process with drift, arXiv preprint arXiv:1611.01119, Arxiv, 2016
We consider the Wiener process with drift
dXt=μdt+σdWt
with initial value problem X0=x0, where x0∈R, μ∈R and σ>0 are parameters. By use values (zk)k∈N of corresponding trajectories at a fixed positive moment t, the infinite-sample consistent estimates of each unknown parameter of the Wiener process with drift are constructed under assumption that all another parameters are known. Further, we propose a certain approach for estimation of unknown parameters x0,μ,σ of the Wiener process with drift by use the values (z(1)k)k∈N and (z(2)k)k∈N being the results of observations on the 2k-th and 2k+1-th trajectories of the Wiener process with drift at moments t1 and t2 , respectively.
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L Labadze, Gogi Pantsulaia, Estimation of the parameters of the Ornstein-Uhlenbeck's stochastic process, arXiv preprint arXiv:1608.04507, Arxiv, 2016
We consider Ornstein-Uhlenbeck process 0
(1 ) t t
t
x x e e
( )
0
,
t
t s
s
e dW
where 0
x , 0, , 0 and Ws
are Wiener process. By using the values ( ) k k N z
of the
corresponding trajectories at a fixed positive moment t, the estimates Tn
and ** Tn
of unknown parameters
0
x and are constructed, where 0
x is an underlying asset initial price and is a rate by which these
shocks dissipate and the variable reverts towards the mean in the Ornstein-Uhlenbeck’s stochastic
process. By using Kolmogorov’s Strong Law of Large Numbers the consistence of estimates Tn
and ** Tn
are proved.
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Gogi Pantsulaia, Givi Giorgadze, Representation of the Dirac delta function in in terms of infinite-dimensional Lebesgue measures, arXiv preprint arXiv:1605.02723, Arxiv, 2016
A representation of the Dirac delta function in C(R∞) in terms of infinite-dimensional Lebesgue measures in R∞ is obtained and some it's properties are studied in this paper.
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Aleks Petre Kirtadze, Gogi Pantsulaia, Nino Rusiashvili, On uniform distribution for invariant extensions of the linear Lebesgue measure, arXiv preprint arXiv:1603.04472, Arxiv, 2016
The concept of uniform distribution in [0,1] is extended for a certain strictly separated maximal (in the sense of cardinality) family (λt)t∈[0,1] of invariant extensions of the linear Lebesgue measure λ in [0.1], and it is shown that the λ∞t measure of the set of all λt-uniformly distributed sequences is equal to 1, where λ∞t denotes the infinite power of the measure λt. This is an analogue of Hlawka's (1956) theorem for λt-uniformly distributed sequences. An analogy of Weyl's (1916) theorem is obtained in similar manner.
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Gogi Pantsulaia, On Moore–Yamasaki–Kharazishvili Type Measures and the Infinite Powers of Borel Diffused Probability Measures on R, Applications of Measure Theory to Statistics, 57-71, Springer, 2016
The paper contains a brief description of Yamasaki's remarkable investigation (1980) of the relationship between Moore-Yamasaki-Kharazishvili type measures and infinite powers of Borel diffused probability measures on ${\bf R}$. More precisely, we give Yamasaki's proof that no infinite power of the Borel probability measure with a strictly positive density function on $R$ has an equivalent Moore-Yamasaki-Kharazishvili type measure. A certain modification of Yamasaki's example is used for the construction of such a Moore-Yamasaki-Kharazishvili type measure that is equivalent to the product of a certain infinite family of Borel probability measures with a strictly positive density function on $R$. By virtue of the properties of equidistributed sequences on the real axis, it is demonstrated that an arbitrary family of infinite powers of Borel diffused probability measures with strictly positive density functions on $R$ is strongly separated and, accordingly, has an infinite-sample well-founded estimator of the unknown distribution function. This extends the main result established in [ Zerakidze Z., Pantsulaia G., Saatashvili G. On the separation problem for a family of Borel and Baire $G$-powers of shift-measures on $\mathbb{R}$ // Ukrainian Math. J. -2013.-65 (4).- P. 470--485 ].
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Gogi Pantsulaia, Why Null Hypothesis Is Rejected for Almost Every Infinite Sample by the Hypothesis Testing of a Maximal Reliability, Applications of Measure Theory to Statistics, 107-118, Springer, 2016
The notion of a Haar null set introduced by Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke, has been used, in the last two decades, in studying exceptional sets in diverse areas, including analysis, dynamic systems, group theory, and descriptive set theory. In the present paper, the notion of “prevalence” is used in studying the properties of some infinite sample statistics and in explaining why the null hypothesis is sometimes rejected for “almost every” infinite sample by some hypothesis testing of maximal reliability. To confirm that the conjectures of Jum Nunnally [17] and Jacob Cohen [5] fail for infinite samples, examples of the so called objective and strong objective infinite sample well-founded estimate of a useful signal in the linear one-dimensional stochastic model are constructed.
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Gogi Pantsulaia, Structure of All Real-Valued Sequences Uniformly Distributed in from the Point of View of Shyness, Applications of Measure Theory to Statistics, 47-56, Springer, 2016
In the paper [G.Pantsulaia, On Uniformly Distributed Sequences on [-1/2,1/2], Georg. Inter. J. Sci. Tech.,4(3) (2013), 21--27], it was shown that μ-almost every element of R∞ is uniformly distributed in [−12,12], where μ denotes Yamasaki-Kharazishvili measure in R∞ for which μ([−12,12]∞)=1. In the present paper the same set is studying from the point of view of shyness and it is demonstrated that it is shy in R∞. In Solovay model, the set of all real valued sequences uniformly distributed module 1 in [−12,12] is studied from the point of view of shyness and it is shown that it is the prevalent set in R∞.
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Gogi Pantsulaia, Objective and Strong Objective Consistent Estimates of Unknown Parameters for Statistical Structures in a Polish Group Admitting an Invariant Metric, Applications of Measure Theory to Statistics, 73-106, Springer, 2016
By using the notion of a Haar ambivalent set introduced by Balka, Buczolich and Elekes (2012), essentially new classes of statistical structures having objective and strong objective estimates of unknown parameters are introduced in a Polish non-locally-compact group admitting an invariant metric and relations between them are studied in this paper. An example of such a weakly separated statistical structure is constructed for which a question asking ”whether there exists a consistent estimate of an unknown parameter” is not solvable within the theory (ZF ) & (DC). A question asking ”whether there exists an objective consistent estimate of an unknown parameter for any statistical structure in a non-locally compact Polish group with an invariant metric when subjective one exists” is answered positively when there exists at least one such a parameter the pre-image of which under this subjective estimate is a prevalent. These results extend recent results of authors. Some examples of objective and strong objective consistent estimates in a compact Polish group {0; 1} are considered in this paper. Primary 62-02; secondary 62D05.
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Gogi Pantsulaia, Calculation of Improper Integrals by Using Uniformly Distributed Sequences, Applications of Measure Theory to Statistics, 1-18, Springer, 2016
We present the proof of a certain modified version of Kolmogorov's strong law of large numbers for calculation of Lebesgue Integrals by using uniformly distributed sequences in (0,1). We extend the result of C. Baxa and J. Schoiβengeier (cf.\cite{BaxSch2002}, Theorem 1, p. 271) to a maximal set of uniformly distributed (in (0,1)) sequences Sf⊂(0,1)∞ which strictly contains the set of sequences of the form ({αn})n∈N with irrational number α and for which ℓ∞1(Sf)=1, where ℓ∞1 denotes the infinite power of the linear Lebesgue measure ℓ1 in (0,1).
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Gogi Pantsulaia, Infinite-Dimensional Monte Carlo Integration, Applications of Measure Theory to Statistics, 19-46, Springer, 2016
By using main properties of uniformly distributed sequences of increasing finite sets in infinite-dimensional rectangles in R∞ described in [G.R. Pantsulaia, On uniformly distributed sequences of an increasing family of finite sets in infinite-dimensional rectangles, Real Anal. Exchange. 36 (2) (2010/2011), 325--340 ], a new approach for an infinite-dimensional Monte-Carlo integration is introduced and the validity of some infinite-dimensional Strong Law type theorems are obtained in this paper. In addition, by using properties of uniformly distributed sequences in a unite interval, a new proof of Kolmogorov's strong law of large numbers is obtained which essentially differs from its original proof.
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Gogi Pantsulaia, THE PROOF OF A CERTAIN VERSION OF KOLMOGOROV STRONG LAW OF LARGE NUMBERS, , , 2016
We present the proof of a certain version of Kolmogorov strong law of large numbers which differs from Kolmogorov’s original proof.
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Luba Tsamalashvili, Special exp-function method for travelling wave solutions of Burger's equation, Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics Volume 30, Tbilisi University Press, 2016
Nonlinear Burger’s equation describes shock waves in liquid and gas. It can be also used to model vehicles density on motor roads. Burger’s equation connects the dissipative uux term with the convectional uux one (see below Eq. (10)). Using the straightforward method used by Tsamalashvili in [1] soliton like exact solutions are obtained for the 2D nonlinear modified Burger’s equation. Employing the special exp-function expansion method Mohyud-Din et al. [2] constructed exact traveling wave solutions for (2+1) - dimensional Burger’s equation. Unfortunately this paper contains numerous wrong results and our main purpose is to revise previously obtained solutions.
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Alexander Kharazishvili, Elements of combinatorial geometry I, 282 p, Georgian National Academy of Sciences., 2016
The presented book is devoted to the certain combinatorial and set-theoretical aspects of the geometry of Euclidean space and consists of two parts. The material of this book is primarily devoted to various discrete geometric structures and, respectively, to certain constructions of algorithmic type which are associated with such structures. Typical questions of combinatorial, discrete and convex geometry are examined and discussed more or less thoroughly. There are indicated close relationships between the questions of geometry and other areas of discrete mathematics.
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Temur Jangveladze, Long-Time Behavior of Solution and Semi-Discrete Scheme for One Nonlinear Parabolic Integro-Differential Equation, Transactions of A.Razmadze Mathematical Institute, V.170, N1, p.47-55, Elsevier BV, 2016
Long-time behavior of solution and semi-discrete scheme for one nonlinear parabolic integro-differential equation are studied. Initial–boundary value problem with mixed boundary conditions are considered. Attention is paid to the investigation of more wide cases of nonlinearity than already were studied. Considered model is based on Maxwell’s system describing the process of the penetration of a magnetic field into a substance.
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Temur Jangveladze, Zurab Kiguradze, Finite Difference Scheme for One Nonlinear Parabolic Integro-Differential Equation, Transactions of A.Razmadze Mathematical Institute, V.170, N3, p.395-401, Elsevier BV, 2016
Initial–boundary value problem with mixed boundary conditions for one nonlinear parabolic integro-differential equation is considered. The model is based on Maxwell system describing the process of the penetration of a electromagnetic field into a substance. Unique solvability and asymptotic behavior of solution are fixed. Main attention is paid to the convergence of the finite difference scheme. More wide cases of nonlinearity that already were studied are investigated.
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Besik Dundua, Mariam Beriashvili, A ConstraintSolver for Equations overSequences and Contexts, Proceedings of the4th International Conferenceon Computer Science, AppliedMathematics and Applications, ICCSAMA 2016. Advances in IntelligentSystems and Computing 453, Springer, Springer Cham, 2016
In this paper we propose a solving algorithm for equational constraints over unranked terms, contexts, and sequences. Unranked terms are constructed over function symbols which do not have fixed arity. For some function symbols, the order of the arguments matters (ordered symbols). For some others, this order is irrelevant (unordered symbols). Contexts are unranked terms with a single occurrence of hole. Sequences consist of unranked terms and contexts. Term variables stand for single unranked terms, sequence variables for sequences, context variables for contexts, and function variables for function symbols. We design an terminated and incomplete constraint solving algorithm, and indicate a fragment for which the algorithm is complete.
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Besik Dundua, B. Miara, Mircea Marin, A Rewrite-based Computational Model for Functional Logic Programming, Proceedings of the7th International Symposium on Symbolic Computation in SoftwareScience, SCSS 2016. EPiC Series, Volume39,pages 95–106, EasyChair, EPiC Series in Computing, 2016
Functional logic programming is an extension of the functional programming style with
two important capabilities: to define nondeterministic operations with overlapping rules,
and to use logic variables in both defining rules and expressions to evaluate. A suitable
model for functional logic programs are conditional constructor-based term rewrite systems
(CB-CTRSs), which can be transformed into an equivalent program in a simpler class of
rewrite systems (the core language) where computations can be performed more efficiently.
Recently, Antoy and Hanus proposed a translation of CB-CTRSs into an equivalent
class of programs where computation can be performed efficiently by mere rewriting. Their
computational model has the limitation of computing only ground answer substitutions
for equations with strict semantics interpreted as joinability to a value. We propose two
adjustments of their computational models, which are capable to compute non-ground
answers.
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Besik Dundua, Temur Kutsia, Mario Florido, Mircea Marin, CLP(H): Constraint Logic Programming for Hedges, Theoryand Practiceof Logic Programming. Volume16, Issue2, pages 141–162, Cambridge University Press, Cambridge University Press: , 2016
CLP(H) is an instantiation of the general constraint logic programming scheme with the constraint domain of hedges. Hedges are finite sequences of unranked terms, built over variadic function symbols and three kinds of variables: for terms, for hedges, and for function symbols. Constraints involve equations between unranked terms and atoms for regular hedge language membership. We study algebraic semantics of CLP(H) programs, define a sound, terminating, and incomplete constraint solver, investigate two fragments of constraints for which the solver returns a complete set of solutions, and describe classes of programs that generate such constraints.
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Besik Dundua, Temur Kutsia, Klaus Reisenberger-Hagmayer, PρLog: Combining Logic Programming with Conditional Transformation Systems, Proceedings of the32ndInternational Conferenceon Logic Programming, ICLP 2016. Vol.52of OpenAccess Series in Informatics (OASIcs). Schloss Dagstuhl,pages 10.1–10.5, oasics, 2016
PρLog extends Prolog by conditional transformations that are controlled by strategies. We give
a brief overview of the tool and illustrate its capabilities.
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Kartlos Joseph Kachiashvili, Archil Prangishvili, Verification in biometric systems: problems and modern methods of their solution, Journal of Applied Statistics, 45(1): 43-62, Journal of Applied Statistics, 2016
The paper deals with the problem of electronic verification of people on the basis of measurement information of a fingerprint reader and new approaches to its solution. The offered method guaranties the restriction of error probabilities of both type at the desired level at making a decision about permitting or rejecting the request on service in the system. On the basis of investigation of real data obtained in the real biometrical system, the choice of distribution laws is substantiated and the proper estimations of their parameters are obtained. Using chosen distribution laws, the normal distribution for measurement results of characteristics of the people having access to the system and the beta distribution for the people having no such access, the optimal rule based on the Constrained Bayesian Method (CBM) of making a decision about giving a permission of access to the users of the system is justified. The CBM, the Neyman–Pearson and classical Bayes methods are investigated and their good and negative points are examined. Computation results obtained by direct computation, by simulation and using real data completely confirm the suppositions made and the high quality of verification results obtained on their basis
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Kartlos Joseph Kachiashvili, David Melikdzhanian, Software for Pollutants Transport in Rivers and for Identification of Excessive Pollution Sources, MOJ Ecology & Environmental Science, 1(1): 1-8, MedCrave Publishing, 2016
The program packages of realization of mathematical models of pollutants transport in rivers and for identification of river water excessive pollution sources located between two controlled cross-sections of the river will be considered and demonstrated. The software has been developed by the authors on the basis of mathematical models of pollutant transport in the rivers and statistical hypotheses testing methods. The identification algorithms were elaborated with the supposition that the pollution sources discharge different compositions of pollutants or (at the identical composition) different proportions of pollutants into the rivers. One-, two-, and three-dimensional advection-diffusion mathematical models of river water quality formation both under classical and new, original boundary conditions are realized in the package. New finite-difference schemes of calculation have been developed and the known ones have been improved for these mathematical models. At the same time, a number of important problems which provide practical realization, high accuracy and short time of obtaining the solution by computer have been solved. Classical and new constrained Bayesian methods of hypotheses testing for identification of river water excessive pollution sources are realized in the appropriate software. The packages are designed as a up-to-date convenient, reliable tools for specialists of various areas of knowledge such as ecology, hydrology, building, agriculture, biology, ichthyology and so on. They allow us to calculate pollutant concentrations at any point of the river depending on the quantity and the conditions of discharging from several pollution sources and to identify river water excessive pollution sources when such necessity arise
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Kartlos Joseph Kachiashvili, Constrained Bayesian Method of Composite Hypotheses Testing: Singularities and Capabilities, International Journal of Statistics in Medical Research, 5(3): 135-167, Canada, Lifescience Global, 2016
The paper deals with the constrained Bayesian Method (CBM) for testing composite hypotheses. It is shown that, similarly to the cases when CBM is optimal for testing simple and multiple hypotheses in parallel and sequential experiments, it keeps the optimal properties at testing composite hypotheses. In particular, it easily, without special efforts, overcomes the Lindley’s paradox arising when testing a simple hypothesis versus a composite one. The CBM is compared with Bayesian test in the classical case and when the a priori probabilities are chosen in a special manner for overcoming the Lindley’s paradox. Superiority of CBM against these tests is demonstrated by simulation. The justice of the theoretical judgment is supported by many computation results of different characteristics of the considered methods.
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Kartlos Joseph Kachiashvili, EDITORIAL: Inference in Clinical Experiments, International Journal of Statistics in Medical Research, 5(3): 133-134, Int. J. Statistics in Medical Research, 2016
Statistical methods all are more widely used in all spheres of human activity. Their importance in medicine and biology especially intensively is developing and increasing since the latest decade of the previous century. The reason of this circumstance consists in especial
complexity of the problems of these domains caused by complexity of their character, by the great number of the parameters included in them and of the factors influencing their. Many of the factors affecting the observation results used for investigation of the problems under study are random by their nature and, hence, the observation results are random. Therefore the study and solution of these problems require the application of the modern methods of probability and mathematical statistics...
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Kartlos Joseph Kachiashvili, Alexander Topchishvili, Parameters estimators of irregular right-angled triangular distribution, Model Assisted Statistics and Applications, 11: 179-184, IOS Press, Amsterdam, Holland, 2016
We obtained and investigated consistent, unbiased and efficient estimators of the parameters of irregular right-angled triangular distribution on the basis of maximum likelihood estimators. Some computation results realized on the basis of simulation of the appropriate random samples demonstrate theoretical outcomes.
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Jemal Antidze, Solving the Problem of the Accents for Speech Recognition Systems, International Journal of Signal Processing Systems, volume 4, number 3, pages 235-238, International Journal of Signal Processing Systems, 2016
Since the speech recognition system has been created, it has developed significantly, but it still has a lot of problems. As you know, any specific natural language may owns about tens accents. Despite the identical word phonemic composition, if it is pronounced in different accents, as a result, we will have sound waves, which are different from each other. Differences in pronunciation, in accent and intonation of speech in general, create one of the most common problems of speech recognition. If there are a lot of accents in language we should create the acoustic model for each separately. When the word is pronounced differently, then the software can become confused and misunderstand (perception) also correctly what is pronounced. The same can also occur, if the human speaks slowly or vice versa quickly, then the program expects. There are any partial decisions (solutions) but they don’t solve all problems. We have developed an approach, which is used to solve above mentioned problems and create more effective, improved speech recognition system of Georgian language and of languages, which are similar to Georgian language. In addition, by the realization of this method, it is available to solve the artificial intelligence issues, such as arrange sound dialogue between computer and human, independent from any accents of any languages.
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Gela Chankvetadze, Lia Kurtanidze, Mikheil Rukhaia, Semi-Automated Construction of Proof Schemata, In: Journal of Applied Mathematics, Informatics and Mechanics, vol. 21 (2), pp. 83–91, Tbilisi University Press, 2016
In this paper we present a goal-directed proof-search algorithm for formula schemata, which is based on a sequent calculus. Usually, sequent calculus inference rules can be applied freely, producing a redundant search space. The standard approaches are extended to formula schemata to get rid of redundancy in a proof-search. A formula schema is a finite representation of an infinite sequence of first-order formulas, thus complete automation of the process is not feasible. Still, there are some (not so trivial) subclasses, where the process can be fully automated.
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Gela Chankvetadze, Lia Kurtanidze, Mikheil Rukhaia, Proof Construction in Unranked Logic, In: Jangveladze, Temur and Kiguradze, Zurab (eds.): Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, vol. 30. pp. 11–14, Tbilisi University Press, 2016
In the paper we study proof construction methods for first-order unranked logic. Unranked languages have unranked alphabet, meaning that function and predicate symbols do not have a fixed arity. Such languages can model XML documents and operations over them, thus becoming more important in semantic web. We present a version of sequent calculus for first-order unranked logic and describe a proof construction algorithm under this calculus. We give implementation details of the algorithm. We believe that this work will be useful for the undergoing work on semantic web logic layer.
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Alexander Kharazishvili, On bijective continuous images of absolute null sets, Ukrainian Math. J., 2016 / 67, no. 8, 1277-1282, Springer, 2016
The images of absolute null sets (spaces) under bijective continuous mappings are studied. It is shown that, in general, these images do not possess regularity properties from the viewpoint of topological measure theory.
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Alexander Kharazishvili, On negligible and absolutely nonmeasurable subsets of uncountable solvable groups, Trans. A. Razmadze Math. Inst., 2016 / 170, no. 1, 69-74, Elsevier , 2016
It is proved that every uncountable solvable group contains two negligible sets whose union is an absolutely nonmeasurable subset of the same group.
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Alexander Kharazishvili, On the cardinal number of the family of all invariant extensions of a nonzero $\sigma$-finite invariant measure, Trans. A. Razmadze Math. Inst., 2016 / 170, no. 2, 200-204, Elsevier , 2016
It is shown that, for any nonzero -finite translation invariant (translation quasi-invariant) measure on the real line R, the cardinality of the family of all translation invariant (translation quasi-invariant) measures on R extending is greater than or equal to , where denotes the first uncountable cardinal number. Some related results are also considered.
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Alexander Kharazishvili, Absolute null subsets of the plane with bad orthogonal projections, Real Anal. Exchange, 2016 / 41, no. 1, 233-243, Michigan State University Press, 2016
Under Martin’s Axiom, it is proved that there exists an absolute null subset of the Euclidean plane R^2, the orthogonal projections of which on all straight lines in
R^2 are absolutely nonmeasurable. A similar but weaker result holds true within the framework of ZFC set theory.
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Alexander Kharazishvili, On the difference between a Vitali-Bernstein selector and a partial Vitali-Bernstein selector, Georgian Math. J., 2016 / 23, no. 3, 387-391, De Gruyter, 2016
It is shown that the difference between a Vitali–Bernstein selector and a partial Vitali–Bernstein selector can be of Lebesgue measure zero and of first Baire category. This result gives an answer to a question posed by G. Lazou.
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Alexander Kharazishvili, Acute triangles in the context of the illumination problem, Ann. Sofia Univ., Fac. Math and Inf., 103, 2016, 39–44., FMI, 2016
We consider strong at-subsets of the Euclidean space Rn and estimate from below
the growth of the maximal cardinality of such subsets (our method essentially differs
from that of [6]). We then apply some properties of strong at-sets to the illumination
problem.
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Roman Janjgava, Miranda Narmania, The solution of some two-dimensional problems of thermoelasticity taking into account the microtemperature, Journal of Thermal Stresses, Taylor&Francis, Philadelphia, 39(1), 57-64, Taylor&Francis, 2016
In this work we consider the two-dimensional system of the differential equations describing the plane statical thermoelastic balance of homogenous isotropic elastic bodies, the microelements of which have microtemperature in addition to the classical displacement and a temperature field. It is construed the general solution of this system of the equations by means of analytic functions of complex variable and solutions of the equation of Helmholtz. The general representation of the solution obtained gives the opportunity to construct the analytical solutions of a number of plane boundary value problems of microthermoelasticity. As an example we consider the boundary value problem for a concentric circular ring.
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Ivane Tsagareli, Solution of the Boundary-Contact Problem of Elastostatics for an Multi-Layer Infinite Cylinder with Double Porosity, YII International Joint Conference of Georgian Mathematical Union and Georgian Mechanical Union, Batumi, September 5-9, Georgian Mathematical Union, 2016
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Ivane Tsagareli, Lamara Bitsadze, Explicit solutions on some problems in the fully coupled theory of elasticity for a circle with double porosity, Bulletin of TICMI, 20 (2), 11-23, Tbilisi University Press, 2016
The purpose of this paper is to consider the two-dimensional version of the fully coupled theory of elasticity for solids with double porosity the and to solve explicitly some boundary value problems (BVPs) of statics for an elastic circle. The explicit solutions of this BVPs are constructed by means of absolutely and uniformly convergent series. The questions on the uniqueness of a solutions of the problems are investigated.
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Lamara Bitsadze, Ivane Tsagareli, Solutions of BVPs in the fully coupled theory of elasticity for the space with double porosity and spherical cavity, Mathematical Methods in the Applied Science, 39 (8), 2136-2145, Wiley, 2016
In this paper, the fully coupled theory of elasticity for solids with double porosity is considered. The explicit solutions of the basic boundary value problems (BVPs) in the fully coupled linear equilibrium theory of elasticity for the space with double porosity and spherical cavity are constructed. The solutions of these BVPs are represented by means of absolutely and uniformly convergent series.
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Lamara Bitsadze, Ivane Tsagareli, The solution of the Dirichlet BVP in the fully coupled theory of elasticity for spherical layer with double porosity, Meccanica, 51 (6), 1457-1463, Springer, 2016
The main goal of this paper is to consider the Dirichlet type boundary value problem (BVP) of the fully coupled equilibrium theory of elasticity for solids with double porosity and to construct explicitly the solution of BVP for a spherical layer in the form of absolutely and uniformly convergent series.
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Lamara Bitsadze, The dirichlet BVP of the theory of thermoelasticity with microtemperatures for microstretch sphere, Journal of Thermal Stresses, 39 (9), 1074-1083, Taylor & Francis, 2016
The present article studies the equilibrium theory of thermomicrostretch elastic solids with microtemperatures. The general solution of the equations for a homogeneous isotropic microstretch thermoelastic sphere with microtemperatures is constructed and the solution of the Dirichlet-type boundary value problem for the sphere in the form of absolutely and uniformly convergent series is obtained.
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Elizbar Nadaraya, Petre Babilua, Grigol Sokhadze, On testing hypotheses of equality distribution densities, Bull. Georgian Natl. Acad. Sci. (N.S.) 10 (2016), no. 3, 27--32., Georgian National Academy of sciences Press, 2016
In the paper the test of homogeneity and goodness-of-fit for checking the hypotheses of equality distribution densities is constructed. The power asymptotics of the constructed test of homogeneity and goodness-of-fit for certain types of close alternatives is also studied.
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Elizbar Nadaraya, Petre Babilua, Grigol Sokhadze, est of a hypothesis on the equality of distribution densities, Ukrainian Math. J. 68 (2016), no. 5, 661--678., Springer , 2016
We construct new criteria for testing the hypotheses that p ≥ 2 independent samplings have identical densities of distribution (hypothesis of homogeneity) or the same well-defined densities of distribution (goodness-of-fit test). The limiting power of the constructed criteria is established for some local “close” alternatives.
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Elizbar Nadaraya, Grigol Sokhadze, On integral functionals of a density, Comm. Statist. Theory Methods 45 (2016), no. 23, 7086--7102., Taylor&Francis, 2016
Estimation of a non linear integral functional of probability distribution density and its derivatives is studied. The truncated plug-in-estimator is taken for the estimation. The integrand function can be unlimited, but it cannot exceed polynomial growth. Consistency of the estimator is proved and the convergence order is established. Aversion of the central limit theorem is proved. As an example an extended Fisher information integral and generalized Shannon's entropy functional are considered.
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Tamaz Kaladze, Khatuna Elbakidze, Oleg Kharshiladze, Luba Tsamalashvili, Generation of Zonal Flow and Magnetic Field by Planetary Waves in the Earth’s Ionosphere, Journal of Applied Mathematics and Physics 4 (02), 487. 2016, SCIRP, 2016
Possibility of generation of large-scale sheared zonal flow and magnetic field by coupled under the typical ionospheric conditions short-scale planetary low-frequency waves is shown. Propagation of coupled internal-gravity-Alfven, Rossby-Khantadze, Rossby-Alfven-Khantadze and collision-less electron skin depth order drift-Alfven waves is revealed and investigated in detail. To describe the nonlinear interaction of such coupled waves with sheared zonal flow the corresponding nonlinear equations are deduced. The instability mechanism is based on the nonlinear parametric triple interaction of the finite amplitude short-scale planetary waves leading to the inverse energy cascade toward the longer wavelengths. It is shown that under such interaction intense sheared magnetic fields can be generated. Appropriate growth rates are discussed in detail.
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Ivane Tsagareli, Solutions of the problems of thermoelastostatics for an circle with double porosity, XXX International Enlarged Sessions of the Seminar of I.Vekua Institute of Applied Mathematics, , Tbilisi University Press , 2016
In the present paper, using absolutely and uniformly convergent series, the boundary value problems of thermoelastostatics for an elastic circle with double porosity are solved explicitly. The question on the uniqueness of a solution of the problem is investigated.
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Maia Svanadze, Plane waves and problems of steady vibrations in the theory of viscoelasticity for Kelvin-Voigt materials with double porosity, Archives of Mechanics, Volume 68, Issue 6, Pages 441-458., Polish Scientific Publishers PWN, 2016
In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with double porosity is considered. Some basic properties of plane harmonic waves are established and the boundary value problems (BVPs) of steady vibrations are investigated. Indeed on the basis of this theory three longitudinal and two transverse plane harmonic waves propagate through a Kelvin–Voigt material with double porosity and these waves are attenuated. The basic properties of the singular integral operators and potentials (surface and volume) are presented. The uniqueness and existence theorems for regular (classical) solutions of the BVPs of steady vibrations are proved by using the potential method (boundary integral equations method) and the theory of singular integral equations.
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Bakur Gulua, One boundary value problem for the plates, Seminar of I. Vekua Institute of Applied Mathematics REPORTS, Volume 42, 3-9, Tbilisi University Press, 2016
In this work we consider equations of equilibrium of the isotropic elastic shell. By means of Vekua’s method, the system of differential equations for thin and shallow shells is obtained, when on upper and lower face surfaces displacements are assumed to be known.
The general solution for approximations N=1 is constructed. The concrete problem is solved.
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Giorgi Kapanadze, Bakur Gulua, One problem of the bending of a plate for a curvilinear quadrangular domain with a rectilinear cut, Seminar of I. Vekua Institute of Applied Mathematics REPORTS Volume 42, 27-33, Tbilisi University Press, 2016
In the present paper we consider the problem of bending of a plate for a curvilinear quadrangular domain with a rectilinear cut. It is assumed that the external boundary of the domain composed of segments (parallel to the abscissa axis) and arcs of one and the same
circumference. The internal boundary is the rectilinear cut (parallel to the Ox-axis). The plate is bent by normal moments applied to rectilinear segments of the boundary, the arcs of the boundary are free from external forces, while the cut edges are simply supported.
The problem is solved by the methods of conformal mappings and boundary value problems of analytic functions. The sought complex potentials which determine the bending of the midsurface of the plate are constructed effectively (in the analytical form). Estimates are
given of the behavior of these potentials in the neighborhood of the corner points.
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Tengiz Meunargia, The isometric system of coordinates and the complex form of the system of equations for the non-shallow and nonlinear theory of shells, Seminar of I. Vekua Institute of Applied Mathematics REPORTS, Volume 42, 47-53, Tbilisi University Press, 2016
In this paper, the 3-D geometrically and physically nonlinear theories of non-shallow shells are considered. The isometrical system of coordinates is of special interest, since in this system we can obtain bases equations of the theory of shells in a complex form. This circumstance makes is possible to apply the methods developed by N. Muskhelishvili and his disciples by means of the theory of functions of a complex variable and integral equations.
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Giorgi Kapanadze, Bakur Gulua, About One Problem of Plane Elasticity for a Polygonal Domain with a Curvilinear Hole, AMIM, 21 (1), 121-129, Tbilisi University Press, 2016
In the present paper we consider a plane problem of elasticity for a polygonal domain with a curvilinear hole, which is composed of the rectilinear segment (parallel to the abscissa axis) and arc of the circumference. The problem is solved by the methods of conformal mappings and boundary value problems of analytic functions. The sought complex potentials are constructed effectively (in the analytical form). Estimates of the obtained solutions are derived in the neighborhood of angular points.
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Bakur Gulua, Solution of Boundary Value Problems of Spherical Shells by the Vekua Method for Approximation N=2, AMIM, 21 (2), 3-15, Tbilisi University Press, 2016
In the present paper we consider the geometrically nonlinear and non-shallow spherical shells, when components of the deformation tensor have nonlinear terms. By means of I. Vekua’s method the system of equilibrium equations in two variables is obtained. Using complex variable functions and the method of the small parameter approximate solutions are constructed for N = 2 in the hierarchy by I. Vekua. Concrete problem has been solved.
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Bakur Gulua, Giorgi Kapanadze, Some Boundary Value Problems for Plane Theory of Elasticity for Doubly-Connected Domain, AMIM, 21 (2), 38-45, Tbilisi University Press, 2016
We consider a plane problem of elasticity for double-connected domain bounded by polygons. The problem is solved by the methods of conformal mappings and boundary value problems of analytic functions. The sought complex potentials are constructed effectively (in the analytical form). Estimates of the obtained solutions are derived in the neighborhood of angular points.
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Bakur Gulua, Roman Janjgava, Miranda Narmania, Derivation of System of the Equations of Equilibrium for Shallow Shells and Plates, Having Double Porosity, AMIM, 21 (2), 16-37, Tbilisi University Press, 2016
We consider the three-dimensional system of the equations of elastic static equilibrium of bodies with double porosity. From this system of the equations, using a method of a reduction of I. Vekua, we receive the equilibrium equations for the shallow shells having double porosity. Further we consider a case of plates of constant thickness in more detail. Namely, the system of the equations corresponding to approximations N=1 it is written down in a complex form and we express the general solution of these systems through analytic functions of complex variable and solutions of the Helmholtz equation. The received general representations of decisions give the opportunity to analytically solve boundary value problems about elastic equilibrium of plates with double porosity.
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Tengiz Meunargia, On the 2-D Nonlinear Systems of Equations for Non-Shallow Shells (E. Reissner, D. Naghdi, W. Koiter, A. Lurie, I. Vekua), AMIM, 22 (2), 64-72, Tbilisi University Press, 2016
I. Vekua constructed several versions of the refined linear theory of thin and shallow shells, containing, the regular processes by means of the method of reduction of 3-D problems of elasticity to 2-D ones. In the present paper, by means of Vekua’s method, the system of differential equations for the Geometrically nonlinear theory non-shallow shells is obtained.
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David Gulua, Jemal Rogava, On the perturbation algorithm for the semi-discrete scheme for the evolution equation and estimation of the approximate solution error using semigroups, Computational Mathematics and Mathematical Physics, volume 56, Issue 7, pp 1269–1292, , 2016
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Tamaz Kaladze, Comment on “Large-scale Alfven vortices” [Phys. Plasmas 22, 122901 (2015)], Physics of Plasmas, v.23, No.3, 034703, American Institute of Physics, 2016
In the recently published paper,1 the authors assert that they have investigated large-scale nonlinear dispersionless Alfven waves and have obtained appropriate vortex structures.
First of all, this conclusion contradicts the generally accepted fact that in nonlinear plasma theory, waves without dispersion at the nonlinear stage should undergo steepening leading to break. The authors did not explain the physical mechanism of self-organization of dispersionless Alfven waves into stationary vortical structures.
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Gia Giorgadze, Giorgi Akhalaia, Valerian Jikia, On the Vortex equation on the complex plane, Proceedings of I. Vekua Institute of Applied Mathematics, Volume 66, 20-33, Tbilisi University Press, 2016
In this paper we consider the vortex equation as a particular case
of Carleman-Bers-Vekua Equation and analyzed solutions space of this equation from the point of view of generalized analytic functions.
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