We have some ideas about connecting the heat conduction equation to the viscous-elastic equation of the beam. These two equations will be related by the temperature dependence of the viscosity coefficient. From the standpoint of continuous body mechanics, we want to connect Hooke's and Archimedes' laws; Also Pascal's law and Saint-Venant's principle.

We present an unranked probabilistic logic, which is an extension of the first-order probability logic with sequence variables and flexible-arity (unranked) function and predicate symbols. The semantics of the logic is defined using Kripke worlds and the strong completeness theorem holds for it. Such a formalism is interesting as it provides very flexible and expressive platform to model various problems coming from real world applications.

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The purpose of this work is to study the nonstationary flow of a mixture of hydrogen and natural gas under high pressure in a natural gas pipeline. The density of a homogeneous mixture of hydrogen and natural gas is determined by an expression averaging the two gas densities at which a polytropic process is allowed. Pressure, temperature and flow rate are considered as three main dependent variables, which are determined by nonlinear hyperbolic partial differential equations. Analytical solutions of one-dimensional partial differential equations describing the isothermal gas flow in inclined and inclined-branched pipelines are presented

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In the talk, some subsets of the Euclidean plane $R^2$ that are Mazurkiewicz type sets with respect to the family of all straight lines lying in $R^2$ are considered. Analogously, subsets of $R^2$ that are Mazurkiewicz type sets with respect to the family of all circumferences lying in $R^2$ are studied. Several statements highlighting the above-mentioned sets from the viewpoint of the Baire property and various measures on $R^2$ are presented

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მოხსენება ეხება $(0,1)$-ზე განსაზღვრული ფუნქციათა ზოგიერთი $A$ კლასის წარმომდგენი უნივერსალური ფუნქციების არსებობის საკითხს. მოვიყვანოთ აუცილებელი განსაზღვრებები: 1) ვიტყვით, რომ $E\subset (0,1)$ სიმრავლე არის $A$ კლასის ცალსახობის სიმრავლე, თუ $A$ კლასის ნებისმიერი $f(x)$ და $g(x)$ ფუნქციებისათვის, რომელთათვისაც $f(x)=g(x)$, როცა $x\in E$ გამომდინარეობს, რომ $f(x)=g(x)$ ნებისმიერი $x\in (0,1)$ -სთვის; 2) ვიტყვით, რომ $(0,1)$-ზე განსაზღვრული $F(x)$ ფუნქცია არის ფუნქციათა $A$ კლასის წარმომდგენი უნივერსალური ფუნქცია, თუ ნებისმიერი $f(x)\in A$ ფუნქციისათვის არსებობს $A$ კლასის ცალსახობის $E$ სიმრავლე ისეთი, რომ $F(x)=f(x)$ ნებისმიერი $x\in E$ -სთვის. შევნიშნოთ, რომ დგინდება $(0,1)$-ზე განსაზღვრულ ფუნქციათა ისეთი კლასის არსებობა, რომელსაც არ გააჩნია წარმომდგენი უნივერსალური ფუნქცია. მოხსენებაში განვიხილავთ $(0,1)$-ზე უწყვეტ ყველა ფუნქციათა $A=C(0,1)$ კლასს, რომლისთვისაც ცხადია, რომ $E$ სიმრავლე არის $C(0,1)$ კლასის ცალსახობის სიმრავლე მაშინ და მხოლოდ მაშინ, თუ $E$ არის $(0,1)$-ში ყველგან მკვრივი სიმრავლე. ზემოთ მოყვანილი შენიშვნის გამო ბუნებრივია შეკითხვა: არსებობს თუ არა $C(0,1)$ კლასის წარმომდგენი უნივერსალური ფუნქცია? ამ კითხვაზე პასუხი დადებითია. კერძოდ, სამართლიანია თეორემა. არსებობს ისეთი $F(x)$ ფუნქცია, რომ $F(x)\in L_p (0,1)$ ნებისმიერი $p>0$-სთვის და $F(x)$ არის $C(0,1)$ კლასის წარმომდგენი უნივერსალური ფუნქცია. შევნიშნოთ, რომ თეორემაში აღნიშნული $F(x)$ ფუნქცია ასევე არის $(0,1)$-ზე განსაზღვრულ წყვეტილ ფუნქციათა ზოგიერთი კლასის წარმომდგენი უნივერსალური ფუნქცია

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The presented article is a direct continuation of the articles [1]-[3] that considered an initial-boundary value problem for the J. Ball integro-differential equation, which describes the dynamic state of a beam. The solution is approximated utilizing the Galerkin method, stable symmetrical difference scheme and the Jacobi iteration method. In the articles [1]-[2] the algorithm has been approved by tests. In the article [3]-[4] and this paper presents the approximate solution to one practical problem. Particularly, the results of numerical computations of the initial-boundary value problem for an iron beam. In the presented article the case where the effective viscosity depends on the temperature is discussed. The results of numerical calculations qualitatively satisfactorily describe the process under consideration References: 1. Papukashvili, Archil; Papukashvili, Giorgi; Sharikadze, Meri. Numerical calculations of the J. Ball nonlinear dynamic beam. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 32 (2018), 47-50. 2. Papukashvili, Archil; Papukashvili, Giorgi; Sharikadze, Meri. On a numerical realization for a Timoshenko type nonlinear beam equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 33 (2019), 51-54. 3. Papukashvili, Archil; Geladze, Giorgi; Vashakidze, Zurab; Sharikadze, Meri. On the Algorithm of an Approximate Solution and Numerical Computations for J. Ball Nonlinear Integro-Differential Equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 36 (2022), 75-78. 4. Papukashvili, Archil; Geladze, Giorgi; Vashakidze, Zurab; Sharikadze, Meri. Numerical solution for J.Ball’s beam equation with velocity – dependent Effective viscosity. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 37 (2023), 35-38.

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Probability theory deals with the challenges posed by uncertainty, while logic is more used for reasoning with perfect knowledge. First-order Probabilistic Logic combines capability of probability and logic. It gives expressive and flexible platform to model and reason problems coming from with Artificial Intelligence (AI). Unranked First-order logic is an variant of First-order logic with function symbols having flexible arity. Such an extension brings flexibility and expressiveness in the language to model and reason with unstructured data. In this talk we propose probabilistic extension of unranked First-order logic. In particular, we discuss syntax, semantic and inference mechanism of the extended formalism – probabilistic unranked logic.

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A multidimensional analogue of Maxwell's system of nonlinear partial differential equations [1] is considered. Numerical experiments have been conducted using decomposition algorithms [2, 3] and machine learning methods [4]. Analysis of the obtained results and comparison of computer experiments with theoretical conclusions is done. Acknowledgements. This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [grant number FR-21-2101]. References 1. Landau, L., Lifschitz E. Electrodynamics of Continuous Media, Course of Theoretical Physics, Moscow, 1957. 2. Abuladze, I.O., Gordeziani, D.G., Dzhangveladze, T.A., Korshiya, T.K. Discrete models for a nonlinear magnetic-field-scattering problem with thermal conductivity. Differential’nye Uravnenyia, 22, 7 (1986), 1119-1129. English translation: Differential Equations, 22, 7 (1986), 769-777 (Russian). 3. Jangveladze, T. Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations. Mem. Differential Equations Math. Phys., 76 (2019), 1-118. 4. Raissi, M., Perdikaris. P., Karniadakis. G. E., Physics informed deep learning (Part I): data-driven solutions of nonlinear partial differential equations. arXiv 1711.10561, 2017; https://arxiv.org/abs/1711.10561.

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One multidimensional model based on Maxwell’s well-known system of nonlinear partial differential equations [1], describing the process of penetration of a magnetic field in a substance is considered. The uniqueness of the solutions of the corresponding initial-boundary problems, the convergence of decomposition method and the finite-difference scheme are studied. The results are an extension of some outcomes obtained in [2, 3]. One multidimensional biological model is also considered [4]. Algorithms of sum approximation [5] and variable directions [6, 7] have been studied for the corresponding initial-boundary problem. Acknowledgements. This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [grant number FR-21-2101]. References 1. Landau, L., Lifschitz E. Electrodynamics of Continuous Media, Course of Theoretical Physics, Moscow, 1957. 2. Abuladze, I.O., Gordeziani, D.G., Dzhangveladze, T.A., Korshiya, T.K. Discrete models for a nonlinear magnetic-field-scattering problem with thermal conductivity. Differential’nye Uravnenyia, 22, 7 (1986), 1119-1129. English translation: Differential Equations, 22, 7 (1986), 769-777 (Russian). 3. Jangveladze, T. Investigation and numerical solution of nonlinear partial differential and integrodifferential models based on system of Maxwell equations. Mem. Differential Equations Math. Phys., 76 (2019), 1-118. 4. Mitchison, G.J. The polar transport of auxin and vein patterns in plants, Philos. Trans. R. Soc. Lond. B Biol. Sci., 295 (1981), 461-471. 5. Dzhangveladze, T.A. Averaged model of sum approximation for a system of nonlinear partial differential equations, Proc. I. Vekua Inst. Appl. Math., 19 (1987), 60–73 (Russian). 6. Jangveladze T.A. The difference scheme of the type of variable directions for one System of nonlinear partial differential equations. Proc. I.Vekua Inst. Appl. Math., 47 (1992), 45-66. 7. Jangveladze, T., Kiguradze, Z., Gagoshidze, M., Nikolishvili, M. Stability and convergence of the variable directions difference scheme for one nonlinear two-dimensional model. International Journal of Biomathematics. 8.5 (2015), 1550057 (21 pages), DOI: 10.1142/S1793524515500576

Due to global warming, the glaciers of the Caucasus (Georgia) have also undergone significant changes. In particular, some of them have disappeared, and the majority are degraded. During the retreat, the area of glaciers decreased, but at the same time the total number of glaciers increased [1]. In general, glaciers play an important role on both a global and regional scale, and their reduction or disappearance would cause significant damage to natural ecosystems and economies. Since glacier-climate interactions are complex, non-linear processes, we use mathematical modeling to predict the adaptation of Georgia's glaciers to ongoing climate change [2,3]. In this paper, for the first time, using mathematical modeling, the process of melting of Buba and Tbilisi glaciers is evaluated against the background of regional climate change. Some simulation results are presented and analyzed.

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The article focuses on the discussion of basic approaches to hypotheses testing in sequential experiments, which are Wald and Berger sequential tests and the test based on Constrained Bayesian Method (CBM). The positive and negative aspects of these approaches are considered and demonstrated on the basis of computed examples.

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Electrical conduction processes in the vasculature have traditionally been described using cable theory, i.e., locally induced signals (membrane potential ) decaying passively along the arteriolar wall. The decay is typically quantified using the length-constant, , derived from cable theory. However, the applicability of cable theory to blood vessels depends on assumptions that are not necessarily fulfilled in small arteries and arterioles. It is known that arterioles consist of at least two cell layers—an endothelial cell (EC), and one or more smooth muscle cell (SMC) layers that are coupled by myoendothelial gap junctions (MEGJ). This work discusses arterioles with two cell layers and studies the change in membrane potential in the EC and SMC layers. The corresponding stationary problems are posed and solved analytically by the method of separation of variables. Numerical modeling of membrane potential propagation is performed using MATLAB software. Membrane isopotential contours, and 2D and 3D graphs corresponding to the obtained numerical results are presented.

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Cyber-Physical Systems, shortly CPS, are networks of controllers that interact or control physical environments. Some examples of CPS are cars, aircrafts, railway systems, smart traffic systems, and the like. Such systems are becoming very complex and hard to get right, and are very expensive to develop and certify. It is very important, that CPS behaves correctly and securely, and only formal mathematical methods can provide strong guarantees about system correctness and security. A candidate for such formalism can be rewriting logic – a simple, general and expressive logic for distributed systems, whose modeling and analysis is supported by the Maude language and tool. In this talk we will speak about different approaches to use the Maude programming language and the Lingua Franca framework for verification of CPS. We underline some difficulties in this process and techniques to overcome them.

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Traveling wave special exact solutions of (2+1)D nonlinear Gardner partial differential equation is obtained. The result is found through hyperbolic secant function and has spatially isolated structural form. Appropriate graphical illustration is given.

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A spongy bone can be considered a multi-porous area with its fissures and pores as the most evident components of a double porous system. The work studies the stress-strain state of a spongy jawbone near the implant under occlusal loading. A mathematical model of the problem is the contact problem of the theory of elasticity between the implant and the jawbone. The problem is solved by using the boundary element methods, which are based on the solutions of Flamant’s (BEMF) and Boussinesq’s (BEMB) problems. The cases of various lengths of implant diameter are considered. Stressed contours (isolines) in the jawbone are drafted and the results obtained by BEMF and BEMB for the different diameter implants are compared.

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The Earth's climate is determined by the complex interactions between the Sun, the oceans, the atmosphere, the cryosphere, the land surface, and the biosphere. The sun is the main driving force for Earth's weather and climate. The influence of solar activity on the global surface of the Earth is determined by temperature variations, which in turn causes instability and is expressed by turbulent effects. Standard approaches to identifying such relationships are often based on correlations between relevant time series. Here we present a new method of Granger causality that can infer/reveal the relationship between two fields. We compare solar activity–climate connections through magnetic turbulence revealed by correlation and Granger causality at different time scales.

The article presents the importance of studying the thermodynamic state of the atmosphere during the formation of convective clouds in the local area. Numerical values of the energy of instability for some regions of Eastern Georgia are given and the change of its numerical values during the day is determined.

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The Earth's climate is determined by the complex interactions between the Sun, the oceans, the atmosphere, the cryosphere, the land surface, and the biosphere. The sun is the main driving force for Earth's weather and climate. The influence of solar activity on the global surface of the Earth is determined by temperature variations, which in turn causes instability and is expressed by turbulent effects. Standard approaches to identifying such relationships are often based on correlations between relevant time series. Here we present a new method of Granger causality that can infer/reveal the relationship between two fields. We compare solar activity–climate connections through magnetic turbulence revealed by correlation and Granger causality at different time scales.

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On the construction of exact and approximate solutions of a number of problems corresponding to the system of equations with some constant coefficients in mathematical physics. For certainty, a number of initial-boundary problems of the spatial classical, moment, thermo-elasticity theory of elasticity are considered, their reduction to the study of a relatively simple operational equation and the efficient construction of the corresponding exact and approximate solutions. Then, the extension of this class of problems to the Cowin-Nunziato model of hollow porous elastic medium, the Kosera porous elastic medium with empty pores, the Kosera medium with two types of porosity, the thermoelastic medium with microtemperature consideration, and various relatively general systems that It was discussed in my works (see Tamaz Vashakmadze-75, pp. 174-194, 2012, compiled by G. Kifiani, TSU publishing house).

The scope of this work is to study some aspects of the formation of the regional climate of the Caucasus (with a specific focus on Georgia) against the background of the impact of mineral aerosols using modelling (the RegCM interactively coupled with a dust module, WRF-Chem, and HYSPLIT models) and satellite data (MODIS, CALIPSO). The annual mean, as well as the error in summer and winter temperatures, standard deviation and correlation coefficient compared to the CRU data were calculated for 8 sub-regions with different orographic and climate properties. The calculation results showed that dust aerosol is an active player in the climatic system of the Caucasus (Georgia). Numerical results showed that the inclusion of dust radiative forcing in the RegCM numerical model brought the simulated summer temperature closer to the observed temperature values. The mean annual temperature increased throughout Georgia in simulations that took into account the direct impact of dust. Calculations using the WRF-Chem and HYSPLIT models revealed that during the study period, aeolian dust was brought into the territory of the South Caucasus (Georgia) equally not only from Africa and the Middle East, but also from Central (Western) Asia deserts, which was not noted earlier

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This poster is a presentation of the interdisciplinary project that aims at formulating unification and matching problems used in probabilistic ontology reasoning, and to search and compare algorithms for their solution. Additionally, when there is no algorithm for solving them, our project aims to study the algebraic structures of degrees induced by Turing and other algorithmic reducibilities.

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We continue the study of thermohydrodynamics and humidity processes based on the numerical model of the mesoscale atmospheric boundary layer (MBLA) developed by us. A new classification of foehns (warm downward wind) is given. An attempt is made to study foehns based on our numerical model. A certain opinion is expressed about the movement of a fluid against a gravitational field in different media.

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The territory of Georgia is rich in solar, wind, hydro, tidal, geothermal and biomass renewable energy sources (can be used for electricity generation, space heating and cooling and water, and for transport), but at present Georgia properly uses only hydro energy. In the context of current regional climate change challenges, Georgia needs cleaner energy from sources that naturally replenish rather than deplete. Wind, thermal and hydrogen energy are among the possible solutions as they are currently considered one of the most promising fuels of the future. In this article, based on a three-dimensional hydrostatic mesoscale model, an air flow over the complex relief of the South Caucasus (Georgia) is studied. Numerical experiments have shown a strong influence of the orographic effects of the Caucasus (the Likhi Ridge) on the movement of air in the troposphere. Besides, the study of the wind regime and statistical characteristics of the Kolkhinsky region for the period 1960-2021 showed that the wind speeds were significant and important for the development of wind farms in Western Georgia. This study discusses also one mathematical model that describes the flow of a mixture of natural gas and hydrogen substances in a pipeline. The distribution of pressure and gas flow through a branched gas pipeline has been studied. In addition, ways to reduce transportation costs are being studied, that is, the economic aspect of various methods of transporting hydrogen using hydrogen gas trailers, liquid hydrogen tanks and hydrogen pipelines of various technical levels is being studied

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In this work we consider the issues of the approximate solutions and the results of numerical computations for the following two practical problems: 1. Non-linear initial-boundary value problem for the J. Ball dynamic beam. 2. Non-linear initial-boundary value problem for the Kirchhoff dynamic string. The presented article is a direct continuation of the articles [1]-[5] that consider the construction of algorithms and their corresponding numerical computations for the approximate solution of nonlinear integro-differential equations for the J. Ball dynamic beam (see [1]-[3]) and for the Kirchhoff dynamic string (see [4]-[5]). References [1] A. Papukashvili, G. Papukashvili, M. Sharikadze. Numerical calculations of the J. Ball nonlinear dynamic beam. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math., 32 (2018), 47-50. [2] A. Papukashvili, G. Papukashvili, M. Sharikadze. On a numerical realization for a Timoshenko type nonlinear beam equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math., 33 (2019), 51-54. [3] A. Papukashvili, G. Geladze, Z. Vashakidze, M. Sharikadze. On the Algorithm of an Approximate Solution and Numerical Computations for J. Ball Nonlinear Integro-Differential Equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math., 36 (2022), 75-78. [4] G. Papukashvili, J. Peradze. A numerical solution of a string oscillation equation. Rep. Enlarged Sess. Semin. I. Vekua Inst. Appl. Math., 23 (2009), 80-83. [5] J. Peradze. A numerical algorithm for the non-linear Kirchhoff string equation. Numer. Math., 102 (2005), 311-342.

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The presentation is devoted to the application of the potential and integral equations methods to the basic and mixed three-dimensional boundary value and boundary-transmission problems of the theory of elasticity. The main goals of the talk are an overview of the results obtained by representatives of the Georgian Mathematical School in the second part of the last century and new developments and extensions of the method to refined mathematical models of solid mechanics taking into account thermal and electro-magnetic physical fields. We treat problems of statics, steady state elastic oscillations and general dynamics for isotropic and anisotropic elastic solids.

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We investigate dynamical problem of zero approximation of hierarchical models for fluids. Applying the Laplace transform technique, we reduce the dynamical problem to the elliptic problem which depends on a complex parameter τ and prove the corresponding uniqueness and existence results. Further, we establish uniform estimates for solutions and their partial derivatives with respect to the parameter τ at infinity and via the inverse Laplace transform show that the original dynamical problem is uniquely solvable.

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In the present talk, the linear coupled model of elastic double-porosity materials are proposed in which the coupled phenomenon of the concepts of Darcy’s extended law and the volume fractions is considered. A two- dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of analytic functions of a complex variable and 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 the coupled theory of elasticity for double-porous bodies.

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Two different approaches were used to construct approximate solutions of the initial-boundary problem for the system of equations corresponding to a two-dimensional nonlinear model [1]. Such type models are studied in many works (see, for example, [2]-[7] and references therein). The first approach used a decomposition method based on an averaged model [3]. An appropriate scheme and necessary algorithms for computer implementation were built. The program was compiled and calculations were made for various tests. The second approach used a variable direction difference scheme [4]. Necessary algorithms for computer implementation were also built for this method. The number of operations was determined for both methods. The time required for the realization of the algorithms and the accuracy of the numerical experiments were compared with each other. An analysis of the obtained results was carried out, and appropriate conclusions were drawn. Acknowledgment This research has been supported by the Shota Rustaveli National Science Foundation of Georgia under the grant FR-21-2101. References [1] G. I. Mitchison, A model for vein formation in higher plants. Proc. R. Soc. Lond. B. volume 207, pages 79-109, 1980. [2] J. Bell, C. Cosner, W. Bertiger, Solution for a flux-dependent diffusion model. SIAM J. Math. Anal. volume 13, pages 758-769, 1982. [3] T. A. Dzhangveladze, Averaged model of sum approximation for a system of nonlinear partial differential equations (Russian). Proc. I.Vekua Inst. Appl. Math. volume 19, pages 60-73, 1987. [4] T. A. Jangveladze, The difference scheme of the type of variable directions for one system of nonlinear partial differential equations. Proc. I.Vekua Inst. Appl. Math. volume 42, pages 45-66, 1992. [5] T. Jangveladze, M. Nikolishvili, B. Tabatadze, On one nonlinear twodimensional diffusion system. Proc. 15th WSEAS Int. Conf. Applied Math. (MATH 10), pages 105-108, 2010. [6] T. Jangveladze, Z. Kiguradze, B. Tabatadze, M. Gagoshidze, Comparison of two methods of numerical solution of Mitchison biological system of nonlinear partial differential equations. International Journal of Mathematics and Computers in Simulation, volume 11, pages 25–31. 2017. [7] T. Jangveladze, Z. Kiguradze, M. Gagoshidze, Economical difference scheme for one multi-dimensional nonlinear system. Acta Mathematica Scientia. volume 39, no. 4, pages 971-988, 2019.

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For certainty, a number of initial-boundary problems of the spatial classical, moment, thermo-elasticity theory of elasticity are considered, their reduction to the study of a relatively simple operational equation and the efficient construction of the corres-ponding exact and approximate solutions. Then the extension of this class of problems to the Cowin-Nunciato model of hollow porous elastic medium, Kosserat porous elastic medium with empty pores, Kosserat medium with two types of porosity, thermoelastic medium with micro-temperature consideration and various relatively general systems discussed in my works will be consider (see ”Tamaz Vashakmadze-75,pp.174-194,” Reduction by G.Kipiani,2012,Tbilisi SU public.).

Consider the Emden-Fowler type difference equation.

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In this talk, a linear mathematical model of thermoviscoelasticity for porous materials is proposed in which the coupled effect of Darcys law and the concept the volume fraction of pores is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations of this model are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed. The uniqueness theorems for the regular (classical) solutions of the BVPs of steady vibrations are proved. The single-layer and double-layer potentials are constructed and the basic properties of these potentials are given. Finally, the existence theorems for classical solutions of the BVPs of steady vibrations are proved by means of the potential method and the theory of singular integral equations.

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Using the tanh-coth method the traveling wave special exact solutions of (1 + 1) and (2 + 1)D nonlinear Gardner partial differential equations are represented. The results are expressed through hyperbolic functions and have spatially isolated structural forms.

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On fluids in angular pipes and wedge-shaped canals Abstract. In the zero approximation of hierarchical models for fluids the full accordance is shown of peculiarities of setting the Dirichlet and Keldysh type boundary conditions by motion of the fluids in pipes of angular cross-sections with the results of experiments carried out by J. Nikuradze in L. Prandtl’s Laboratory at University of Göttingen. Jeffery-Hamel flow is the flow between two planes that meet at an angle was analyzed by Jeffery (1915) and Hamel (1916). We consider the flow between two surfaces that meet at the edge a dihedral (angle), whose sides are the tangents of the surfaces at the edge of dihedral.

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The presented discourse serves as a direct extension of the research papers [1, 2], which delve into the investigation of an initial-boundary value problem associated with Kirchhoff’s integrodifferential equation. This mathematical model effectively characterizes the dynamic behaviour of a string. To seek an approximate solution for this problem, a combined approach involving a Galerkin method, a stable symmetric finite difference scheme, and a Picard-type iterative method is employed. In article [1], the algorithm is tested using a simple test example, providing the error solely for the difference method. However, this work considers a more complex test example that allows us to assess the errors for both the difference method and the Galerkin method. Numerical computations are performed to validate the proposed approach, and the resulting findings are presented in both tabular and graphical formats. References [1] G. Papukashvili and J. Peradze, A numerical solution of string oscillation equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 23 (2009), 80–83. [2] J. Peradze, A numerical algorithm for the nonlinear Kirchhoff string equation. Numer. Math. 102 (2005), no. 2, 311–342.

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We consider incompressible viscous fluid flow for the small Reynolds number in the infinite domains. The velocity components of the flow satisfy the Stokes linear system with the equation of continuity and suitable initial-boundary conditions. The steady and unsteady cases are considered. The novel exact solutions for the axial fluid flow over the ellipsoid and the countable number of discs are obtained.

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Despite the fact that hydrogen in nature is not replenished naturally and is not depleted (by analogy with renewable energy sources), there is a growing interest around the world in using hydrogen for electricity generation or in industry, transport and other areas as a highly efficient energy source. Currently, Georgia uses only hydro, wind and geothermal energy from renewable energy sources and has good opportunities for producing and transporting hydrogen. Indeed, Kazakhstan, Turkmenistan and Azerbaijan are planning to produce “green” and “blue” hydrogen (having a modern production infrastructure for petrochemicals and huge resource potential) and develop the infrastructure and operational components of the “Middle Corridor” for its transportation using the TRACECA route through Georgia and Turkey to EU countries. While efforts are being made in the long term to build a dedicated hydrogen infrastructure (pipeline), blending hydrogen into the existing gas pipeline network is a more promising strategy for transporting hydrogen in the short term. Thus, studying the behavior of mixed flow in a pipeline is relevant to the analysis of several potential problems that arise when mixing hydrogen in natural gas networks. This article focuses on exploring how much hydrogen can be integrated into a gas pipeline from an operational point of view. Namely, on the basis of one mathematical model describing the flow of a mixture of natural gas and hydrogen substances in a pipeline, the distribution of pressure and gas flow through a branched gas pipeline was analytically obtained. Some aspects of the production and transportation of hydrogen as a highly efficient source of energy on the territory of Georgia under the conditions of climate change are discussed

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In this talk, we propose extending set matching to similarity relations. In this way, we incorporate some background knowledge into solving techniques with similarity relations. Although our set terms are interpreted as (finite) classical sets, their elements (arguments of set terms) might be related to each other by a similarity relation, which induces also a notion of similarity between set terms. We design a matching algorithm and study its properties. It can be useful in applications where the exact set matching techniques need to be relaxed to deal with quantitative extensions of equality such as similarity relations.

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In this talk we discuss sequent calculus for unranked probabilistic logic. We show that the calculus is sound and complete.

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We consider an alternative approach to investigate three-dimensional exterior mixed boundary value problems (BVP) for the steady state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, Ω− ⊂ R^3, has a compact boundary surface S = ∂Ω−, which is divided into two disjoint parts, the Dirichlet part S_D and the Neumann part S_N , where the displacement vector (the Dirichlet type condition) and the stress vector (the Neumann type condition) are prescribed respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right-hand sides of the resulting boundary integral equations system are vectorfunctions coinciding with the given Dirichlet and Neumann data of the problem in que stion. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate L_2-based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector-functions belonging to the Sobolev space [W^1_2,loc(Ω−)]^3 and satisfying the Sommerfeld–Kupradze radiation conditions at infinity.We also show that the obtained matrix boundary integral operator is invertible in the Lp-based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses C^α-Hölder continuity property in the closed domain Ω− with α =1/2 − ε, where ε > 0 is an arbitrarily small number.

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It is well known the existence of non-measurable sets on the real line R (Vital’s set, Bernstein’s set, etc., see, for example). This report will present relationship between countable equid composability of sets and existence a nonmeasurability set with respect to Lebesgue measure on real line R. Since any two (bounded or unbounded) point sets of R with nonempty interiors are countably equid composable, we get that there exists a Lebesgue non-measurable set on R.

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Investigated model is based on the well-known system of Maxwell’s equations and represents some of its generalizations. Such type models are studied in many works (see, for example, [1–6] and references therein). The one-dimensional case with a three-component magnetic field is considered. The asymptotic behavior of solution for initial-boundary value problem as time variable tends to infinity is studied. The question of linear stability of the stationary solution of the system and the possibility of the Hopf-type bifurcation is investigated. A finite-difference scheme is constructed. The convergence of this scheme is studied and an estimate of the error of the approximate solution is obtained. Corresponding numerical experiments are carried out. Acknowledgments This research has been supported by the Shota Rustaveli National Science Foundation of Georgia under the grant # FR-21-2101. References [1] T. A. Dzhangveladze, Stability of the stationary solution of a system of nonlinear partial differential equations. (Russian) Current problems in mathematical physics, Vol. I (Russian) (Tbilisi, 1987), 214–221, 481–482, Tbilis. Gos. Univ., Tbilisi, 1987. [2] T. Jangveladze and M. Gagoshidze, Hopf bifurcation and its computer simulation for onedimensional Maxwell’s model. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 30 (2016), 27–30. [3] T. Jangveladze, Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations. Mem. Differ. Equ. Math. Phys. 76 (2019), 1–118. [4] T. Jangveladze, Some properties of the initial-boundary value problem for one system of nonlinear partial differential equations. Bull. TICMI 25 (2021), no. 2, 137–143. [5] T. Jangveladze, Finite difference scheme for one system of nonlinear partial differential equations. Bull. Georgian Natl. Acad. Sci. (N.S.) 16 (2022), no. 2, 7–13. [6] Z. V. Kiguradze, On the stationary solution for one diffusion model. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 16 (2001), no. 1-3, 17–20.

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The two-dimensional system of nonlinear partial differential equations is considered. This system arises in the process of vein formation of young leaves [7]. There are many works where this and many models describing similar processes are also presented and discussed (see, for example, [1, 2, 8, 9] and references therein). Investigation and numerical solution of such type systems are discussed in many papers (see, for example, [1, 3–6] and references therein). In our note, the averaged model of sum approximation is used [3] and the variable directions difference scheme is also considered [4]. Comparison of numerical experiments of the proposed methods is done. Acknowledgments This research has been supported by the Shota Rustaveli National Science Foundation of Georgia under the grant # FR-21-2101. References [1] J. Bell, C. Cosner and W. Bertiger, Solutions for a flux-dependent diffusion model. SIAM J. Math. Anal. 13 (1982), no. 5, 758–769. [2] H. Candela, A. Martí́nez-Laborda and J. L. Micol, Venation Pattern Formation inArabidopsis thalianaVegetative Leaves. Developmental Biology 205 (1999), no. 1, 205–216. [3] T. A. Dzhangveladze, An averaged model of summary approximation for a system of nonlinear partial differential equations. (Russian) Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 19 (1987), 60–73. [4] T. A. Jangveladze, The difference scheme of the type of variable directions for one system of nonlinear partial differential equations. Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 47 (1992), 45–66. [5] T. Jangveladze, Z. Kiguradze and M. Gagoshidze, Economical difference scheme for one multi-dimensional nonlinear system. Acta Math. Sci. Ser. B (Engl. Ed.) 39 (2019), no. 4, 971–988. [6] T. Jangveladze, M. Nikolishvili and B. Tabatadze, On one nonlinear two-dimensional diffusion system. Proc. 15th WSEAS Int. Conf. Applied Math. (MATH 10), (2010), 105–108. [7] G. J. Mitchison, A model for vein formation in higher plants. Proc. R. Soc. Lond. B. 207 (1980), no. 1166, 79–109. [8] P. Prusinkiewicz, S. Crawford, R. S. Smith, K. Ljung, T. Bennett, V. Ongaro and O. Leyser, Control of Bud Activation by an Auxin Transport Switch. Proc. Nat. Acad. Sci. 106(41) (2009), 17431–17436. [9] C. J. Roussel and M. R. Roussel, Reaction–diffusion models of development with statedependent chemical diffusion coefficients, Progress Biophys. Molecular Biology 86 (2004), no. 1, 113–160.

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One type of model of nonlinear parabolic integro-differential equations is considered. The analogous models partially are derived, on one hand, from the description of real diffusion processes and on the other hand, in the generalization of well-known equations and systems of equations, the study of which devoted many scientific papers (see, for example, [1-8] and references therein). Models of such types still yield to the investigation for special cases. In this direction, the latest and rather complete bibliography can be found in the following monographs [6, 7]. In our research uniqueness, stability and asymptotic behavior of the solutions of the initial-boundary value problems are studied. The finite-difference scheme is constructed and its convergence property is established. The approximate algorithm based on this scheme is constructed. Numerical implementation with various experiments for different values of the input parameters is performed to validate the theoretical conclusions. Acknowledgments This research has been supported by the Shota Rustaveli National Science Foundation of Georgia under the grant FR-21-2101. References [1] M. M. Aptsiauri, T. A. Jangveladze and Z. V. Kiguradze, Asymptotic behavior of the solution of a system of nonlinear integro-differential equations. (Russian) Differ. Uravn. 48 (2012), no. 1, 70–78; translation in Differ. Equ. 48 (2012), no. 1, 72–80. [2] T. Chkhikvadze, On one system of nonlinear partial differential equations. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 36 (2022), 19–22. [3] T. Dzhangveladze, An Investigation of the First Boundary-Value Problem for Some Nonlinear Parabolic Integro-differential Equations. (Russian) Tbilisi State University, Tbilisi, 1983. [4] D. G. Gordeziani, T. A. Dzhangveladze and T. K. Korshiya, Existence and uniqueness of the solution of a class of nonlinear parabolic problems. (Russian) Differentsial’nye Uravneniya 19 (1983), no. 7, 1197–1207; translation in Differ. Equ. 19 (1984), no. 7, 887–895. [5] F. Hecht, T. Jangveladze, Z. Kiguradze and O. Pironneau, Finite difference scheme for one system of nonlinear partial integro-differential equations. Appl. Math. Comput. 328 (2018), 287–300. [6] T. Jangveladze, Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations. Mem. Differ. Equ. Math. Phys. 76 (2019), 1–118. [7] T. Jangveladze, Z. Kiguradze and B. Neta, Numerical Solutions of Three Classes of Nonlinear Parabolic Integro-Differential Equations. Elsevier/Academic Press, Amsterdam, 2016. [8] T. Jangveladze, Z. Kiguradze, B. Neta and S. Reich, Finite element approximations of a nonlinear diffusion model with memory. Numer. Algorithms 64 (2013), no. 1, 127–155.

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In this report, it is extended the classical theory (for the linear case developed by E. Goursat, H. Weyl, J. L. Walsh, S. Bergman, G. V. Koloson, N. Muskhelishvili, L. Bers, I. Vekua and so on) of finding general solutions and some boundary value problems of partial differential equations by applying complex analysis. We developed the method of solving system of essentially non- linear DEs when with Laplace and biharmonic operators, DEs containing composition for example of Laplace and Monge–Ampére operator. The method gives possibility to solve some boundary value problems as well. Then this method will be applied to the solution of boundary value problems corresponding to refined theories in enlarged sense for elastic plates and shells. In this direction, in the terminology of real analysis are constructed full numerical analogies to refined theories are proved the convergence of corresponding iteration processes Batumi, September 4 –9, 2023.

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We describe the semantics of CLP(MS): constraint logic programming over multiple similarity relations. Similarity relations are reflexive, symmetric, and transitive fuzzy relations. They help to make approximate inferences, replacing the notion of equality. Similarity-based unification has been quite intensively investigated, as a core computational method for approximate reasoning and declarative programming. In this talk we consider solving constraints over several similarity relations [1], instead of a single one. Multiple similarities pose challenges to constraint solving, since we can not rely on the transitivity property anymore. Existing methods for unification with fuzzy proximity relations (reflexive, symmetric, non-transitive relations) do not provide a solution that would adequately reflect particularities of dealing with multiple similarities. To address this problem, we develop a constraint solving algorithm for multiple similarity relations, prove its termination, soundness, and completeness properties. We integrate the solving algorithm into constraint logic programming schema and study semantics of obtained CLP(MS)

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We consider a new alternative potential method to investigate a mixed boundary value problem (BVP) for the Lamé system of elasticity in the case of the three-dimensional bounded domain Ω ⊂ R^3, when the boundary surface S = ∂Ω is divided into two disjoint parts, S_D and S_N , where the Dirichlet and Neumann type boundary conditions are prescribed respectively for the displacement vector and the stress vector. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudo-differential equations which contain neither extensions of the Dirichlet or Neumann data, nor the Steklov-Poincaré type operator. Moreover, the right-hand sides of the resulting pseudo-differential system are vectors coinciding with the Dirichlet and Neumann data of the problem under consideration. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L_2-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space [W^1_2(Ω)]^3 and representability of solutions in the form of a linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. Using a special structure of the obtained pseudo-differential matrix operator, it is also shown that the operator is invertible in the Lp-based Besov spaces with 4/3 < p < 4, which under appropriate boundary data implies C^α-Hölder continuity of the solution to the mixed BVP in the closed domain Ω with α =1/2 − ε, where ε > 0 is an arbitrarily small number.

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The present article is a continuation of the previously published papers [1]-[3], which examine the initial-boundary value problem for J. Ball's integro-differential equation. The equation models the dynamic behaviour of a beam. To obtain an approximate solution, a combination of the Galerkin method, a stable symmetric difference scheme, and the Jacobi iteration method is utilized. In papers [1]-[2], the numerical algorithm is validated using numerical samples. The present paper, along with [3], focuses on the application of the numerical solution to a practical problem. In particular, the numerical results of the initial-boundary value problem for a specific iron beam are presented, where the effective viscosity of the material depends on its velocity. The results are summarized in a tables and graphics. References 1. Papukashvili, Archil; Papukashvili, Giorgi; Sharikadze, Meri. Numerical calculations of the J. Ball nonlinear dynamic beam. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 32 (2018), 47-50. 2. Papukashvili, Archil; Papukashvili, Giorgi; Sharikadze, Meri. On a numerical realization for a Timoshenko type nonlinear beam equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 33 (2019), 51-54. 3. Papukashvili, Archil; Geladze, Giorgi; Vashakidze, Zurab; Sharikadze, Meri. On the Algorithm of an Approximate Solution and Numerical Computations for J. Ball Nonlinear Integro-Differential Equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 36 (2022), 75-78.

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The thermohydrodynamic foundations for the development of foehns are considered. A new classification of hair dryers is given. The possibility of laboratory simulation of foehn heat release is considered. Certain recommendations are given for the regulation of foehn processes in the real meteorological conditions. We continue the study of foehn processes on the basis of our numerical model of the mesoscale 2-dimensional boundary layer of the atmosphere

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The talk concerns 55 years long history of Ilia Vekua Institute of Applied Mathematics of Ivane Javaxishvili Tbilisi State University. The Institute was founded by Georgian mathematician and mechanist Ilia Vekua on October 29, 1968. The aim of the Institute was to carry out research on important problems of applied mathematics, to involve University professors, teachers and students in research activities on topical problems of applied mathematics in order to integrate mathematics into the study processes and research, and to implement mathematical methodologies and calculating technology in the non-mathematical fields of the University. In 1978,the Institute was named after its founder and first director Ilia Vekua. In December, 2006 - May, 2009 the Institute was acting at the Faculty of the Exact and Natural Sciences. In June, 2009 - September, 2016 the Institute was directly subordinated to the University Administration. Since the end of September, 2016 the Institute has a status of the Independent Scientific-Research Institute. At present, the Institute successfully continues and develops activities launched by his founder in the following four main scientific directions:  Mathematical problems of mechanics of continua and related problems of analysis;  Mathematical modelling and numerical mathematics;  Discrete mathematics and theory of algorithms;  Probability Theory and mathematical Statistics. The institute sees its mission as threefold:  Carrying out fundamental and practical scientific research in applied mathematics, mathematical and technical mechanics, industrial mathematics and informatics, undertaking state and private sector contracts to provide expert services;  Offering the university a high-level computer technology base for University professors and teachers, research employees and students undertaking their scientific research activities;  Supporting PhD and post-graduate students to attain scientific grants, as well as through employment within the Institute and participation in scientific conferences.

In the paper, boundary value problems for rectangular plates with voids having two circular holes are approximately solved. Stress concentration factors are calculated for various tension- compression boundary value problems. The basic two-dimensional equations for plates are obtained from the three-dimensional Cowin-Nunziato equations by the I. Vekua method. To construct approximate solutions of boundary value problems, general representations of the solution and the method of fundamental solutions are used

The present talk deals with linear model of elastic double-porosity materials in which the coupled phenomenon of the concepts of Darcy’s law and the volume fractions is considered. For the plane deformation the corresponding system of differential equations is written in a complex form and its general solution is presented with the use of analytic functions of a complex variable and a solutions of the Helmholtz equations. The boundary value problems are solved for a circle.

The issue of establishing von Kármán's nonlinear system of differential equations will be discussed in relation to the works of August Föpll, Timoshenko, Lev Landau, Ciarlet, Antman, Podio-Guidugli. The incompleteness of the corresponding equations will be shown in the sense that the first dependence of the von Kármán model, in which the main term is the biharmonic operator with respect to bending, is an equation, while the dependence containing the biharmonic operator of the Airy function is the Saint-Venant-Beltrami compatibility condition and not an equation. This fact is in full accordance with this with the opinions expressed by Truesdel regarding this model. A three-dimensional analogue of the spatial variable with a residual term will be presented. By truncation of the error, a non-stationary anisotropic nonlinear parameter-dependent system of differential equations of the von Kármán -Reissner-Mindlin type will be obtained.

Along with the perturbation method for linear operator equations, an alternative method of its solution is also considered; the design is performed not only when the basic system is a qualitative row, but also when we use any complete system defined on the area (for example, for a one-dimensional interval - a system of orthogonal polynomials). The quantity for each finite part of which the solving algorithm can efficiently obtain an approximate solution

Renewable energy is energy derived from sources that naturally replenish and do not deplete. The most popular such energy sources are: solar, wind, hydro, tidal, geothermal and biomass energies. The territory of Georgia is rich in such renewable energy sources (which can be used for electricity generation, space heating and cooling and water, and for transport), but at present Georgia properly uses only hydro and geothermal energy. In the face of the current challenges of climate change, the world needs cleaner (renewable) energy and hydrogen is one possible solution as hydrogen is currently considered one of the most promising fuels of the future. Turkmenistan and Azerbaijan plan to become a leader in the production of "green" and "blue" hydrogen (having a modern production infrastructure for petrochemistry and a huge resource potential) and its transportation along the TRACECA route through Georgia and Turkey to the EU countries. Thus, the study of the behavior of a mixture of natural gas and hydrogen substances when moving through pipelines has become an urgent task of our time and has attracted the attention of a number of scientists. This article discusses one mathematical model that describes the flow of a mixture of natural gas and hydrogen substances in a pipeline. The distribution of pressure and gas flow through a branched gas pipeline has been studied and presented. In addition, ways to reduce transportation costs are being studied, that is, the economic aspect of various methods of transporting hydrogen using hydrogen gas trailers, liquid hydrogen tanks and hydrogen pipelines of various technical levels is being studied. Acknowledgements. The research is funded by Shota Rustaveli National Scientific Foundation Grant No. FR-22-18445.

In the zero approximation of hierarchical models for fluids the full accordance of peculiarities of setting of the Dirichlet and Keldysh type boundary conditions by motion of the fluids in pipes of angular cross-sections with the results of experiments carried out by J. Nikuradze in L. Prandtl's Laboratory at University of Göttingen.

We investigate dynamical problem of zero approximation of hierarchical models for fluids [Jaiani, G. Mathematical Hierarchical models for fluids. Book of Abstracts of XIII Annual International Meeting of the Georgian Mechanical Union, (2022), p. 151]. Applying the Laplace transform technique, we reduce the dynamical problem to the elliptic problem which depends on a complex parameter τ and prove the corresponding uniqueness and existence results. Further, we establish uniform estimate for solutions and their partial derivatives with respect to the parameter $\tau$ at infinity and via the inverse Laplace transform show that the original dynamical problem is uniquely solvable.

The problem of the plane theory of viscous elasticity based on the Kelvin-Voigt model for a doubly-connected plate bounded by a circle and a convex polygon is concidered. Constant normal stresses (pressure) are acting at the points of the circumference, and constant normal stresses (or constant normal displacements) are given on the sides of the hole, and the lateral stresses are zero. Based on the theory of analytic functions, the problem of searching for complex potentials is reduced to the Riemann-Hilbert problem, and by solving the latter, the mentioned potentials are effectively constructed (in analytical form). Their estimates in the vicinity of the vertices of the corners are given and different specific cases - triangle, rectangle and rectilinear cut - are discussed.

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This poster is a presentation of the fundamental research project, that aims to develop an unranked probabilistic logic, study its properties and introduce reasoning method for it. It shows preliminary results obtained during the project preparation and objectives of the project, what is planned to be achieved.

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This poster is a presentation of the fundamental research project, that addresses the problem of developing novel symbolic techniques for supporting automated or semi-automated reasoning activities in theories modulo proximity and similarity relations.

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The glaciers of the Caucasus (Georgia) have undergone significant changes against the background of global warming. Most of them have disappeared, and some have suffered degradation. The glacier area has decreased during the retreat, but at the same time the total number of glaciers has increased. Generally the glaciers play a major role in formation the water balance of the region and their reduction or disappearance poses significant damage to the natural ecosystems and economy. This article presents an analysis of the change in the surface area of the glacier using multitemporal data sets for the Greater Caucasus, based on manual digitization of large-scale (1:50,000) topographic maps of the 1960s. and satellite images of 1964 (Corona), 1986 (Landsat 5) and 2014 (Landsat 8, ASTER). The paper deals with major meteorological factors operating on glaciers and the melting of direct solar radiation on the basis of the melting energy model of the Enguri basin glacier. Modern climate change is characterized by fluctuations in the balance of radiative energy in the lower troposphere, which determines the process of fluctuations in glaciers (melting of the thickness). Since the interaction between glacier and climate is a complex non-linear process, we use mathematical modeling to predict the future adaptation of Georgia's glaciers to current climate changes. With the help of a two-dimensional mathematical model of the dynamics of changes in the thickness of the glaciers, the configuration of the upper surface of the Caucasus glaciers were studied. Some typical problems of mathematical and numerical modeling of glaciers are discussed. For the first time, with the help of mathematical modeling, the process of melting of the Caucasus glaciers (Kazbeg 5030m) was estimated. Some simulation results are presented and analyzed.

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Dust aerosols represent one of the main pollutants on the territory of Georgia and impact on regional climate. In this study, the WRF/Chemistry model with dust module was used to study dust particles transportations on the territory of Georgia from the Sahara and Sahel in Africa, Arabian and ar-Rub’ al-Khali deserts located in the Middle East, Kyzylkum, Karakum and Great Salty in the Central Asia. The results of calculations have shown the WRF model was able to well simulate dust aerosols transportation on the territory of Caucasus in conditions of a complex relief of the environment (verified with CALIPSO and MODIS satellite products and HYSPLIT model). In addition we have executed sets of 30 years simulations (1985–2014) with and without dust effects by RegCM 4.7 model with 16.7 km resolution over the Caucasus domain and with 50 km resolution encompassing most of the Sahara, the Middle East, the Great Caucasus with adjacent regions. Results of calculations have shown that mineral dust aerosol influences on temperature and precipitations (magnitudes) spatial and temporally inhomogeneous distribution on the territory of Georgia. According to results of comparisons of the simulated dust aerosol optical depth seasonal distributions against to the observed ones gave a good agreement. Also dust radiative forcing inclusion has improved simulated summer time temperature, and seasonal distribution of simulated precipitation, but gives over estimation in annual total precipitation. Results of calculations have shown that dust aerosol is an inter-active player in the climate system of Georgia

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In the talk we discuss relation between the Fuchsiam systems of differential equations and Riemann–Hilbert boundary value problem with piecewise con- stant transition functions on the Riemann sphere.In particular, we prove that, for any Fuchsian system there exists a rational matrix function whose partial indices coincide with the splitting type of the canonical vector bundle induced from the Fuchsiansystem. From this we obtain solution of the Riemann–Hilbert boundary value problem for piecewise constant matrix func- tion in terms of holomorphic sections of vector bundle and we give algorithmfor calculating Lp-partial indices for the piecewise constant matrix functions induced from monodromy representation of the Fuchsian system. The results given in the talk based on the work [1] and [2]. Acknowledgment: This work was supported by the EU through the H2020-MSCA-RISE-2020 project EffectFact, Grant agreement ID: 101008140. References [1] Giorgadze, G. (2022). On the factorization and partial indices of piecewise constant matrix functions.Transactions of A. Razmadze Mathematical Institute, 176(3), 367–372. [2] Giorgadze, G., Gulagashvili, G. (2022). On the splitting type of holomorphic vector bundles induced fromregular systems of differential equation. Georgian Mathematical Journal, 29(1), 25–35.

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We consider a new alternative potential method to investigate a mixed boundary value problem (BVP) for the Lam ́e system of elasticity in the case of three-dimensional bounded domain Ω ⊂ R^3, when the boundary surface S = ∂Ω is divided into two disjoint parts, S_D and S_N , where the Dirichlet and Neumann type boundary conditions are prescribed respectively for the displacement vector and the stress vector. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations which contain neither extensions of the Dirichlet or Neumann data, nor the Steklov-Poincar ́e type operator. Moreover, the right hand sides of the resulting pseudodifferential system are vectors coinciding with the Dirichlet and Neumann data of the problem under consideration. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L_2-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space [W^1_ 2 (Ω)]^3 and representability of solutions in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that the operator is invertible in the Lp-based Besov spaces with 4/ 3 < p < 4, which under appropriate boundary data implies C^α-H ̈older continuity of the solution to the mixed BVP in the closed domain Ω with α =1/ 2 − ε, where ε > 0 is an arbitrarily small number.

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Axial symmetry occurs in various physical processes such as fluid flow in pipes, atmospheric vortices, astrophysical fluids. We study the axisymmetric homogeneous incompressible Newtonian fluid flow of large viscosity for the small Reynolds number over the axis of symmetry in the infinite area. Such flows are called Stokes flows or creeping flows. Creeping fluids (such as some oils, polymers) are widely used in industrial processes and microelectromechanical systems (MEMS). Velocity components of the flow satisfy the Stokes system (STS) with the equation of continuity and suitable initial-boundary conditions. The Stokes system is the Stokes approximation of the Navier-Stokes Equations (NSE) for the creeping flows and represents the linear system of parabolic equations. We consider STS in the cylindrical polar coordinates under the action of certain pressure. The existence of the bonded solutions of this system is proved. By the separation of variables the novel exact solutions of the Stokes system for the specific pressure are obtained in the unsteady case. The profiles of the velocity and the vortex are constructed by means of “MAPLE”.

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The present report is devoted to some aspects of geometrical and set-theoretical definetions of equidecomposability of point sets. Connections between the notions of finite equidecomposability and countable equidecomposability of point sets (figures) are shown. In particular: (a) if sets X and Y are finitely equidecomposable, then they are also countably equidecomposable, but the converse assertion does not hold; (b) if in the space R^n some sets X and Y are such that \lamda_n(X)>0 and \lamda_n(Y)=0, then these sets are not countably equidecomposable under the group of all affine transformations of R^n ; (c) in R^n there exist two sets X and Y such that card(X)=card(Y)=c and X is not countably equidecomposable with Y, under a sufficiently large group of transformations of R^n; (d) in R^n two points sets are countably equidecomposable if both of them have interior points. The latter implies that in the space R^n there exists a non-measurable set with respect to the Lebesgue measure.

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In the talk the following two related questions are considered and answered: 1) Let $d$ be an arbitrary fixed non-negative real number and let $n$ be an arbitrary fixed natural number such that $n\geq 3$. Is there a non-negative real number $r_{n,d}$ having the property: if $S$ is a subset of the Euclidean plane with $\text{card}(S)=n$ such that, for every straight line $L_{a,b}$ passing through two distinct points $a$ and $b$ of $S$, there exists a point $c$ of $S$ such that $a \neq c$ and $b \neq c$ and the distance between $L_{a,b}$ and $c$ does not exceed $d$, then there exists a straight line $L$ in the plane such that the distance from an arbitrary point of $S$ to $L$ is less than or equal to $r_{n,d}$? 2) Let $N_3$ be the set of all natural numbers greater than 2, $d$ be an arbitrary fixed positive real number, and let $n$ be an arbitrary fixed natural number such that $n\geq 3$. Is there a function of two variables $\f(m.y)$ acting from $N_3 \times (0,+\infty)$ into $(0,+\infty)$, satisfying $\lim\limits_{y\rightarrow 0+}f(m.y)=0$ for every $m\in N_3$, and if $S$ is a subset of the plane with $\text{card}(S)=n$ such that, for every straight line $L_{a,b}$ passing through two distinct points $a$ and $b$ of $S$ there exists a point $c$ of $S$ such that $a \neq c$ and $b \neq c$ and the distance from $c$ to $L_{a,b}$ does not exceed $d$, then there exists a straight line $L$ in the plane such that the distance from an arbitrary point of $S$ to $L$ is less than or equal to$f_{n,d}$? Also, an interrelation between the presented questions and the Sylvester-Gallai well-known theorem (see, e.g., [1], [2], [3]) is considered. . References 1. H. Hadwiger and H. Debrunner, Combinatorial Geometry in the Plane. Translated by Victor Klee. With a new chapter and other additional material supplied by the translator, Holt, Rinehart and Winston, New York, 1964. 2. A. Kharazishvili, Elements of Combinatorial Geometry, Part I, Tbilisi, 2016. 3. J. J. Sylvester, Question 11851, Educational Times, 59(1893), p. 98.

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The existence of almost everywhere convergent and divergent Rademacher series such that they are universal in the sense of convergence to any given continuous function on the everywhere dense set is established.

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The current article is a continuation of the previously published papers [1]-[3], which examine the initial-boundary value problem for J. Ball's integro-differential equation. The equation models the dynamic behaviour of a beam. To obtain an approximate solution, a combination of the Galerkin method, a stable symmetric difference scheme, and the Jacobi iteration method is utilized. In papers [1]-[2], the numerical algorithm is validated using numerical samples. The present paper, along with [3], focuses on the application of the numerical solution to a practical problem. In particular, the numerical results of the initialboundary value problem for a specific iron beam are presented, where the effective viscosity of the material depends on its velocity. The results are summarized in a table. References: 1. Papukashvili, A., Papukashvili, G., Sharikadze, M. Numerical calculations of the J. Ball nonlinear dynamic beam. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 32 (2018), 47-50. 2. Papukashvili, A., Papukashvili, G., Sharikadze, M. On a numerical realization for a Timoshenko type nonlinear beam equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 33 (2019), 51-54. 3. Papukashvili, A., Geladze, G., Vashakidze, Z., Sharikadze, M. On the Algorithm of an Approximate Solution and Numerical Computations for J. Ball Nonlinear Integro-Differential Equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 36 (2022), 75-78.

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A three–step method for a nonlinear integro-differential hyperbolic equation which describes the behavior of a dynamic string is presented. The method has been tested on an example.

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The problem of testing directional hypotheses is examined using the consideration of the basic and alternative hypotheses in pairs, allowing implementing computation easily and faster with guaranteed reliability. The concept of mixed directional false discovery rate (mdFDR) is used for the decision rule optimality. The fact of guaranteeing the quality of a decision (in the developed approach) on the desired level theoretically is proved and practically is demonstrated by computation of practical examples. The developed method is applied for testing multiple hypotheses that guarantee the restriction of the total mdFDR on the desired level. It is also shown that the offered method can be used for solving the problems of intersection-union, union-intersection hypotheses. The offered method is adapted for testing large numbers of the subsets of individual hypotheses at testing multiple hypotheses that saves computational time and resources. Reliability and convenience of the developed method for big data are demonstrated. References 1. Bahadur, R. R. (1952). A property of the t-statistics. Sankhya, 12, 79-88. 2. Bansal, N. K., Hamedani, G. G. & Maadooliat, M. Testing Multiple Hypotheses with Skewed Alternatives, Biometrics, 72, 2 (2016), 494-502. 3. Kachiashvili K. J. The Methods of Sequential Analysis of Bayesian Type for the Multiple Testing Problem. Sequential Analysis, 33, 1 (2014), 23-38. 4. Kachiashvili, K. J. (2018) Constrained Bayesian Methods of Hypotheses Testing: A New Philosophy of Hypotheses Testing in Parallel and Sequential Experiments. Nova Science Publishers, Inc., New York, 361 p. 5. Kachiashvili K. J., Kachiashvili J. K. & Prangishvili I. A. (2020) CBM for Testing Multiple Hypotheses with Directional Alternatives in Sequential Experiments. Sequential Analysis: Design Methods and Applications. (ID: 1727166 DOI:10.1080/07474946.2020.1727166).

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Hydrogen is currently considered one of the most promising fuels of the future. It is expected to be used in a wide variety of applications such as the generation and storage of electricity, automotive fuels and reactive devices, various industries and even our domestic energy needs [1]. At present, the problems of efficient production, storage and transportation of gaseous hydrogen are the main focus of many researchers around the world. The study of the behavior of a mixture of natural gas and hydrogen substances during flow in pipelines has become an urgent task of our time and has attracted the attention of a number of scientists [1-4]. This work is devoted to one mathematical model describing the flow of a mixture of natural gas and hydrogen substances in a pipeline. A quasi-nonlinear system of two-dimensional partial differential equations is considered, which describes the unsteady flow of a mixture of natural gas and hydrogen substances in a pipe. The distribution of pressure and gas flow through a branched gas pipeline has been studied. Some results of numerical calculations of a mixture of natural gas and hydrogen in a gas pipeline are presented. Acknowledgements. The research was funded by Shota Rustaveli National Scientific Foundation Grant No. FR-22-18445. 1. Ball, M., Basile, A., Veziroglu, T.N. Compendium of Hydrogen Energy: Hydrogen Use, Safety and the Hydrogen Economy; Woodhead Publishing: Cambridge, UK, 2015. 2. Hosseini, S.F., Wahid, M.A. Hydrogen production from renewable and sustainable energy resources: Promising green energy carrier for clean development. Renew. Sustain. Energy Rev. 2016, 57, 850–866. 3. Davitashvili, T. On liquid phase hydrates formation in pipelines in the course of gas nonstationary flow. E3S Web of Conferences 230, 01006 (2021), DOI: https://doi.org/10.1051/e3sconf/202123001006.

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Since the early days of Artificial Intelligence logical and probabilistic methods have been independently used in order to solve tasks that require some sorts of intelligence. Probability theory deals with the challenges posed by uncertainty, while logic is more often used for reasoning with perfect knowledge. Considerable efforts have been devoted to combining logical and probabilistic methods in a single framework, which influenced the development of several formalisms and programming tools.
All probabilistic logic formalisms studied so far permit only individual variables, that can be instantiated by a single term. On the other hand, theories and systems that use also sequence variables (these variables can be replaced by arbitrary finite, possibly empty, sequences of terms) and unranked symbols (function and/or predicate symbols without fixed arity) have emerged. The unranked term is a first-order term, where the same function symbol can occur in different places with different number of arguments. Unranked function symbols and sequence variables bring a great deal of expressiveness in language. Therefore, it is actual to study extension of probabilistic logic with sequence variables and flexible-arity function and predicate symbols.
This talk is a presentation of the fundamental research project, that aims to develop an unranked probabilistic logic, study its properties and introduce reasoning method for it. We discuss preliminary results obtained during the project preparation and objectives of the project, what is planned to be achieved.

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Cardiovascular diseases are still the leading factors of mortality worldwide. Particularly noteworthy, major reason of death is caused by the heart failure, for example, due to a heart attack and development of a fatal arrhythmia. The direct cause of fatal cardiac arrhythmias is still not completely investigated, however, in many cases the cause can be traced to a failure of the cardiac action potential to propagate correctly. Remarkably, the propagation of action potential is still not completely understood in spite of many years of investigation. Therefore, the study remains a critical topic of many modern scientific studies. The aim of the current work is to investigate the prevalence of action potentials in cardiac tissue using the cable equation. In the variation of the cable equation, developed by Lord Kelvin for modeling the propagation of electrical signals of underwater telegraphs, the passive one-dimensional cable equation is obtained, which is a model of monodomain and bidomain, which describes the electrical behavior of the cell membrane of the heart tissue and the propagation of the action potential. A homogeneous representation of heart tissue includes a large number of identical cells, which can be imagined as two interconnected spaces - intracellular and extracellular. Cells are connected to each other by gap junctions. The paper discusses 1D model of continuously connected myocytes. Here, due to the assumption of continuity, the electrical behavior in the tissue is average for many cells, so we will study the behavior of the transmembrane potential for a single cell. For monodomain, in the absence of current at the beginning and end of the cable (cell), numerical modeling in Matlab is carried out. Figures of the corresponding contours (isolines), 2D and 3D graphs of the obtained numerical results are presented.

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We consider the axi-symmetric incompressible unsteady fluid flow over the axis of symmetry for the small Reynolds number. The velocity components of the flow satisfy the nonlinear Navier - Stokes equations (NSE) with the suitable initial-boundary conditions. For the small Reynolds number NSE can be reduced to the Stokes linear system (STS) . We have studied the Stokes system in the axi-symmetric case when the pressure depends on time exponentially. By the separation of variables the exact solutions of STS are obtained.

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მოხსენებაში განსახილავი სხეული არის დრეკადი კოსერას გარემო სიცარიელეებით. ბრტყელი დეფორმაციის შემთხვევის შესაბამისი განტოლებების ორგანზომილებიანი სისტემა ჩაწერილია კომპლექსური ფორმით და მისი ზოგადი ამონახსნი წარმოდგენილია კომპლექსური ცვლადის ორი ანალიზური ფუნქციისა და ჰელმჰოლცის განტოლების ორი ამონახსნის გამოყენებით. ზოგადი წარმოდგენის საფუძველზე ამოხსნილია კონკრეტული სასაზღვრო ამოცანები წრიული რგოლისათვის.

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განხილულია ბლანტი დრეკადობის ბრტყელი თეორიის ამოცანა წრიული ფირფიტისათვის მრავალკუთხა ხვრელით კელვინ-ფოიგტის მოდელის საფუძველზე. იგულისხმება, რომ ფირფიტის გარე საზღვარზე მოქმედებენ ნორმალური მკუმშავი ძალვები (წნევა), ხოლო ხვრელში ჩადგმულია შედარებით დიდი ზომის ხისტი შაიბა ისე, რომ საზღვრის წერტილთა ნორმალური გადაადგილებები ღებულობენ მუდმივ მნიშვნელობებს და ხახუნის ძალები ნულის ტოლია. კონფორმულ ასახვათა და ანალიზურ ფუნქციათა სასაზღვრო ამოცანების თეორიის მეთოდების საფუძველზე საძიებელი კომპლექსური პოტენციალები აგებულია ეფექტურად (ანალიზური ფორმით). მოყვანილია აღნიშნული პოტენციალების შეფასება კუთხის წვეროების მახლობლად. განხილულია ზღვრული შემთხვევები (მართკუთხედი, სწორხაზოვანი ჭრილი).

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We consider tension-compression problems for rectangular porous aluminum plates with one or two circular holes. The corresponding two-dimensional system of equilibrium equations is obtained from the linear three-dimensional Cowin-Nunziato model by the method of successive differentiation. Boundary value problems are solved by an approximate method, for which general representations of the solution of a system of equilibrium equations and the method of fundamental solutions are used. Stress concentration factors on the contours of the hole are calculated.

A multidimensional analogue of one non-linear two-dimensional system of partial differential equations describing some biological processes is studied [1]. The averaged model of sum approximation [2] and the variable direction difference scheme [3] for an initial-boundary problem are considered. Various numerical experiments have been conducted and a comparative analysis of the obtained results is given. Acknowledgements. This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [grant number FR-21-2101]. References [1]. Mitchison, G.J. The polar transport of auxin and vein patterns in plants. Philos. Trans. R. Soc. Lond. B Biol. Sci., 295 (1981), 461-471. [2]. Dzhangveladze, T.A. Averaged model of sum approximation for a system of nonlinear partial dierential equations. Proc. I. Vekua Inst. Appl. Math., 19 (1987), 60-73 (Russian). [3]. Jangveladze, T., Kiguradze, Z., Gagoshidze, M., Nikolishvili, M. Stability and convergence of the variable directions dierence scheme for one nonlinear twodimensional model. International Journal of Biomathematics. 8, 5 (2015), 1550057 (21 pages), DOI: 10.1142/S1793524515500576.

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Along with the perturbation method for linear operator equations, an alternative method of its solution is also considered; the design is performed not only when the basic system is a qualitative row, but also when we use any complete system defined on the area (for example, for a one-dimensional interval - a system of orthogonal polynomials). The quantity for each finite part of which the solving algorithm can efficiently obtain an approximate solution

განვიხილოთ მეორე რიგის გადახრილ არგუმენტიანი ემდენ-ფაულერის ტიპის სხვაობიანი განტოლება. დადგენილია ამონახსნების რხევადობის ახალი ტიპის საკმარისი პირობები.

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Using the exp-function method the traveling wave special exact solutions of the (2+1)D nonlinear Zakharov-Kuznetsov partial differential equation in an electron-positron-ion plasma are represented. The results are expressed in the forms of hyperbolic, trigonometric, exponential and rational functions and have spatially isolated structural forms. Traveling wave velocity is defined as the function of dynamic parameters.

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In the zero approximation of hierarchical models for fluids [1] the full accordance of peculiarities of setting of the Dirichlet and Keldysh type boundary conditions by motion of the fluids in pipes of angular cross-sections with the results of experiments carried out by J. Nikuradze (see [2] and also [3]) in L. Prandtl's Laboratory at University of Göttingen. References 1. Jaiani, G. Mathematical Hierarchical models for fluids. Book of Abstracts of XIII Annual International Meeting of the Georgian Mechanical Union, (2022), p. 151. 2. Nikuradze, J. Untersuchungen über turbulente. Strömungen in nichtkreisförmigen Rohren. Ing. Arch. B. I. 1930. S306. 3. Kavtaradze, R. Johann (Iwane) Nikuradze. Mythos and Realitaet. Tbilisi, (2023), 154-155 (in Georgian).

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In this talk we extend simply typed lambda calculus with regular types and study properties of the extended formalism. Moreover, we construct higher-order unification procedure for regularly typed lambda terms, and prove soundness and completeness theorems.

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Along with the perturbation method for linear operator equations, an alternative method of its solution is also considered; the design is performed not only when the basic system is a qualitative row, but also when we use any complete system defined on the area (for example, for a one-dimensional interval - a system of orthogonal polynomials). The quantity for each finite part of which the solving algorithm can efficiently obtain an approximate solution.

The measure extension problem is one of the most important question in measure theory. It is known that there exist various measures on the real line R which strictly extend the classical Lebesgue measure λ on R and are invariant under the group of all isometric transformations of R. An interesting direction in measure theory is concerned with the investigation of properties of various (countably-additive) extensions of initial measures. In this connection, there are some well-known methods of extending invariant measures: Marczewski’s method; the method of Kodaira and Kakutani; the method of Kakutani and Oxtoby; the method of surjective homomorphisms. In the present talk we discuss, several methods of extending invariant and quasi- invariant measures. Moreover, we will demonstrate several classes of measures with a different cardinality number.

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The present work stress-strain state of the homogeneous isotropic body bounded by coordinate lines of the parabolic coordinate system is studied, when on parabolic border normal or tangential stress is given. Analytical solution is obtained by the method of separation of variables. Using the MATLAB software, the numerical results are obtained of some specific problems and relevant graphs are presented.

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Reasoning with incomplete, imperfect information is very common in human communication. For such problems, exact equality/equivalence is replaced by its approximation. This kind of reasoning is a highly nontrivial task and remains an important issue in applications of artificial intelligence. Modeling the incomplete and imprecise information is achieved using so called proximity relations, which are reflexive and symmetric, but not necessarily transitive relations. When we have transitivity, we get so called similarity relation, i.e. fuzzy equivalence relation. While similarity-based unification and crisp set unifications are separately well-studied techniques, their combinations has attracted less attention. In this talk we define similarity-based crisp set unification problem and discuss possible solutions.

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Constrained Bayesian method (CBM) is used for testing asymmetrical hypotheses. The direct application of all statements of CBM allows us to make decisions on the desired levels of reliability. There is proven that mixed directional false discovery rates (mdFDR) are restricted on the desired levels at the suitable choice of restriction levels at different statements of CBM. The computation of concrete exa mples confirms the correctness of theoretical results for different statements of CBM. References [1] N. K. Bansal and R. Sheng, Bayesian decision theoretic approach to hypothesis problems with skewed alternatives. J. Statist. Plann. Inference 140 (2010), no. 10, 2894–2903. [2] K. J. Kachiashvili, Constrained Bayesian Methods of Hypotheses Testing: A New Philosophy of Hypotheses Testing in Parallel and Sequential Experiments. Nova Science Publishers, Inc., New York, 2018

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The present report is devoted to various definitions of quidecomposability of sets. The connection between finitely equidecomposable and countable equidecomposable sets will be shown. In particular: (a) if X and Y are finitely equidecomposable, then they are also countable equidecomposable; (b) in R^n there exist two sets X and Y with λ_n(X) > 0 and λ_n(Y) = 0, which are not countable equidecomposable under than of all affine translations of R^n; (c) in R^n there exist two sets X and Y such that card(X) = card(Y ) = c and X is not countably equidecomposable with Y.

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The neutral differential equation is a mathematical model of such system whose behavior at a given moment depends on the velocity and state of the system in the past. Many real processes are described by neutral differential equations. In the present work is considered the quasi-linear neutral differential equation with the two types controls, where one control function is piecewise-continuous and the second control function is measurable. The theorem about continuity of solution on the initial data is proved. Here, under the initial data we mean the collection of delay parameter; initial function ; initial vector and control functions.

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A system of fourth-order nonlinear integro-differential equations is considered. The second-order analogous models partially are derived, on one hand, from the description of real diffusion processes and on the other hand, in the generalization of well-known equations and systems of equations, the study of which devoted many scientific papers (see, for instance, [2, 3, 4] and references therein). Such type higher order models are also studied in some other works (see, for instance, [1, 3, 5] and references therein). In our research uniqueness and stability of the solution of the initialboundary value problem for fourth-order models are studied. The approximate algorithm based on the finite-difference scheme is constructed and corresponding numerical experiments are fulfilled. Acknowledgements This research has been supported by the Shota Rustaveli National Science Foundation of Georgia under the grant FR-21-2101. References [1] T. Chkhikvadze, On one nonlinear integro-differential parabolic equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 35 (2021), 19–22. [2] D. G. Gordeziani, T. A. Dzhangveladze and T. K. Korshiya, Existence and uniqueness of a solution of certain nonlinear parabolic problems. (Russian) Differential’nye Uravnenyia 19 (1983), no. 7, 1197–1207; translation in Differ. Equ. 19 (1984), no. 7, 887–895. [3] T. Jangveladze, Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations. Mem. Differ. Equ. Math. Phys. 76 (2019), 1–118. [4] T. Jangveladze, Z. Kiguradze and B. Neta, Numerical Solutions of Three Classes of Nonlinear Parabolic Integro-Differential Equations. Elsevier/Academic Press, Amsterdam, 2016. [5] T. Paikidze, On one system of fourth-order nonlinear integro-differential parabolic equation. Rep. Enlarged Sess. Semin. I. Vekua Inst. Appl. Math. 36 (2022) (accepted).

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A one-dimensional system of nonlinear partial differential equations is considered. Such type models are based on the well-known Maxwell system and are studied in many works (see, for instance, [1–5] and references therein). The asymptotic behavior of solution for initial-boundary value problem as time variable tends to infinity is studied. The question of linear stability of the stationary solution of the system and the possibility of the Hopf-type bifurcation is investigated. Finite difference scheme is constructed and corresponding numerical experiments are carried out. Acknowledgements This research has been supported by the Shota Rustaveli National Science Foundation of Georgia under the grant FR-21-2101. References [1] T. A. Dzhangveladze, Stability of the stationary solution of a system of nonlinear partial differential equations. (Russian) Current problems in mathematical physics, Vol. I (Russian) (Tbilisi, 1987), 214–221, 481–482, Tbilis. Gos. Univ., Tbilisi, 1987. [2] T. Jangveladze and M. Gagoshidze, Hopf bifurcation and its computer simulation for onedimensional Maxwell’s model. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 30 (2016), 27–30. [3] T. Jangveladze, Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations. Mem. Differ. Equ. Math. Phys. 76 (2019), 1–118. [4] Z. Kiguradze, On the stationary solution for one diffusion model. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 16 (2001), no. 1-3, 17–20. [5] N. Mzhavanadze, On one nonlinear diffusion system. Rep. Enlarged Sess. Semin. I. Vekua Inst. Appl. Math. 36 (2022) (accepted).

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An initial-boundary value problem is posed for the J. Ball integro-differential equation, which describes the dynamic state of a beam (see [1]). A physical model that J. Ball uses in the article [1] is taken from the handbook of Engineering Mechanics written by E. Mettler (see [2]). For this model, he wrote the corresponding initial-boundary value problem for the integro-differential equation of beam [1]. The presented article is a direct continuation of the articles [3, 4] that consider the construction of algorithms and their corresponding numerical computations for the approximate solution of nonlinear integro-differential equations of the Timoshenko type. In particular, in this work, it is considered an initial-boundary value problem for the J. Ball integro-differential equation, which describes the dynamic state of a beam [1]. The solution is approximated utilizing the Galerkin method, stable symmetrical difference scheme and the Jacobi iteration method. In the articles [3,4] the algorithm is approved by tests. This paper presents the approximate solution to one practical problem. Particularly, the results of numerical computations of the initial-boundary value problem for an iron beam are represented in the tables. Issues of the initial-boundary value problem of the iron beam are studied for the following meanings of parameters: spatial, temporal, mathematical algorithm and physical nature of the beam. References [1] J. M. Ball, Stability theory for an extensible beam. J. Differential Equations 14 (1973), 399–418. [2] E. Mettler, Dynamic buckling. In Handbook of engineering mechanics (S. Flügge, Ed.), Chapter 62, McGraw-Hill, New York, 1962. [3] A. Papukashvili, G. Papukashvili and M. Sharikadze, Numerical calculations of the J. Ball nonlinear dynamic beam. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 32 (2018), 47–50. [4] A. Papukashvili, G. Papukashvili and M. Sharikadze, On a numerical realization for a Timoshenko type nonlinear beam equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 33 (2019), 51–54.

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This paper proposes a simple mathematical model of the dynamics of change in the thickness of glaciers in the Caucasus, based on the integration of nonlinearly generated differential equations. To some extent the model takes into account the change in the mass balance of the glacier due to the direct solar radiation. A scheme similar to the Lax–Wendroff scheme is used to numerically solve the nonlinear PDE. Some typical problems inherent in mathematical and numerical modeling of glaciers are discussed. For the first time, the process of melting of some glaciers in the Caucasus has been assessed using mathematical modeling. Some simulation results are presented and analyzed

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The strong wind regime and statistical characteristics of the Imereti region were researched according to the data of the Kutaisi Meteorological Station. For the period 1960–2021, the wind speeds are divided into intervals of 5 m/s, and for each interval the wind speed recurrence rate is studied by months. The paper presents the percentage distribution of wind speed gradations and the change in their average values over the years and months. It has been determined that in terms of energy, the main range of wind speed for the Kutaisi region is 16–20 m/s. It should also be mentioned that the wind values at intervals of 20–25 m/s in summer are minimal compared to the wind values in other seasons. But from an energy point of view, this is less important because a wind speed interval of 16–20 m/s ensures maximum efficiency of wind energy use. Thus, from an energy point of view, speeds of such magnitude are essential, which ensure the automatic mode of the wind farms and are an important basis for the development of wind power plants in western Georgia.

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Probability theory deals with the challenges posed by uncertainty, while logic is more used for reasoning with perfect knowledge. Probabilistic logic combines capability of probability and logic. It gives expressive and flexible platform to model and reason problems coming from with Artificial Intelligence (AI). Unranked predicate logic is an variant of predicate logic with function symbols having flexible arity. Such an extension brings flexibility and expressiveness in the language to model and reason with unstructured data. In this talk we propose probabilistic extension of unranked predicate logic. In particular, we discuss syntax, semantic and inference mechanism of the extended formalism – probabilistic unranked logic.

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For a one-dimensional case of the well-known Maxwell system, the questions of linear and global stability of a stationary solution of an initial-boundary value problem and approximate solution are studied (see, for instance, [1-5] and references therein). A finite-difference scheme obtained by splitting with regard to physical processes is also discussed. For the first time, such a question was studied in the article [5]. Implementations were carried out using the machine learning method. Comparison of computer experiments with theoretical conclusions is presented. Acknowledgement This research has been supported by the Shota Rustaveli National Science Foundation of Georgia under the grant FR-21-2101. References [1] T.A. Dzhangveladze, Stability of the stationary solution of a system of nonlinear partial differential equations. Sovremennye problemy matematicheskoi fiziki. (Russian) (Proceeding of All-Union Sympozium. The Modern Problems of Mathematical Physics). Tbilisi, 1 (1987), 214-221. [2] Z.V. Kiguradze, On the stationary solution for one diffusion model. Rep. Enlarged Sess. Semin. I. Vekua Inst. Appl. Math. 16 (2001), 17-20. [3] T. Jangveladze, Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations. Mem. Differ. Equ. Math. Phys. 76 (2019), 1-118. [4] M. Gagoshidze, System of nonlinear one-dimensional Maxwell's equations and its approximate solution. Rep. Enlarged Sess. Semin. I. Vekua Inst. Appl. Math. 35 (2021), 35-38. [5] I.O. Abuladze, D.G. Gordeziani, T.A. Dzhangveladze and T.K. Korshiya, Discrete models for a nonlinear magnetic-field-scattering problem with thermal conductivity. (Russian) Differential’nye Uravnenyia, 22, 7 (1986), 1119-1129. English translation: Differential Equations, 22, 7 (1986), 769-777.

In the present paper, issues of an approximate solution of the singular integral equation and pair systems of integral equations containing fixed-singularity are studied. The studied integral equations are obtained from the anti-plane problems of the elasticity theory for a composite (piece-wise homogeneous) orthotropic (in particular, isotropic) plane slackened by crack when it reaches or intersects the dividing boundary at the right angle ([1], [2]). Algorithms of an approximate solution are designed by the collocation method, namely the method of discrete singularities ([3]). In both cases, (when the crack reaches or crosses the dividing border) behaviour of the solutions is studied and the stress intensity coefficients at the ends of the crack are calculated. Consider a body composed of two isotropic materials (copper and aluminum). Results of numerical computations are demonstrated. According to the obtained results, hypothetical predictions of the propagation of crack are made. . References 1.Papukashvili A. (2004) Anti-plane problems of theory of elasticity for piecewice-homogeneous orthotropic plane slackened with cracks. Bull. Georgian Acad. Sci., 169, 2: 267-270. 2.Papukashvili A., Davitashvili T., Vashakidze Z. (2015) Approximate solution of anti-plane problem of elasticity theory for composite bodies weakened by cracks by integral equation method. Bull. Georg. Natl. Acad. Sci., 9, 3: 50-57. 3.Belotserkovsky S. M., Lifanov I. K. (1992) Method of Discrete Vortices. (G. P. Cherepanov & V. A. Khokhryakov Trans.), Boca Raton, Florida, United States: CRC Press, LLC.

Based on the numerical model of the mesoscale boundary layer of the atmosphere, a number of humidity processes are imitated: Simultaneous existence of fog and cloud; Combined vertical complex of fog and cloud; To study the process of cloud conception-dispersion from a synergistic point of view; A relatively new classification of fions; Cloud and fog cluster simulation; Use of "merging" two clouds to cause precipitation; Active influence on fog by means of artificial heat sources and downstream currents.

The present talk is devoted to a concise survey of scientific, pedagogical, and educational activities of the outstanding Georgian mathematician and mechanist Ilia Vekua. Biographical data are also given. The talk was prepared on the occasion of his 115 anniversary of his birth and 45 anniversary of his death.

We construct mathematical hierarchical models in Eulerian coordinates for a Newtonian viscous fluid flow in the prismatic shell-like 3D domains. The peculiarities of angular domains are discussed.

In [1] by G. Jaiani is constructed mathematical hierarchical models in Eulerian coordinates for a Newtonian viscous fluid flow in the prismatic shell-like 3D domains. The present talk is devoted to investigation of a motion of an incompressible fluid in the zeroth approximation of constructed hierarchical models. References [1] George Jaiani. Hierarchical models for fluids. ZAMP, submitted for publication

The present talk deals with the basic boundary value problems of the plane theory of elasticity for a circular ring with double voids. general solution of the governing system of equations of the plane strain is represented by means of two analytic functions of the complex variable and two solutions of Helmholtz equations. Using the obtained solutions, the problems of the plane theory of elasticity for a circular ring are solved analytically.

The report considers the case of plane deformation for elastic bodies with voids. For perforated rectangular domains, various boundary value problems are approximately solved and stress concentration coefficients are calculated. Approximate solutions are constructed using the general solution of the corresponding system of equations and the method of fundamental solutions.

The problem of finding an equal-strength contour inside a viscoelastic square plate is considered on the outer boundary of which the same tensile forces with given principal vectors (or constant normal displacements) act, and the inner boundary (the desired equal-strength contour) is free from external forces. The equal strength of the desired contour lies in the fact that at each point of the contour the tangential normal stress takes on the same values. To solve the problem, methods of conformal mappings and boundary value problems of analytic functions are applied. The equation of the desired contour, as a function of point and time, is constructed efficiently (in an analytical form).

We consider 2D and 3D incompressible unsteady fluid flow over the rhombus and over the prism with the rhomboidal cross-section correspondingly. The velocity components of the flow satisfy the nonlinear Navier - Stokes equations (NSE) with the suitable initial-boundary conditions and with the specific pressure. We supposed that near sharp edges the velocity components are non-smooth and by the methods of mathematical physics we have obtained novel exact solutions of NSE.

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In the paper the unsteady incompressible fluid flow in a prismatic pipe is studied for the low Reynolds number. The linearized Navier-Stokes equation (the Stokes equation) is considered with the suitable initial-boundary conditions. It is assumed that the pressure depends on time exponentially. The Stokes equation is reduced to the system of linear integral equations with the weakly singular kernel. The existence and uniqueness of the solutions of those equations is proved and the approximate solutions are obtained by means of the conformal mapping and the step-wise approximation method.

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For the controlled neutral functional differential equation whose right-hand side is linear with respect to prehistory of the phase velocity the continuous dependence of a solution on the initial data and on the nonlinear term in right-hand side of equation is investigated. The perturbation nonlinear term and initial data are small in the integral and standard sense, respectively. Under initial data we mean the collection of delay parameters, the initial vector and function, the control function.

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We consider 2D incompressible unsteady fluid flow over the rectangle and between two similar rectangles. The velocity components of the flow satisfy the nonlinear Navier - Stokes equationს (NSE) with the suitable initial-boundary conditions . We suppose that near sharp edges the velocity components are non-smooth and by the methods of mathematical physics we obtain exact solutions of NSE for the specific pressure.

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In this paper we consider reasoning with incomplete, imperfect information, which is very common in human communication. For such problems, exact equality/equivalence is replaced by its approximation. This kind of reasoning is a highly nontrivial task and remains an important issue in applications of artificial intelligence. Modeling the incomplete and imprecise information is achieved using so called tolerance relations, which are reflexive and symmetric, but not necessarily transitive relations. This idea goes back to Poincaré, who viewed tolerance as the notion of fundamental importance in distinguishing mathematics applied to the physical world from ideal mathematics. In this paper we discuss several tolerance relations, starting from crisp and ending with fuzzy tolerance relations.

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If M is a given class of σ-finite measures on E, then all real-valued functions f defined on E can be of the following three categories: absolutely nonmeasurable functions with respect to M , relatively measurable functions with respect to M and absolutely (or universally) measurable functions with respect to M . In the presented talk we consider various families of the measures, their characterizations in the sense of the set theory and the cardinality of some classes of measures be presented

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Based on known trigonometric formulas, a decomposition formula is constructed for the cosine operator function when the argument is the sum of two bounded operators. The error of the n-th approximation is estimated in a Banach space. The case when the number of summands is more than two is also considered, Such an algorithm is proposed that allows us to obtain a 2p + 2-order decomposition formula from the 2p-order of decomposition one (p > 1 is a natural number).

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In this talk we survey uncertainty reasoning in predicate logic, where formulas are interpreted over [0,1] interval.e discuss different calculi for reasoning with uncertainty, in particular probabilized sequent calculus and natural deduction

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One-dimensional, two models based on Maxwell's well-known system of nonlinear partial differential equations [1], describing the process of penetration of a magnetic field in a substance are considered. The uniqueness of the solutions of the corresponding initial-boundary value problems and the convergence of the finite-difference schemes are studied, which are an extension of some of the results obtained in [2, 3]. References 1. Landau, L., Lifschitz E. Electrodynamics of Continuous Media, Course of Theoretical Physics, Moscow, 1957. 2. Abuladze, I.O., Gordeziani, D.G., Dzhangveladze, T.A., Korshiya, T.K. Discrete models for a nonlinear magnetic-field-scattering problem with thermal conductivity. Differential’nye Uravnenyia, 22, 7 (1986), 1119-1129. English translation: Differential Equations, 22, 7 (1986), 769-777 (Russian). 3. Jangveladze, T. Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations. Mem. Differential Equations Math. Phys., 76 (2019), 1-118.

The asymptotic behavior, as time variable tends to infinity, of a solution for a nonlinear diffusion system [1, 2] is considered. It is shown that the stationary solution of the system is linearly stable, and the possibility of the Hopf-type bifurcation is observed. References 1. Dzhangveladze, T.A. Stability of the stationary solution of a system of nonlinear partial differential equations, Sovremennye problemy matematicheskoi fiziki. (Proceeding of AU-Union Sympozium. The Modern Problems of Mathematical Physics). Tbilisi, 1 (1987), 214-221 (Russian). 2. Jangveladze, T. Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations. Mem. Differential Equations Math. Phys., 76 (2019), 1-118.

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Using I. Vekua’s dimension reduction method, the title governing equations are constructed when either $$\rho = \rho_0 + \tilde\rho(x_1, x_2, x_3, t),\;\; ρ_0 = const,\;\; \tilde\rho << \rho_0 $$ or $$\rho = \rho(x_1, x_2, t),$$ where $\rho$ is the density, $x_1, x_2, x_3, t$ are Eulerian variables.

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We consider shallow incompressible barotropic fluid occupying non-Lip-schitz, in general, prismatic domains within the scheme of small displacements linearized with respect to the rest state (see [1], [2]). The well-posedness of BVPs for the fluid occupied symmetric prismatic shell like domains in the zero-th approximation of hierarchical models under the reasonable BCs in velocities at the cusped edge and given velocities at the non-cusped edge are studied. References: 1. Chinchaladze, N., Jaiani, G.: Hierarchical mathematical models for solid–fluid interaction problems (in Georgian). Materials of the International Conference on Non-classic Problems of Mechanics, Kutaisi, Georgia, 25-27 October, Kutaisi 2, 59-64 (2007) 2. Jaiani, G.: Cusped Shell-like Structures, SpringerBriefs in Applied Science and Technology, Springer-Heidelberg-Dordrecht-London-New York, 2011, 84 p.

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In this talk we consider several properties of the algebraic curves, which are of interest from geometric, combinatorial, algebraic, and number-theoretical points of view and somehow illustrates the role of algebraic curves in various mathematical topics.

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In this report we consider some boundary value problems for a circular plate. The plate is the elastic material with voids. The state of plate equilibrium is described by the system of differential equations that is derived from three dimensional equations of equilibrium of an elastic material with voids (Cowin-Nunziato model) by Vekua’s reduction method. Its general solutions are represented by means of analytic functions of a complex variable and solutions of Helmholtz equations. The problems are solved analytically by the method of the theory of functions of a complex variable when the components of the displacement vector or the components of the stress tensor are given.

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Using the dimensional reduction method developed by Ilia Vekua, the main two-dimensional equilibrium equations for shallow shells consisting of a binary elastic mixture were obtained in. We consider the case when the shell consists of porous mixture, and each of its components are characterized by different volume functions.

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The problem of finding an equally strong contour inside a rectangular viscoelastic plate according to the Kelvin-Voigt model is considered. It is assumed that normal compressive forces with given principal vectors (or constant normal displacements) are applied on the sides of the rectangle by means of linear absolutely rigid punches, and the unknown part of the boundary (the equally strong contour) is free from external forces. The equal strength of the sought-for contour lies in the fact that at each point of the contour the tangential normal stress takes the constant value (generally it depends on both the point and the time). To solve the problem, methods of conformal mappings and boundary value problems of analytic functions are used, and the equation of the desired contour, as a function of point and time, is constructed efficiently (in an analytical form).

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We show that every universal first-order safety property can be compiled into a universal invariant of a first-order transition system using quantifier-free substitutions only. We apply this insight to prove that every universal first-order safety property is decidable for large classes of stratified guarded first-order transition systems.

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We present a number of observations concerned with the so-called invertible polynomials introduced and studied in a series of papers on mathematical physics and singularity theory. Specifically, we consider real versions of invertible polynomials and investigate invariants of the associated isolated hypersurface singularities. By the very definition such a polynomial is weighted homogeneous and its gradient vector field grad f has an isolated zero at the origin hence its index ind0grad f is well defined. This index, referred to as the gradient index of polynomial, is our main concern. In particular, we give an effective estimate for the absolute value of the gradient index ind0grad f in terms of the weighted homogeneous type of f and investigate its sharpness. For real invertible polynomials of two and three variables, we give the whole set of possible values of the gradient index. As an application, in the case of three variables we give a complete list of possible topological types of Milnor fibres of real invertible polynomials, which generalizes recent results of L.Andersen ”On real isolated singularities.” I. rXiv: 2110. 04407 [math.AG], 2021 on the topology of isolated real hypersurface singularities. Finally, we present a few open problems and conjectures suggested by our results.

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The presented article is a direct continuation of the articles [1]-[2] that consider the construction of algorithms and their corresponding numerical computations for the approximate solution of nonlinear integro-differential equations of the Timoshenko type. In particular, in this work, it is considered an initial-boundary value problem for the J. Ball integro-differential equation, which describes the dynamic state of a beam (see, [3]). The solution is approximated utilizing the Galerkin method, stable symmetrical difference scheme and the Jacobi iteration method. In the articles [1]-[2] the algorithm has been approved by tests. This paper presents the approximate solution to one practical problem. Particularly, the results of numerical computations of the initial-boundary value problem for an iron beam are represented in tables and graphics. References 1. Papukashvili Archil, Papukashvili Giorgi, Sharikadze Meri. Numerical calculations of the J. Ball nonlinear dynamic beam. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 32 (2018), 47-50. 2. Papukashvili Archil, Papukashvili Giorgi, Sharikadze Meri. On a numerical realization for a Timoshenko type nonlinear beam equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 33 (2019), 51-54. 3. Ball J. M. Stability theory for an extensible beam. J. Differential Equations 14 (1973), 399-418.

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Based on a three-dimensional hydrostatic mesoscale model, an air flow over the complex relief of the South Caucasus (Georgia) is modeled under the conditions of non-stationary large-scale background processes. Numerical experiments have shown a strong influence of orographic effects on air movement in the troposphere. In particular, it is shown that when an air flow of synoptic scales moves, the vertical amplitudes of the mesoscale flow and the deviation of the air velocity vector along the Likh Ridge increase significantly. Besides, the strong wind regime and statistical characteristics of the Rioni River region were researched for the period 1960-2021. The wind speeds are divided into intervals of 5 m/s, and for each interval the wind speed recurrence rate is studied by months. It has been determined that in terms of energy, the main range of wind speed for the Kutaisi region is 16-20 m/s. Thus, from an energy point of view, speeds of such magnitude are essential, which ensure the automatic mode of the wind farms and are an important basis for the development of wind power plants in western Georgia

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It is considered the principle ways of using stochastic analyses in some problems of thermodynamics, descriptive by equations with random parameters for variety mass and volume, for example the adiabatic equation for real gas is considered

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The report will establish the necessary and sufficient conditions for the sequence obtained as a result of the matrix transformation of a sequence of partial sums of Walsh-Fourier series to be convergent in norm.

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The The maximum principle for Carleman-Vekua irregular equations has been formulated and approved, the coefficients of which belong to sufficiently wide range of functions. These spaces represent an extension of the classical spaces introduced by Vekua and a generalization of examples of functions he has explored. Find the classes of Irregular Equations of Carlemagne-Vekua for which the principle of maximum is not fulfilled. The definitions of the spaces of these functions and their properties are given.

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In the bipolar system of coordinates exact solutions of two dimensional static boundary value problems of elasticity are constructed for homogeneous isotropic bodies occupying domains bounded by coordinate lines of bipolar coordinates. These represent boundary value problems of elastic equilibrium of eccentric circular rings, half-planes with circular holes, etc. In the bipolar coordinates are written the equilibrium equation system and Hooke's law. The requirement of static equilibrium of the external load at each circular boundary of the region is not taken into account in this work. This requirement, which significantly limits the range of tasks to be solved, usually appears in works devoted to the above problems. In addition, the process of obtaining exact (analytical) solutions becomes much easier compared to the traditional approach. Exact solutions are obtained using the method of separation of variables.

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The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in L_1 space and in CW space in terms of modulus of continuity and matrix transform variation. Moreover, we show the sharpness of our result. We also discuss some properties of the maximal operator t^∗(f) of the matrix transform of the Walsh–Fourier series. As a consequence, we obtain the sufficient condition so that the matrix transforms tn(f) of the Walsh–Fourier series are convergent almost everywhere to the function f. The problems listed above are related to the corresponding Lebesgue constant of the matrix transformations. The paper sets out two-sides estimates for Lebesgue constants. The proven theorems can be used in the case of a variety of summability methods. Specifically, the proven theorems are used in the case of Cesàro means with varying parameters.

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In 1914, S. Mazurkiewicz presented a transfinite construction of a subset A of the Euclidean plane R^2, having the following extraordinary property: every straight line in R^2 meets A in exactly two points. Mazurkiewicz set has a difficult and interesting descriptive structure. For instance, these three results are well known: there exists a Mazurkiewicz set which is of Lebesgue measure zero and of first Baire category; there exists a Mazurkiewicz set which is Lebesgue nonmeasurable and does not have Baire property; if there exists a Mazurkiewicz set which is analytic in R^2 , then it is Borel in R^2. In the presented project proposal we will be investigate the measurability properties of Mazurkiewicz sets with respect to the class M(R^2). of all nonzero sigma-finite translation invariant measures on R^2. We claim to establish a connection between Mazurkiewicz sets and absolutely non-measurable sets. Earlier we have shown that there exists a translation invariant measure μ on R extending the standard Lebesgue measure and such that all Sierpinski sets are measurable with respect to μ.

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In the talk theorems concerning the existence of independent families of convex subsets of the Euclidean plane of various cardinalities are considered. Among the mentioned theorems are ones related to the existence of independent families of polygons and independent families of quasi-polygons. Theorems related to the structure of constituents of finite independent families of convex and compact subsets of the Euclidean plane are also presented.

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http://www.rmi.ge/eng/QUALITDE-2021/Shavadze_Tadumadze_workshop_2021.pdf

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Filippov’s type theorems on the existence of an optimal element are given for the nonlinear optimal control problems with delays in the phase coordinates and commensurable delays in controls.

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The integro-differential equations are applied in many branches of science, such as physics, engineering, biochemistry, etc. A lot of scientific works are dedicated to the investigation and numerical resolution of integro-differential models. One type of nonlinear integro-differential parabolic model is obtained at the mathematical simulation of processes of electromagnetic field penetration into a substance. The purpose of this talk is to analyze degenerate one-dimensional case of such type models. Unique solvability and convergence of the constructed semi-discrete scheme with respect to the spatial derivative and fully discrete finite difference scheme are studied.

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The limiting distribution of the integral square deviation of kerneltype nonparametric estimator of Poisson regression function is established. The test of the hypothesis testing about Poisson regression function is constructed. The question of consistency of the constructed test is studied. The power asymptotic of the constructed test is also studied for certain types of close alternatives

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Different kind of Constrained Bayesian method (CBM) for testing directional hypotheses is considered. There is shown that the direct application of all statements of CBM allows us to make decisions on the desired levels of reliability. Theoretically is proved that mixed directional false discovery rates (mdFDR) are restricted on the desired levels at the suitable choice of restriction levels at different statements of CBM. These facts are demonstrated by computation of concrete examples for different statements of CBM.

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Logical methods are core in automated reasoning, where it is vital to represent knowledge in a format understood by computers. The widest practical application of knowledge representation is in so-called Web 3.0, which is a collection of different kinds of technologies and the core among them is the semantic web. Nowadays, on the logic layer of the semantic web, the formal languages are considered, which are based on unranked alphabets. This means that functional and/or predicate symbols do not have a fixed arity. Such languages can naturally model XML documents and operations over them and many more. One of the most interesting formalisms, based on an unranked alphabet, is Common Logic, which is used to exchange information between different systems and networks. This language is more and more often used in knowledge representation and ontologies are written on it (see e.g. COLORE and OntoHub). Although the ontology languages are standardized by W3C (e.g. OWL), there are still many problems remaining. One of the most important problems is related to, so-called, fuzzy ontologies. These are ontologies, where information is vague, imprecise and incomplete. Such kind of ontologies have applications in many different areas, such as medicine, biology, e-commerce just to name a few. We attempt to solve knowledge modelling problems related to vague and incomplete information by introducing unranked fuzzy logic and corresponding reasoning method. In this talk we present a first-order many-valued logic with sequence variables and flexible-arity function and predicate symbols. We develop a tableau method for this logic and identify some terminating fragments.

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Ulf electromagnetic planetary waves can self-organize into vortex structures (monopole, dipole or into vortex chains). They are often detected in the plasma media, for instance in the magnetosheath, in the magnetotail and in the ionosphere. Large scale vortices may correspond to the injection scale of turbulence, so that understanding their origin is important for understanding the energy transfer processes in the geospace environment. In a recent work, the THEMIS mission has detected vortices in the magnetotail in association with the strong velocity shear of a substorm plasma flow (Keiling et al., J. Geophys. Res., 114, A00C22 (2009), doi:10.1029/2009JA014114), which have conjugate vortices in the ionosphere. By analyzing the THEMIS data for that event, we find that several vortices can be detected together with the main one, and that the vortices indeed constitute a vortex chain. The study is carried out by analyzing both the velocity and the magnetic field measurements for spacecraft C and D, and by obtaining the corresponding hodograms. It is found that both monopolar and bipolar vortices may be present in the magnetotail. The comparison of observations with numerical simulations of vortex formation in sheared flows is also discussed.

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For an optimal problem containing neutral differential equation with the two type controls and whose right-hand side is linear with respect to prehistory of the phase velocity, existence theorems of optimal element are proved. Under element we imply the collection of delay parameters, initial vector and control functions.

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Glaciers are one of the main indicators of current climate change, as the interaction between a glacier and climate is a complex non-linear process. Climate change characterized by fluctuations in the balance of radiation energy in the lower troposphere, which determines the process of fluctuating glaciers (melting of thickness). In addition, the abundant glacial ensemble with its special properties contributes to some extent to climate variation. Therefore, in mathematical modeling of the dynamics of glaciers, it is not easy to take into account these nonlinear processes completely. In this article, a two-dimensional mathematical model of the dynamics of changes in the thickness of glaciers in the Caucasus has been developed, based on the integration of nonlinear partial differential equations, which in turn is ensured by a change in the equilibrium mass of the glacier due to direct solar radiation. The configuration of the upper surface of the glacier is predicted by solving the continuity equation. A scheme, similar to the Lax-Wendroff scheme, is used to solve numerically the nonlinear PDE model. Some typical problems of mathematical and numerical modeling of glaciers are discussed. For the first time, the process of melting of some glaciers in the Caucasus has been assessed using mathematical modeling. Some simulation results are presented and analyzed

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The 2D viscous flow with large viscosity and low Reynolds number (Re<<1) is considered in the half-plane. Solutions of the Stokes system are obtained, when the pressure is harmonic function. The profiles of free boundaries for the different pressures are constructed. For 2D perfect fluid the free boundary problem is reduced to the singular integral equation with the Weierstraß kernel for unknown free boundary. The equation is solved numerically. The profiles of waves with peaks are constructed.

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The present talk deals with plane strain problem for linear elastic materials with double voids. general solution of the governing system of equations of the plane strain is represented by means of two analytic functions of the 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 with double voids.

The paper considers the problem of the plane theory of viscoelasticity for a doubly-connected domain bounded by convex polygons. It is assumed that absolutely smooth rigid punches are applied to the outer boundary while the inner polygon has a smooth washer whose dimensions are slightly different from the dimensions of the rectangle so that the boundary points receive constant normal movements without friction. The problem consists of determining the corresponding complex potentials characterizing the equilibrium of the plate by the Kelvin-Voigt model.

The anti-plane problem of the elasticity theory for a composite (piecewise homogeneous) orthotropic (in particular, isotropic) plane weakened by a crack is reduced to a system of singular integral equations containing a fixed-singularity with respect to characteristic functions of disclosure of crack when the crack intersects the dividing border of interface with the right angle. In contrast to the above works, we present in the presented paper that the characteristic features of crack opening have a weak peculiarity at the dividing line. A set of vectors characteristic of opening both cracks near the dividing boundary We took a modulus of approximately equal to 0. To solve the obtained system, we used the method of discrete singularity . Relevant new algorithms are built and numerical calculations are performed for two specific isotropic bodies (copper and aluminum). Tension intensity coefficients near the crack ends are calculated, the behavior of the characteristic features of the crack opening is studied, and a hypothetical prediction of crack propagation is made.

The talk is dedicated to a problem for non-homogeneous piezoelectric elastic rod is studied in the case when constitutive coefficients vary from zero as power functions of spatial variable z. The well-posedness of initial-boundary value problem is studied. The conditions on the volume force components $F_x$ and $F_y$ , which guarantee the strain state under consideration, are established.

Analytical (exact) solution of two-dimensional problems of elasticity in the area bounded by a hyperbola is constructed in the elliptic coordinates. A special kind of internal boundary value problem is set and solved in area, when both parts of hyperbola are lines and non-homogenous symmetry or antisymmetry conditions are given on it, while non-homogenous conditions, such as stresses or displacements, are given on eta=pi/2. Exact solutions are obtained with the method of separation of variables. 2D and 3D graphs for some numerical values of some boundary value problems are given and considered. The graphs are obtained by using computer software MATLAB.

The present paper consider the 3D linear theory of thermoelasticity for materials in which the particles are subjected to classical displacement, temperature and mass diffusion fields and whose microelements possess microtemperatures and microconcentrations. A basic thermoelastic problems for solids with diffusion, microtemperatures, and microconcentrations encounter in many engineering applications such as satellite problems, aircraft landing on water or land, returning space vehicles, the oil extraction. Therefore, the study basic problems of thermoelasticity for materials with diffusion, microtemperatures and microconcentrations have mean considerable attention. In the present paper is presented an explicit solutions of the Dirichlet type BVP for an isotropic sphere and the Neumann type BVP for the full space with spherical cavity with diffusion, microtemperatures and microconcentrations. The obtained solutions are given by means of the harmonic, bi-harmonic and meta-harmonic functions. For the harmonic functions the Poisson type formulas are obtained. The bi-harmonic and meta-harmonic functions are presented as absolutely and uniformly convergent.

The report discusses the process of disseminating some methods of complex analysis for a certain of nonlinear tasks in continuum mechanics. The report consists of two parts. The first one is a purposeful review of the well-known results, as we are essentially relying on the methodology presented in this section when discussing nonlinear problems. This part of the report is common. The second part discusses the reduction of some substantially nonlinear models of continuum mechanics to two-dimensional spatial variables with initial boundary tasks. The main part of the system of reduced differential equations can be presented together the second-order linear and Monge-Ampère operators as we can see in, or the fourth-order linear biharmonic type operator and the composition of Laplace and Monge-Ampère operators. The construction of the solution of the discussed problems is carried out according. As for the distribution of conformal mapping for these tasks, the invariance form of non-linear members is important in transformation. In this respect it is fair in particular the following theorem: Suppose one area is conformingly reflected over another by means of two conjugate harmonic functions. Then the Monge-Ampére operator (Poisson brackets too) retains the invariant form if the first-order private derivatives of the transformer functions are equal to the accuracy of the mark. We dedicate this report to the memory of Professors Zaur Samsonia, Vakhtang Zhgenti and Ramaz Abdulayev.

2D nonlinear Navier-Stockes equation with the appropriate initial-boundary conditions is considered in a plane with the rectangular hall. By the methods of Classical Analysis exact solutions vanishing at infinity are obtained.

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The initial-boundary value problem for one-dimensional system of nonlinear partial differential equations with the mixed boundary condition is considered. It is proved that in some cases of nonlinearity there exists a 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 as $\psi$ passes through $\psi_{c}$ then the Hopf type bifurcation may take place.

The estimate for the Bernoulli regression function is constructed. The question of its consistency and asymptotic normality is studied. Testing hypothesis is constructed on the form of the Bernoulli regression function. Also, the test is constructed for the hypothesis on the equality Bernoulli functions. The question of consistency of the constructed tests is studied.

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The nonlinear Maxwell system is considered, which describes the propagation of magnetic field in the medium and the temperature change at the expense of Joule heating and heat conductivity. If there is not heat conductivity this system may be rewritten in the integro-differential form [1]. Some aspects of the investigation and numerical solution of this partial differential system and the above-mentioned integro-differential analogs are studied in many works (see, for example, [1-3] and references therein). Our aim was to investigate and numerical solution of the one-dimensional version of the Maxwell system and its integro-differential analogs. Especially, the semi-discrete and finite difference scheme for initial-boundary value problems for some kind of nonlinearities are constructed and investigated. The asymptotic behavior of solutions is also studied. The corresponding numerical experiments are done. References [1] D. Gordeziani, T. Dzhangveladze, T. Korshija, Existence and uniqueness of a solution of certain nonlinear parabolic problems, Differ. Uravn. 19, N7 (1983), 1197–1207 (Russian). English translation: Differ. Equ., 19, N7 (1983), 887–895. [2] T. Jangveladze, Z. Kiguradze, B. Neta, Numerical Solution of Three Classes of Nonlinear Parabolic Integro-Differential Equations, Elsevier/Academic Press, Amsterdam, 2016. [3] T. Jangveladze, Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations, Mem. Differential Equations Math. Phys. 76, (2019), 1–118.

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The problem of testing complex hypotheses with respect to the equal parameters of normal distribution using the constrained Bayesian method is discussed. Hypotheses are tested using maximum likelihood and Stein’s methods. The optimality of decision rule is shown by the criteria: the mixed direction false discovery rate, the false discovery rate, the Type I and Type II errors, under the conditions of providing the desired level of constraint. The algorithms for the implementation of the created methods and the programs for their realization are given. Simulation results show the correctness of the theoretical results and their superiority over the classical Bayes method.

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According to well-known set-theoretical results, there are models of ZFC in which all projective subsets of R are well-behaved from the point of view of descriptive set theory, in particular, they all are Lebesgue measurable. For instance, the Axiom of Projective Determinacy (PD) implies prominent regularity properties of the projective sets: Lebesgue measurability, the Baire property, the perfect subset property and the Ramsey property. This implies that definable absolutely nonmeasurable functions on R can only exist in certain models of ZFC without substantial large cardinals. Assuming that there exists a well-ordering of R whose graph is ∆1 2-subset of the plane, there exists a Vitali set in R which is a ∆1 2-subset of R. Consequently, under this assumption, there are projective sets which are Lebesgue non-measurable and do not have the Baire property. Recall that this assumption holds true in Gödel’s Constructible Universe L. (cf. [2], [4]). In [1] we have shown, that there exists a model of ZF, such that there is no well-ordering of the reals but there is a Hamel basis. A. Miller has shown that in L, there is Π1 1-Hamel basis. It is an old result by S. Feferman that the existence of Vitali sets doesn’t imply that there is a well-ordering of the reals (cf. [2], [4]). Still in ZFC, there is a Mazurkiewicz set which is simultaneously a Hamel basis, cf. [3]. In joint work with R. Schindler we formulate a sufficient criterion for a model of ZF to have a Mazurkiewicz set

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Solving equations between logic terms is a fundamental problem with many important applications in mathematics, computer science, and artificial intelligence. It is needed to perform an inference step in reasoning systems and logic programming, to match a pattern to an expression in rule-based and functional programming, to extract information from a document, to infer types in programming languages, to compute critical pairs while completing a rewrite system, to resolve ellipsis in natural language processing, etc. Unification and matching are well-known techniques used in these tasks. Unification (as well as matching) is a quite well-studied topic for the case when the equality between function symbols is precisely defined. This is the standard setting. There is quite some number of unification algorithms whose complexities range from exponential to linear. Besides, many extensions and generalizations have been proposed. Those relevant to our interests are unification with sequence variables and flexible-arity (unranked) function and predicate symbols. Unranked fuzzy logic is an extension of first-order Lukasiewicz product logic with sequence variables and unranked function and predicate symbols. In this talk we consider unification problem for this logic. Unranked fuzzy unification is divided into two parts: unranked unification and solving sets of linear inequalities. Unranked unification, and thus unranked fuzzy unification, is non-terminating in general, but there are some well-known terminating classes. We discuss challenges of unranked fuzzy unification and identify corresponding terminating fragments.

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Anti-plane problems of the elasticity theory, composed for the orthotropic plane, weakened by cracks, when a crack reaches the interface, are reduced to the following integral equation containing a fixed singularity to the unknown characteristic function of the crack disclosure (see: [1, 2]). The range of singularity of the solution at the point t = 0 is also dependent on the elasticity constants of the materials α ∈ (0, 1). However at the point, t = 1 The range of singularity of the solution is constant, namely, β = 0.5. The integral equation is solved by the collocation method, in particular, the method of discrete singularities [3]. The corresponding algorithms are designed and implemented. The results of numerical computations are presented. References [1] A. Papukashvili. Antiplane problems of theory of elasticity for piecewice homogeneous orthotropic plane slackened with cracks. Bull. Georgian Acad. Sci, 169(2):267–270, 2004. [2] A. Papukashvili, T. Davitashvili, and Z. Vashakidze. Approximate solution of antiplane problem of elasticity theory for composite bodies weakened by cracks by integral equation method. Bulletin of the Georgian National Academy of Sciences, 9(3), 2015. ISSN 01321447. [3] S.M. Belotserkovskii and I.K. Lifanov. Numerical methods in singular integral equations and their application in aerodynamics. Elasticity Theory, and Electrodynamics [in Russian], Moscow, 1985.

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A numerical model of the full cycle of cloud and fog genesis in the mesoboundary layer of atmosphere has been created. The critical values of relative humidity at which the formation of humidity processes takes place have been determined. A numerical model of the distribution of aerosol from an instantaneous point source into the mesoboundary layer of the atmosphere has been created. The time intervals at which the deposition of aerosol on the earth’s surface begins and ends have been determined. The formation of smog is simulated based on the synthesis and “overlay” of the two above models.

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In the talk two versions of Sylvester’s well-known problem for the Euclidean plane are considered. Also, several variants of this problem for higher-dimensional Euclidean spaces are presented. One problem related to the original version of Sylvester’s problem is posed. A solution of this problem is obtained as one of the consequences of the following statement: Theorem. Let $n \geq 5$ be an arbitrary, fixed natural number and $a$ and $b$ be any two fixed positive real numbers. There exists a subset $S$ of the Euclidean plane $\alpha$, such that $card (S) = n$ and for every straight line $L_{U,V}$ passing through arbitrary two distinct points $U$ and $V$ of $S$ there exists a point $W_{L_{U,V}}$ of $S$ such that the Euclidean $d(W_{L_{U,V}}, L_{U,V})$ distance between $\{W_{L_{U,V}}\}$ and $L_{U,V}$ satisfies the following inequality: $d(W_{L_{U,V}}, L_{U,V}) < a$. Also, for any straight line $L \subset\alpha$ there exists a point $Z_L$ of $S$ such that the Euclidean $d(Z_L, L)$ distance between ${Z_L}$ and $L$ satisfies the following inequality: $d(Z_L, L) > b$.

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Unification is a process to identify two expressions by replacing variables of each expression by other expressions. More specifically, if we have two terms s and t, the aim is to find a substitution σ (mapping from variables to terms ), such that sσ and tσ are identical terms. In this case, we say the substitution is an unifier of s and t. In this talk we discuss unification problem in τ -logic [1, 2, 3]. In particular, we define notions of substitution, unifier, and most general unifier. We construct a sound and complete algorithm, which takes as an input s and t terms of τ -logic and returns a substitution σ such that sσ = tσ. We proved that, the algorithm always terminates and computed σ is a most general unifier for s and t. References [1] C. L. Chang and R. C. T. Lee, Symbolic Logic and Mechanical Theorem Proving. Computer Science and Applied Mathematics. Academic Press, New York-London, 1973. [2] Kh. Rukhaia, Description of the formula mathematical τ theory with substitution operators. (Russian) In Studies in Mathematical Logic and Theory of Algorithm of I. Vekua Institute of Applied Mathematics, pp. 58–74, 1985. [3] Sh. S. Pkhakadze, Some Questions of Notation Theory. (Russian) Izdat. Tbilis. Univ., Tbilisi, 1977

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The problem of the approximation of solutions of stationary linear as well as nonlinear Schrödinger equations is considered. The linear Schrödinger equation is studied in the areas of polygonal configuration. By means of the conformal mapping method 2D linear Schrödinger equation is replaced by the approximate elliptic equation. The initial area is mapped on the rectangle, the solutions of the elliptic equation in the rectangle are obtained. The cases of hexagonal and pentagonal areas are considered. The results has applications to quantum mechanics, Biophysics and Chemistry. Also, the multi-dimensional cubic non-linear Schrödinger equation is considered in the infinite area. The equation is replaced by the approximate non-linear elliptic equation. The exact solutions of this equation vanishing at infinity are obtained. These solutions represent solitary waves.

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Artificial Intelligence is developing using statistical and logical methods (machine learning, automated deduction, natural language processing, etc.). Logical methods are core in automated reasoning, where it is vital to represent knowledge in a format understood by computers. One of the main tools to represent knowledge is ontologies, which are a collection of logic-based formal language sentences. These sentences are used by automated reasoning programs to extract new knowledge and answer the given questions. The widest practical application of knowledge representation is in so-called Web 3.0. It is a collection of different kinds of technologies and the core among them is the semantic web. The main purpose of these technologies is to describe the semantic content of the web, i.e. their meaning and sense, in the format understood by computers. Such descriptions are called ontologies and the languages, on which these descriptions are written the ontology languages. Although the ontology languages are standardized by W3C, there are still many problems remaining. One of the most important problems is related to, so-called, unranked fuzzy ontologies, where information is vague, imprecise and incomplete. The following simple example illustrates this: assume an ontology of clothes is given, with three instances of coat: X has property “long” with value 0.8; Y has property “long” with value 0.4; and Z has no property “long” provided at all. Note that not having the property means that the value of this property is unknown and cannot be assumed that its value is 0. To model such ontologies, we need concepts like fuzziness and unrankedness. Under the term fuzziness we mean a multi-valued logic, where truth values instead of 0 (false) and 1 (true) can be any real number from the interval [0,1]; and under the term unrankedness we mean that functional and/or predicate symbols do not have a fixed arity. Without unranked formalism, we cannot directly omit the property for Z. Having a (terminating) reasoning method over unranked fuzzy language will broaden knowledge engineering capabilities in different fields like medicine, biology, e-commerce, etc.

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The tension-compression oscillation problem is investigated in the zero approximation of governing system for Kelvin-Voigt plates with variable thickness, using I. Vekua’s dimension reduction method.

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Using I. Vekua’s [1], [2] dimension reduction method, governing systems are derived and in the Nth approximation boundary value problems are set for Kelvin-Voigt plates with variable thickness. The ways of investigation of boundary value problems are indicated. In addition tension-compression and bending problems are investigated in the zeroth approximation of hierarchical models. [1] I.N. Vekua, Shell Theory: General methods of construction, Pitman Advanced Publishing Program, Boston-London, Melbourne, 1985. [2] G. Jaiani, Cusped Shell-Like Structures, Springer, Heidelberg-Dordrecht- London-New York, 2011.

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We consider a system of differential equations of Pfaffian type on a Riemann surface and develop for it an analog of the monodromy theory of Fuchsian systems of differential equations. Namely, let X be a compact Riemann surface and G a compact Lie group. Let $f_0$ be a solution of system under consideration in a neighborhood $U\subset X$ of the point $z_0$. After continuation of $f0$ along a path circling around a singular point $z_i$ system determines a monodromy representation. For such system, Riemann-Hilbert monodromy problem is formulated.

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We consider a boundary value problem for an infinite plate with a circular hole. The plate is the elastic material with voids. The hole is free from stresses, while unilateral tensile stresses act at infinity. The state of plate equilibrium is described by the system of differential equations that is derived from threeimensional equations of equilibrium of an elastic material with voids (Cowin-Nunziato model) by Vekua’s reduction method. its general solution is represented by means of analytic functions of a complex variable and solutions of Helmholtz equations. The problem is solved analytically by the method of the theory of functions of a complex variable. This is a joint work with R. Janjgava, T. Kasrashvili and M. Narmania.

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Analogues of the well-known Kolosov-Muskhelishvili formulas for homogeneous equations of statics in the case of elastic materials with double voids are obtained. It is shown that in this theory the displacement and stress vector components are represented by two 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 with double voids.

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The limiting distribution of the integral square deviation of kerneltype nonparametric estimator of Poisson regression function is established. The test of the hypothesis testing about Poisson regression function is constructed. The question of consistency of the constructed test is studied. The power asymptotic of the constructed test is also studied for certain types of close alternatives

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In the paper the fluid flow of large viscosity and low Reynolds number is considered in large reservoirs. It is assumed that viscosity is large and the linearized 2D Navier-Stokes equation (Stokes system) is studied in the rectangular area partly filled with the heavy fluid. The case of the solenoidal body force is considered. The solutions of the Stokes system are obtained with the appropriate initial-boundary conditions. It is proved that for the given pressure the solution is uniquely defined. The profiles of free surfaces are constructed for the different pressure.

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The estimate for the Bernoulli regression function is constructed using the Bernstein polynomial. The question of its consistency and asymptotic normality is studied. Testing hypothesis is constructed on the form of the Bernoulli regression function. Also, the test is constructed for the hypothesis on the equality Bernoulli functions. The question of consistency of the constructed tests is studied

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The estimate for the Bernoulli regression function is constructed using the Bernstein polynomial. The question of its consistency and asymptotic normality is studied. Testing hypothesis is constructed on the form of the Bernoulli regression function. Also, the test is constructed for the hypothesis on the equality Bernoulli functions. The question of consistency of the constructed tests is studied

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Consistent, unbiased and efficient estimators of the parameters of some irregular probability distributions such as: Right-Angled Triangular, Triangular, Trapezoidal, Antimodal-I, Antimodal-II, Truncated Relay and Beta distributions are considered. For the Beta distribution an iteration algorithm of computation of sought for estimators is developed. Some computation results, realized on the basis of simulation of the appropriate random samples, demonstrate the correctness of the offered theoretical outcomes.

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The present paper is devoted to construct explicit solutions of the quasi-static boundary value problems (BVPs) of coupled theory of thermoelasticity for a porous elastic sphere and for a space with a spherical cavity. In this research the regular solution of the system of equations for an isotropic porous material is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The basic BVPs for a sphere and for a space with a spherical cavity are solved explicitly. The obtained solutions are given by means of the harmonic, bi-harmonic and meta-harmonic functions. For the harmonic functions the Poisson type formulas are obtained. The bi-harmonic and meta-harmonic functions are presented as absolutely and uniformly convergent series.

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In 1995 TICMI (http://www.viam.science.tsu.ge/ticmi) was founded by initiation of Prof. George Jaiani with the support of EMS (European Mathematical Society) and personally of its first President Prof. Friedrich Hirzebruch, who beforehand on 16.11.1995 invited G. Jaiani to Bonn at Max-Planck Institute in order to discuss aims and prospective role of the centre in Caucasian independent republic of Georgia. He was the first who used the acronym TICMI with words “TICMI is a good idea” during the above mentioned discussions and instruct EMS secretary of that time Prof. E.C. Lance to carry out the corresponding actions from the side of EMS.TICMI is based in the I. Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University.

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We reduce the problem to computation of partial indices of a rational matrix function and after such reduction we use the algorithm of factorization of the rational matrix function by rank of moment matrices

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The limiting distribution of the integral square deviation of kerneltype nonparametric estimator of Poisson regression function is established. The test of the hypothesis testing about Poisson regression function is constructed. The question of consistency of the constructed test is studied. The power asymptotic of the constructed test is also studied for certain types of close alternatives

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We consider 3D problem for the non-stationary Stokes flow in the infinite cylindrical and prismatic pipes when their cross section is an arbitrary simply-connected region bounded with the piecewise smooth line. We admit that the pressure can be controlled and depends on time exponentially. The linear Stokes system for the incompressible fluid is considered with the appropriate initial-boundary conditions. By means of the integral equation method the system is equivalently reduced to the system of integral equations with the weakly singular kernel. The existence and uniqueness of solution is obtained, if the power at the exponent satisfies certain conditions. The exact solutions are obtained by means of the stepwise approximation method. Several examples are given.

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Using the special exp-function method traveling wave exact solutions of the (2+1)D nonlinear Zakharov-Kuznetsov type partial differential equation are obtained. It is shown that such solutions can be expressed through hyperbolic, trigonometric, exponential, and rational functions and have spatially isolated structural (soliton-like) forms. Revision of previously obtained solutions is discussed.

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Initial-boundary value problem with nonlocal boundary conditions [1, 2] for one nonlinear integro-differential equation is considered. Integro-differential models of this type are based on the system of Maxwell equations and are studied in many works (see, for example, [3] and references therein). References 1. Bitsadze, A.V., Samarskii, A.A. On some simple generalizations of linear elliptic boundary problems. (in Russian) Doklady Akademii Nauk, 185, 4 (1969), 739-740. 2. Steklov, V.A. Fundamental Problems in Mathematical Physics. (in Russian, ed. V.S. Vladimirov). Nauka, Moscow, 1983, 67. 3. Jangveladze, T. Investigation and numerical solution of nonlinear partial differential and integrodifferential models based on system of Maxwell equations. Mem. Differential Equations Math. Phys., 76 (2019), 1-118.

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We consider the Dirichlet type boundary value problem in the coupled linear quasi-static theory of elasticity, when on the boundary the displacement vector, changes of volume fraction of pores and the change of fluid pressure in pore network are given. The explicit solution of the Dirichlet boundary value problem is given as absolutely and uniformly convergent series.

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Random measures and connected with them questions of absolute continuity under nonlinear transformations in abstract Hilbert space are considered. Estimation of solution of first order Differential Equations with random measures parameters are given.

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In the present paper the static two-dimensional problems for an elastic material with voids are consider. The corresponding system of differential equations is written in a complex form and its general solution is presented with the use of two analytic functions of a complex variable and a solution of the Helmholtz equation. The boundary value problems are solved for a circle and a circular ring when on the boundary the displacement vector or the stress tensor and changes of volume fraction of pores given.

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The punch problem of the viscous half-plane with a friction is considered. However, it is based of the Kelvin-Voigt model. Using the methods of complex analysis which in the plane theory of elasticity are developed by acad. N. Muskhelishvili and his followers, the unknown complex potentials, that describe the equilibrium of a half-plane, are constructed effectively (analytically). Two specific examples of the outline of a punch base are considered when it represents an arc of a parabola with a radius of great curvature, or an ellipse arc whose half-axis is small in the direction of the Oy axis. In these cases, using the theory of residuals, the integral in the solution is constructed explicitly.

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In the report the integral equation of the third kind which are associated with the irregular Bers-Carleman-Vekua equations when the coefficients of equations belong to a wide class of functions will be discussed. It is shown that the irregular Bers-Carleman-Vekua equation and the corresponding integrated equations from this class have only a trivial solution in the class of analytic functions of two variables. The existence of a wide class of integral equations is proved, where the non-trivial solution does not existand it is shown that for such equations the Fredholm alternative does not hold.

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The report is concerned with some aspects of the theory of equidecomposability of elementary figures from the measure-theoretical standpoint. Two polyhedrons in R3 are equidecomposable if the first of them can be cut into finitely many polyhedrons which can be reassembled to yield the second one. Obviously, any two equidecomposable polyhedrons have the same volume. Theorem. Among the solution of the Cauchy functional equations one can meet those ones which are nonmeasurable with respect to every translation invariant measure on the real line R, extending the Lebesgue measure.

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One of the main tools in knowledge representation is ontology, which is a collection of logic-based formal language sentences. These sentences are used by automated reasoning programs to extract new knowledge and answer the given questions. Although ontology languages are standardized by W3C, there are still many problems remaining. One of the most important problems is related to, so called, fuzzy ontologies. These are ontologies, where information is vague and imprecise. Fuzzy ontologies are obtained by integrating fuzzy logic with ontologies. Such kind of ontologies have applications in many different fields, such as medicine, biology, e-commerce and the like. In this talk, we develop an unranked fuzzy logic and a tableau method for reasoning over such logic. The novelty of our approach is that we will extend many-valued logics with sequence variables and flexible-arity function and predicate symbols. The unranked fuzzy language and corresponding reasoning method will broaden the knowledge engineering capabilities in different fields.

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The anti-plane problem of the elasticity theory for a composite (piecewise homogeneous) orthotropic (in particular, isotropic) plane weakened by a crack, when the crack intersects an interface or eaches this one with the right angle, is studied by the integral equation method. When the crack reaches the interface, the problem is changed with a singular integral equation containing a fixed-singularity, but when the crack intersects the dividing border of interface – system (pair) of singular integral equations containing a fixed-singularity concerning characteristic functions of disclosure of crack. The behaviour of the solutions is studied. The present paper develops new computational algorithms for the approximate solution of the above-mentioned problems by the collocation (in particular by a discrete singularity) method ([3]). The algorithms are carried out in various specific practical problems. Numerical results are presented. In the case of loads of different quantities on the crack, the stress intensity factors at the ends of the crack are calculated, which allows us to make a hypothetical prediction about the crack spread.

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Characterizing present climate conditions and providing future climate projections at a regional scale is an extremely difficult task as it involves additional uncertainties while reducing, a spatial scale of Global Climate Models (GCMs) simulated climate parameters. Decreasing in spatial accuracy of GCMs simulated climate variables occurs from continental to local scale using statistical downscaling (SD) or dynamical downscaling (DD) techniques. There is a gap in most studies, specifically focused on estimating the uncertainty of downscaling results due to different statistical methods, as well as in creating ensembles from different GCM and SD methods at several sites in Georgia. In this article, a climate change parameter such as temperature has been investigated by SD and DD methods with an emphasis on SD

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The main aim of the present comments is by quotations, brought from competent publications, to emphasize the importance of hierarchical models, their purpose and requirements for them.

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A numerical model of the full cycle of cloud and fog genesis in the mesoboundary layer of atmosphere has been created. A numerical model of the distribution of aerosol from an instantaneous point source into the mesoboundary layer of the atmosphere has been created. The formation of smog is simulated based on the synthesis and "overlay" of the two above models.

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It is known that many elastic materials under finite deformations are deformed without noticeable change in volume. Such materials are incompressible elastic materials. An example of a homogeneous isotropic incompressible body is a rubber body. The boundary value problems in the present work are considered for an incompressible (rubber) confocal elliptic ring in an elliptic coordinate system. For incompressible bodies, equilibrium equations and Hook’s Law are written in the elliptic coordinate system, boundary value problems are set and explicit (analytical) solutions are presented with two harmonic functions, which are obtained by a method of separation of variables. The present work considers the boundary value problems for a confocal elliptic semi-ring , when the conditions of discontinuous continuation (of symmetry or asymmetry) of solutions are given on boundaries alfa=0 and alfa=pi . Following the discontinuous continuation of solutions, the solutions of the boundary value problems for a whole (closed) confocal elliptic ring are obtained. The boundary value problems for a confocal elliptic semi-ring are given with the superposition of the internal and external problems of a semi-ellipse. The numerical results of the concrete problems are obtained and corresponding graphs are constructed.

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Existing approaches to testing statistical hypotheses are discussed. Their characterization and comparison with each other are given. There are discussed ideas that underlie existing approaches and basic methods of hypothesis testing. Disadvantages of existing methods are shown, related to the perspective of using these methods at the modern level, caused by the unprecedented increase in data volume and increased demands on the reliability of the decision made.

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In 1914, S. Mazurkiewicz presented a transfinite construction of a subset of the Euclidian plane , having the extraordinary property. More precisely, a set is called a Mazurkiewicz subset of if for every straight line in (see [3]). As is well-known, from the existence of a well-ordering of the reals follows the existence of Mazurkiewicz sets. Mazurkiewicz sets may have a complicated descriptive set theoretical structure in there is a Mazurkiewicz set which is nowhere dense and of Lebesgue measure zero and there is also a Mazurkiewicz set which is Lebesgue nonmeasurable and does not have the Baire property (see [2]). The current presentation adds information about Mazurkiewicz sets in models of with no well-ordering of the reals. Arnold Miller has shown that there exists a model of ZF with an infinite Dedekind-finite set of reals in which there is a Mazurkiewicz set (see. [4], [5]). His model is the forcing extension of the Cohen-Halpern-Levy model obtained by adding Cohen reals. We shall prove here that there is a Mazurkiewicz set already in the Cohen-Halpern-Levy model. (see [1]) In fact, we shall present a general sufficient criterion for a Mazurkiewicz set to exist and show that it applies in the Cohen-Halpern-Levy model

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This report solves a static two-dimensional problem for an elastic plane with voids, in which an elastic circle from a different material with voids is inserted. Special representations of a general solution of a system of differential equations (1) are constructed via elementary (harmonic, biharmonic, and metaharmonic) functions which make it possible to reduce the initial system of equations to equations of simple structure and facilitate the solution of the initial problems. Solutions are written explicitly in the form of absolutely and uniformly converging series.

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For the non-linear Charney–Obukhov differential equation, the initial-boundary problem (with the periodic boundary conditions) in the rectangular domain is considered. For the stated problem, a symmetrical, semi-discrete scheme of an approximate solution is designed, which is locally linear. The order of an approximation of this scheme is O(tau^2) (tau is the step for the temporal variable). The error estimate of the approximate solution is obtained in terms of L2 -norm for vortex, while for the stream function the error is estimated in terms of both W - norm and C - norm.

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For the non-linear Charney–Obukhov differential equation, the initial-boundary problem (with the periodic boundary conditions) in the rectangular domain is considered. For the stated problem, a symmetrical, semi-discrete scheme of an approximate solution is designed, which is locally linear. The order of an approximation of this scheme is $O\left( \tau^2 \right)$ ($\tau$ is the step for the temporal variable). The error estimate of the approximate solution is obtained in terms of $L_2$ -norm for vortex, while for the stream function the error is estimated in terms of both $W_2^1$ - norm and $C$ - norm

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The limit distribution of an integral square deviation between two kernel type of Nadaraya–Watson estimators of the Bernoulli regression function for the group data is established.

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It the presented talk is shown that the cardinality of the continuum is not real-valued measurable if and only if there exists no nonzero σ-finite diffused measure μ on the real line such that all Vitali sets (respectively all Bernstein sets) are μ-measurable.

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In the articles [1] and [2] some results concerning with geometric realizations of families of sets are presented. In the talk statements related to the above-mentioned results are discussed. Namely, theorems related: • to the possibility of geometric realization of any finite family of subsets of the nonempty universal set by a family of subsets of the Euclidean plane whose each member is representable as a finite union of semi-open rectangles; • to the finitness of any independent family of subsets of the Euclidean plane such that the boundary of each member of the family is a non-reducible algebraic curve of degree not more than some fixed natural number; • to the existence of a countably infinite independent family of convex polygons in the Euclidean plane; • to the non-existence of an uncountable independent family of convex polygons in the Euclidean plane; • to the existence of an uncountable independent family of convex quasi-polygons (convex compacts with nonempty interiors and with the boundaries representable as a countable union of line segments) in the Euclidean plane are considered. References: 1) COMBINATORIAL PROPERTIES OF FAMILIES OF SETS AND EULER-VENN DIAGRAMS, A. Kharazishvili and T. Tetunashvili, Proc. A. Razmadze Math. Inst. 146(2008), 115–119; 2) ON SOME COMBINATORIAL PROBLEMS CONCERNING GEOMETRICAL REALIZATIONS OF FINITE AND INFINITE FAMILIES OF SETS, ALEXANDER KHARAZISHVILI AND TENGIZ TETUNASHVILI, Georgian Mathematical Journal, Volume 15 (2008), Number 4, 665-675.

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Over the past two decades, Georgia has faced increasingly heavy rainfall, hail and flooding, which especially devastated Kakheti wine region in Southern Georgia, causing severe damage to hundreds of vineyards. Since 2015, 85 anti-hail missile systems have been installed to protect entire Kakheti region, however, for the effective use of a modern anti-hail system, it became necessary to timely forecast extreme weather events of a regional and local scale. Thus, this article aims to develop timely forecasting of strong convection, dangerous precipitation and hail using modern weather forecasting models and radar technologies in Georgia. For this reasons various combinations of the physics parameterization schemes of the WRF-ARW model, the ARL READY system and the data of the modern meteorological radar Meteor 735CDP10 are used to predict the thermodynamic state of the atmosphere and assess the possible level of development of convective processes. The analysis of the calculated results showed that the variants of the microphysics parametrization schemes of the WRF model lead to significant variability in precipitation forecasts on complex terrain. Meanwhile, the upper-air diagrams of the READY system clearly showed the instability of the atmosphere for the cases discussed. Some results of these calculations are presented and analysed in this paper.

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In this talk is analyzed degenerate one-dimensional integro-differential model. Unique solvability and convergence of the constructed semi-discrete scheme with respect to the spatial derivative and fully discrete finite difference scheme are studied.

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Constrained Bayesian Method (CBM) of testing statistical hypotheses and its applications to different types of hypotheses are considered. It is shown that CBM is a new philosophy in statistical hypotheses theory, incorporating philosophies of Fisher, Neyman-Pearson, Jefery and Wald. Different kinds of hypotheses are tested at simultaneous and sequential experiments using CBM: simple, complex, directional, multiple, Union-Intersection and Intersection-Union. Obtained results clearly demonstrate an advantage of CBM in comparison with the listed approaches.

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Nowadays, when the emphasis is on alternative means of energy, natural gas is still used as an efficient and convenient fuel both in the home (for heating buildings and water, cooking, drying and lighting) and in industry together with electricity. In industrial terms, gas is one of the main sources of electricity generation in both developed and developing countries. Pipelines are the most popular means of transporting natural gas domestically and internationally. The main reasons for the constipation of gas pipelines are the formation of hydrates, freezing of water plugs, pollution, etc. It is an urgent task to take timely measures against the formation of hydrates in the pipeline. To stop gas hydrate formation in gas transporting pipelines, from existing methods the mathematical modelling with hydrodynamic method is more acceptable. In this paper the problem of prediction of possible points of hydrates origin in the main pipelines taking into consideration gas non-stationary flow and heat exchange with medium is studied. For solving the problem the system of partial differential equations governing gas non-stationary flow in main gas pipeline is investigated. The problem solution for gas adiabatic flow is presented.

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ρLog is a system for rule-based programming implemented in Mathematica, a state-of-the-art system for computer algebra. It is based on the usage of (1) conditional rewrite rules to express both computation and deduction, and of (2) patterns with sequence variables, context variables, ordinary variables, and function variables, which enable natural and concise specifications beyond the expressive power of first-order logic. Rules can be labeled with various kinds of strategies, which control their application. Our implementation is based on a rewriting-based calculus proposed by us, called ρLog too. We describe the capabilities of our system, the underlying ρLog calculus and its main properties, and indicate some applications.

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The Stokes Flow in a pipes with an arbitrary cross-section is studied. The Stokes equations with the appropriate initial-boundary conditions are considered. The problem is reduced to the system of linear integral equations with the weakly singular kernel.The existence and uniqueness of solutions is proved. The approximate solutions are obtained by means of the step-wise approximation method.

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One of the main tools in knowledge representation is ontology, which is a collection of logic-based formal language sentences. These sentences are used by automated reasoning programs to extract new knowledge and answer to the given questions. Although ontology languages are standardized by W3C, there are still many problems remaining. One of the most important problems is related to, so called, fuzzy ontologies. These are ontologies, where information is vague and imprecise. Fuzzy ontologies are obtained by integrating fuzzy logic with ontologies. Such kind of ontologies have applications in many different fields, such as medicine, biology, e-commerce and the like. In this paper, we develop an unranked fuzzy logic and study some of its properties. The novelty of our approach is that we will extend many-valued logics with sequence variables and flexiblearity function and predicate symbols. To the best of our knowledge, such formalisms are not yet studied in the literature. The unranked fuzzy language will broaden the knowledge engineering capabilities in different fields.

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The estimate for the Bernoulli regression function is constructed using the Bernstein polynomial. The question of its consistency and asymptotic normality is studied. Testing hypothesis is constructed on the form of the Bernoulli regression function. Also, the test is constructed for the hypothesis on the equality Bernoulli functions. The question of consistency of the constructed tests is studied.

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An Analytic Representation Formula of Solution of the Perturbed Delay Controlled Differential Equation with discontinuous and Continuous Initial Conditions is obtained. There is considered three cases when the variation of the initial moment occurs from left or from right or from both side.

The limit distribution is established for an integral mean-square deviation of a nonparametric generalized kernel-type estimate of the Bernoulli regression function. A test criterion is constructed for the hypothesis on the Bernoulli regression function. The question of consistency is considered, and for some close alternatives the asymptotics of test power behavior is investigated.

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A class of increasing sequences of natural numbers n_k is found for which there exists a function f in L[0,1) such that the subsequence of partial WalshFourier sums (S_(n_k)(f)) everywhere.

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Localized boundary-domain integro-differential equations (LBDIDE) systems associated with the Dirichlet and Robin boundary value problems (BVP) for the stationary heat transfer partial differential equation (PDE) with a variable coefficient are obtained and analyzed. Localization is performed by a non-smooth parametrix represented as the product of a global parametrix and the characteristic function of a ball centered at a reference point. The equivalence of the LBDIDE systems to the original variable-coefficient BVPs and unique solvability of the LBDIDE systems in appropriate Sobolev spaces are the main results of the present paper.

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Studying random functions of sets and their distributions is often more convenient with their characteristic functions. Here, some properties of random measures are studied, which can be easily proved by their characteristic functionalities. In particular, the necessary and sufficient conditions for the additivity of random functions of sets are proved.

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In this work we consider the two-dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Some BVPs are solved for a circular ring.

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The notion of volume is closely tied with several important geometric topics, such as equidecomposability theory, dissections of figures into finitely many ones of prescribed type, etc. One of the problems, arising here, is to extend an elementary volume on the Euclidean space to a volume on the same space, defined on maximally large class of figures. This problem is solved within the framework of the modern theory of invariant measures and its solution depends on the purely algebraic properties of a basic group of transformations of the Euclidean space. The present report is devoted to some aspects of the close connections between the elementary theory of volume and general methods of the theory of invariant measure. Theorem. Let G Dn contain an everywhere dense set of transformation of Rn . Then no G -measure is defined on the family of all subsets of Rn . At the same time, it may happen that there are G -volumes defined on the family of all bounded subsets of Rn .

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This paper is concerned to solve effectively the basic 2D boundary value problems (BVPs) for transversely isotropic elastic half-plane with double porosity. For finding explicit solutions of the basic BVPs is used the potential method and the theory of integral equations. For all problems, are constructed Fredholm type integral equations and the Poisson type formulas are obtained.

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We present the Riemann-Hilbert type boundary value problem for generalized analytic vectors in plane domains bounded by smooth curves and give the Noetherity conditions of the problem. As a model problem of the boundary value problems of the theory of generalized analytic vectors we consider the Riemann-Hilbert problem.

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In the present work, using absolutely and uniformly convergent series, the boundary value problems of quasi-statics 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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The properties of the numerical sequence ${(an)^{1/n}}$ with real parameter $a>0$ were studied, that leads to the one more interpretation of the asymptotic behavior of the Edmund Landau’s function $\pi n$ (the number of primes not exceeding $n$).

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General procedure of identification of non-linear regression dependences is offered below. It is developed with the purpose of overcoming two basic difficulties not only of regression analysis but also all modern mathematics: non-linearity and multidimensionality of a problem [1, 3]. The universal algorithm of determination of the intervals containing unknown regression parameters with probability close to unit, is developed. The quality of identification of regression dependences depends on the successful determination of these intervals. The method is suitable for the rather wide class of non-linear regressions at passive experiment. It considerably reduces the time necessary for solving identification problems and provides necessary reliability. The obtained results are also correct at active experiment at some hardening of imposed restrictions on the nature of noises [2].

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http://www.viam.science.tsu.ge/enlarged/2020/abstracts.pdf

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One of the main tools in knowledge representation is ontology, which is a collection of logic-based formal language sentences. These sentences are used by automated reasoning programs to extract new knowledge and answer the given questions. Although ontology languages are standardized by W3C, there are still many problems remaining. One of the most important problems is related to, so called, fuzzy ontologies. These are ontologies, where information is vague and imprecise. Fuzzy ontologies are obtained by integrating fuzzy logic with ontologies. Such kind of ontologies have applications in many different fields, such as medicine, biology, e-commerce and the like. In this talk we present a research project “Unranked Fuzzy Logic and its Applications” and discuss its preparatory research topic. In particular, we present main concepts of unranked and fuzzy logics. The goal of the project is to combine these two formalisms and find their application in semantic web, solving some of the problems, related to ontologies with vague and imprecise knowledge.

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The tension-compression dynamical problems are set and investigated in the zero approximation of governing system for Kelvin-Voigt plates with variable thickness

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The present work studies the elastic equilibrium of a plane deformed hyperbolic cylinder. Thus, in elliptical coordinate system internal boundary value problems of elasticity for domain with hyperbolic boundary are considered. The equilibrium equations system and Hooke’s law are written in the elliptic coordinates and the analytical solution of two-dimensional problems of elasticity is constructed in the region bounded by the coordinate lines of the elliptic coordinate system. Internal boundary value problems of elastic equilibrium of a homogeneous isotropic body with a hyperbolic boundary are presented for cases when normal or tangential stresses are given on the hyperbolic boundary. The exact solution is obtained using the method of separation of variables. The graphs for the numerical results of some test problems are presented.

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Certain classes of number sequences are introduced and a criterion of uniqueness of multiple function series is established via application of these classes. Also, various properties of sets of uniqueness of these series are established.

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The interaction of glacier-climate is a complex, non-linear process. The energy balance of the atmosphere, the glacier-atmosphere interface (meteorological conditions above the glacier) and the physical properties of the glacier themselves determine the process of glacier fluctuation (thickness melting). Also, the glacier ensemble, with its specific properties, contributes to climate change to some extent. Therefore, it is not easy to fully take these processes into account in mathematical models of glacier dynamics. In this article was developed a two-dimensional mathematical model of the dynamics of change in the thickness of the Caucasus glaciers, based on the integration of non-linear partial differential equations, which in turn provided a change in the equilibrium mass of the glacier. For the first time, the melting process of some Caucasus glaciers has been assessed by mathematical modeling. Some results of modeling are presented and analyzed

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Modeling of multidimensional random series with the help of computer is widely used to solve many application tasks. All existing methods of modeling of random sequences involve the existence of certain a priori information: multidimensional distribution function or spectral density, vector of mathematical expectation, covariance functions, etc., which are usually unknown and are determined on the basis of observation results. Errors in estimates, of course, affect the accuracy of modeling results. The stationary Gaussian series is fully determined by the given covariance matrix. Dependence determining the number of observations, necessary for modeling series (1) with given accuracy is given in the work. Covariance matrices, necessary for modeling, are determined by these observation results.

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This paper is concerned to study quasi-static boundary value problems of coupled theory of elasticity for porous sphere and for porous space with a spherical cavity. The general solution of the system equations in the coupled linear quasi-static theory of elasticity for a isotropic porous materials is constructed by means of the elementary (harmonic, bi-harmonic and meta harmonic) functions. The Dirichlet type BVPS for sphere and for space a with a spherical cavity are solved explicitly. The obtained solutions are represented as absolutely and uniformly convergent series.

The punch problem of the viscous half-plane with a friction is considered. However, it is based of the Kelvin-Voigt model. Using the methods of complex analysis which in the plane theory of elasticity are developed by acad. N. Muskhelishvili and his followers, the unknown complex potentials, that describe the equilibrium of a half-plane, are constructed effectively (analytically).

In the report elastic materials with voids is considred. Basic equations are written for a plane in the case of approximation N=1 of Vekua's theory. The general solutions are represented by means of analytic functions and solutions of the Helmholtz equations. Some boundary value problem for a circular plate and an infinite plate with a circular hole is solved.

In the monographs of Vekua [Vekua, L: Shell Theory: General Methods of Construction, Pitman Advanced Publishing Program,287 pp., Boston-London-Melbourne (1985), 11 Vekua, Generalized Analytical Function,M.:Fizmatgiz, 1959(in Russian)] were formulated two involving problems: 1. for elastic thinwalled structures the problem of satisfaction of Aoundary conditions when the generalized stress vector is given on the surfaces for elastic plates and shells. 2.The development of theory of find general solutions of linear partial differential equations of the complex variable for the nonlinear case. We consider the problem of satisfaction of boundary conditions when the generalized stress vector is given on the surfaces for elastic plates and shells. This problem was open also both for refined theories in the wide sense and hierarchical type models. This one for hierarchical models was formulated by Vekua. In nonlinear cases the bending and compression-extension did not split and for this aim we cited von Kármán type system without variety of ad hoc assumptions since, in the classical form of this system one of them represents the condition of compatibility but it i not an equilibrium equation. This problem was open also both for refined theories in the wide sense and hierarchical type models. Thus, we created the mathematical theory of refined theories both in linear and nonlinear cases for anisotropic nonhomogeneous elastic plates and shells, approximately satisfying the corresponding system of partial differential equations and boundary conditions on the surfaces. The optimal and convenient refined theory might be chosen easily by selection of arbitrary parameters, preliminarily a few necessary experimental measurements have been made without using any simplifying hypotheses. The same problem is solved for hierarchical models too. Extended the classical theory (for linear case developed by Goursat, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and so on) of finding the general solution of partial differential equations of complex analysis for the essential nonlinear differential equations.

The anti-plane problem of the elasticity theory for a composite (piecewise homogeneous) orthotropic (in particular, isotropic) plane weakened by a crack is reduced to a system of singular integral equations containing a fixed-singularity with respect to characteristic functions of disclosure of crack when the crack intersects the dividing border of interface with the right angle ([1], [2]). The method of discrete singularity ([3]) is applied to finding a solution of the obtained system. The corresponding new algorithm is constructed and realized. In this work, the behavior of the solutions is studied. The results of computations are represented. As we have mentioned a main objective was studying of behavior of solution in a vicinity at the ends of cracks and a finding the coefficients of intensity of stress. Consider a body composed of two isotropic materials (copper and aluminum). As we consider linear tasks of the elasticity theory so increment or diminution loading will lead to proportionally increment or diminution of values of relevant solutions. The last fact gives possibility to make the hypothetical forecasts about developments of a crack. References 1.Papukashvili A., Antiplane problems of theory of elasticity for piecewise-homogeneous orthotropic plane slackened with cracks. Bulletin of the Georgian Academy of Sciences, 169, N2, 2004. p. 267-270; 2.Papukashvili A., Davitashvili T., Vashakidze Z. Appro¬xi¬¬mate solution of anti-plane problem of elasticity theory for composite bodies weakened by cracks by integral equation method. Bulletin of the Georgian Academy of Sciences, Vol.9., No.3, 2015. p. 50-57. 3.Belotserkovski S.M., Lifanov I.K. Numerical methods in the singular integral equations and their application in aerodynamics, the elasticity theory, electrodynamics. Moscow, ”Nauka”, 1985. p. 256. (in Russian).

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დიდია ილია ვეკუას სამეცნიერო მემკვიდრეობა. მისი შრომები მათემატიკისა და მექანიკის აქტუალურ პრობლემებს ეხება. იგი იკვლევდა თეორიული და გამოყენებითი მათემატიკის მრავალი დარგის საკითხებს, რომელთა შორის განსაკუთრებული ადგილი უკავია კერძოწარმოებულიანი განტოლებების თეორიას. მან ელიფსური განტოლებებისათვის დაამუშავა ამონახსნთა ზოგადი წარმოდგენების თეორია და შესაძლებელი გახადა მისი გამოყენება სასაზღვრო ამოცანების შესასწავლად, რასაც მიუძღვნა მონოგრაფია “ელიფსურ განტოლებათა ამოხსნის ახალი მეთოდები”, რომელსაც მიენიჭა სტალინური პრემია. მნიშვნელოვანი წვლილი შეიტანა ერთგანზომილებიან სინგულარულ ინტეგრალურ განტოლებათა თეორიაში. განიხილა ფუნქციათა თეორიის ერთობ ზოგადი სასაზღვრო ამოცანა, ახლებურად გაიაზრა მისი ბუნება, გამოიკვლია ამოხსნადობის საკითხი და მოგვცა ინდექსის გამოსახულება, რომელიც საფუძვლად დაედო ღრმა განზოგადებებს ელიფსური დიფერენციალური ოპერატორების ინდექსის პრობლემასთან დაკავშირებით. დაამუშავა განზოგადებულ ანალიზურ ფუნქციათა თეორია, რომელიც კლასიკურ ანალიზურ ფუნქციათა თეორიის არსებითი გაფართოებაა.

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In this tutorial, we will introduce post-quantum cryptography with applications in real world. Then, we will focus on lattice- based cryptosystems and their formal analysis with automated softwares. In the security side, we will analyze the methods that are resistant to well-known attacks. We will discuss about how to write a code to analyze lattice-based cryptosystems formally. Also, we will provide some examples by considering NTRU-based cryptosystems.

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The estimate for the Bernoulli regression function is constructed using the Bernstein polynomials. The question of its consistency and asymptotic normality is studied. Testing hypothesis is constructed on the form of the Bernoulli regression function. Besides, the test is constructed for the hypothesis on the equality Bernoulli functions. The question of consistency of the constructed tests is studied.

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The existence of optimal elements is proved for the two-stage variational and optimal problems .

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Let X be a subset of the plane and p is a straight line in R^2 , X called an Uniform subset of R^2 with respect to p if for each p' parallel to p we have card(p' \cap X) less or equal 1. The study of the measurability properties of the uniform subsets was interesting question for our research. In measure theory is well known the standard concept of measurability of sets and functions with respect to a fixed measure on E. We introduce a concept of measurability of sets and functions not a just with respect to a fixed measure , but with a respect to the classes of the measures, which are defined on a different sigma- algebras on a base space E . In particular, it is demonstrated some important measurability properties of the uniform subsets and applications of the set-theoretical methods in the study of the measure theory. References 1. Kharazishvili A. One property of Hamel bases. Bull. Acad. Sci. GSSR, 95, (1979). 2. Kharazishvili A. Questions in the theory of sets and in measure theory. TSU, Tbilisi, (1978).

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2D Stokes system for the viscous incompressible Fluid with the free boundary is studied in the infinite area. Non-stationary case is considered. We admit that the pressure is a harmonic function and can be controlled.The Stokes system is reduced to the system of Fredholm integral equations . The existence of solutions of this system is proved and the approximate solution is obtained. The profiles of free boundaries are constructed for the different pressure.

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Investigation and numerical solution of the nonlinear integro-differential equation of parabolic type is considered. Integro-differential models of this type are based on the system of Maxwell equations. They appear in various diffusion problems and are studied in many works (see, for example, [1, 2] and references therein). Unique solvability, asymptotic behavior of the solution of the initial-boundary value problem and convergence of the finite-difference scheme are given. References 1. Jangveladze T., Kiguradze Z., Neta B. Numerical Solution of Three Classes of Nonlinear Parabolic Integro-Differential Equations. Elsevier/Academic Press, Amsterdam, (2015). 2. Jangveladze T. Investigation and Numerical Solution of Nonlinear Partial Differential and IntegroDifferential Models Based on System of Maxwell Equations. Mem. Differential Equations Math. Phys., 76, (2019).

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PrhoLog [1] is a programming language based on RhoLog calculus[2]. In this talk we consider an extension of Prholog with probabilistic strategies. Such an extension supports probabilistic programming, which is useful for randomize systems modeling. We studied semantics of the extended PrhoLog and possible applications.

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Currently, regional climate change is an objective reality, and the melting of mountainglaciers is one of the best evidence of this. For example, most glaciers in the Himalayas, the Alps and the Caucasus Mountains have halved over the past 70 years. At the same time, billions of people in these regions depend on glaciers (which are natural mountain water storage) for drinking water, agriculture, industry, and power generation. On the other hand, glacier deformation trends foresee what might happen in the lowlands, where most of the population lives. That is why the physical components of glaciers are currently being fully studied globally and regionally. In this article, a simple model of the glacier, which defines the basis for the deformation of the glacier, is used to study the melting processes of the Caucasus glaciers. Namely, the shape of the incomepressible non-Newtonian liquid of the glacier is modeled on the basis of a one-dimensional nonlinear high-order PDE model with advection, diffusion, and source. The configuration of the upper surface of the glacier is predicted by solving the continuity equation. A scheme similar to the Lax-Wendroff scheme is used to numerically solve the PDE nonlinear model. Some characteristic problems of mathematical and numerical modeling of glaciers are discussed

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The anti-plane problem of the elasticity theory for a composite (piecewise homogeneous) orthotropic (in particular, isotropic) plane weakened by a crack is reduced to a system of singular integral equations containing a fixed-singularity with respect to characteristic functions of disclosure of crack when the crack intersects the dividing border of interface with the right angle. The method of discrete singularity is applied to finding a solution of the obtained system. The corresponding new algorithm is constructed and realized. In this work, the behavior of the solutions is studied. The results of computations are represented.

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In case of (0,0) approximation of hierarchical models of bars for piezoelectric transversely isotropic cusped bars is investigated static and vibration problems.

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The limit distribution of the integral square deviation of a nonparametric estimator of the Bernoulli regression function for two samples is investigated.

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The problem of Luzin and the theorem of Sierpinski have found interesting connections with the measure extension problem. The study of the measur- ability properties of uniform sets is an interesting topic for our research. In measure theory it is well known the standard concept of measurability of sets and functions with respect to a fixed measure μ on a base (ground) set E. Now we introduce the concept of measurability of sets and functions not with respect to a fixed measure μ, but with respect to certain classes of measures, which are defined on different σ-algebras of subsets of base space E. (see [1], [2]). Let E be a set and let M be a class of measures on E (in general, we do not assume that measures belonging to M are defined on the one and same σ-algebra of subset of E). Definition. • We say that a function f : E → R is absolutely (or universally) measurable with respect to M if f is measurable with respect to all measures from M. • We say that a function f : E → R is relatively measurable with respect to M if there exists at least one measure μ from M such that f is μ- measurable. • We say that a function f : E → R is absolutely nonmeasurable with respect to M if there exists no measure μ from M such that f is μ- measurable. In particular, the graph of a function φ : R → R, which yields a positive solution of Luzin’s problem, is an absolutely nonmeasurable subset of E = R2 with respect to the class of all nonzero σ-finite measures on R2 that are invariant under the group of all isometries of R2. Theorem. There exists a uniform subset of R2 which is a Hamel basis of R2. References 1. A. Kharazishvili, One property of Hamel bases, Bull. Acad. Sci. GSSR, 95, 2 (1979), 2. A. Kharazishvili Questions in the theory of sets and in measure theory , TSU, Tbilisi, 1978

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The problem of testing directional hypotheses is examined considering the basic and alternative hypotheses in pairs. For optimality of the decision rule, the concept of mixed directional false discovery rate (mdFDR) is used. The developed method is applied for testing multiple hypotheses. The fact of guaranty of the quality of made decision on the desired level for the developed approach theoretically is proved and practically is demonstrated by computation of a concrete examples. There is also shown that the offered method is a private case of union-intersection and intersection-union hypotheses testing problems.

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In the talk the existence of point sets of special type is shown. One of the direct corollaries of the existence of these sets is that Sylvester’s well-known theorem on collinearity of points cannot be strengthened in a certain sense.

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The static equilibrium of elastic materials with voids is considered. 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 two analytic functions of a complex variable and one 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 body with voids. The concrete boundary value problems are solved.

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Mathematical and numerical simulation of the non-classical problems, namely problems of localization of stresses and displacements in the elastic body, are obtained by the boundary element method. The current work examines two localization problems, which have the following physical sense: on the middle point of the segment lying inside a body parallel to the border half plane in first case a point force is applied, and we must find such value of the normal stress along the section of the border half plane, which will cause this point force, while in the second case, there is given a vertical narrow deep trench outgoing of this point, and we must find such value of the normal stress along the section of the border half plane, which will result in such a pit. By using MATLAB software, the numerical results are obtained and corresponding graphs are constructed.

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The equation in variations is obtained for the controlled differential equation with delay. As application, on the bases of the basic theorem an approximate solution is constructed for the perturbed equation.

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Systems of nonlinear partial differential equations are describing many real processes. The present talk is devoted to one of such mathematical model arising in the investigation of the veinformation in leaves of higher plants and is represented as the two-dimensional nonlinear partial differential system. The convergence of the solution of initial-boundary value problem of the regularized system to corresponding solution of the given model is discussed.

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Our model study shows that the reversible transition between a passive mode and an active mode causes super-Gaussian transport dynamics, observed in various experiments. We find the non-Gaussian character of the matter's displacement distribution is essentially determined by the population ratio between active and passive motion. Interestingly, under a certain population ratio of the active and passive modes, the displacement distribution changes from sub-Gaussian to super-Gaussian as time increases. The mean-square displacement of our model exhibits transient superdiffusive dynamics, yet recovers diffusive behavior at both the short- and long-time limits.

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The present talk is devoted to construction of hierarchical models for piezoelectric nonhomogeneous porous elastic and viscoelastic Kelvin-Voigt prismatic shells on the basis of linear theories. Using I. Vekua’s dimension reduction method, governing systems of partial differential equations are derived and in the Nth approximation of hierarchical models BVPs and IBVPs are set. In the N = 0 approximation, considering, e.g., elastic plates of a constant thickness, governing systems mathematically coincide with the governing systems of the plane strain corresponding to the basic three-dimensional (3D) linear theory up to a separate equation for the out of plane component of the displacement vector. The ways of investigation of BVPs and IBVPs, including the case of cusped prismatic shells, are indicated and some preliminary results are presented. Antiplane deformation of piezoelectric nonhomogeneous materials in the threedimensional formulation and in N = 0 approximation is analysed. Some BVPs are solved in explicit forms in concrete cases. The aim of the present talk is also to draw the attention to problems to be solved connected with cusped shell-like elastic and viscoelastic piezoelectric bodies with voids and with related nonclassical BVPs and IBVPs for partial differential equations with order and type degeneracy. The development of the corresponding numerical methods and numerical calculations on computers are especially challenging.

In the present talk a criterion of the convergence in measure of a sequence of functions is formulated. An interrelation between this criterion and Lebesgue and F. Reisz well known theorems related to the convergence in measure of a sequence of functions is considered.

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Microphysics (MP) and cumulus parameterization schemes (CPSs) options in the WRF model may result in significant variability in precipitation prediction and this inconstancy especially highlighted in those studies that deal with predictions of warm season convective events (WSCE) over the mountains territories. The problem of MP and CPSs options for warm season precipitations prediction taking into account specifications of the concrete area has been widely explored in the scientific literature but this question is open yet. In this article a comparison study of three cumulus parameterization and five micro physics schemes of WRF v.3.6 model is conducted on the background of two exceptional precipitation events occurred on the territory of eastern Georgia during warm season of 2015. Three set of domains with horizontal grid-point resolutions of 19.8, 6.6 and 2.2 km are chosen to study precipitations formation over the territory of eastern Georgia taking into consideration influence of the Caucasus complex topography. Predicted accumulated total (24 h) precipitations are evaluated by careful examination of meteorological radar and radio zoned data and its comparison with simulated fields. Some results of the numerical calculations executed for two particulate cases of warm season convective events are presented and analyzed

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In this study, the WRF/Chemistry model with dust module is used to study dust particles short term transportations to the Caucasus from the neighbouring deserts. The results of calculations have shown the WRF model was able to well simulate dust aerosols transportation on the territory of Caucasus in conditions of a complex relief of the environment. Also the effect of the dust forcing on the Caucasus (Georgia) regional climate change with a regional climate model interactively coupled to a dust model is studied. Toward this purpose the sets of 30 years simulations (1985–2014) with and without dust effects by RegCM 4.7 model with 16.7 km resolution over the Caucasus domain and with 50 km resolution encompassing most of the Sahara, the Middle East, the Great Caucasus with adjacent regions was executed. Results of calculations have shown that dust aerosol is an inter-active player in the climate system of Georgia. Mineral dust aerosol influences on temperature and precipitations (magnitudes) spatial and temporally inhomogeneous distribution on the territory of Georgia and generally has been agreed with MODIS satellite data. According to results of comparisons of the simulated dust aerosol optical depth seasonal distributions against to the observed ones gave a good agreement. Also dust radiative forcing inclusion has improved simulated summer time temperature, as it was tend to warm bias mostly on continental territories of Georgia. Also there was observed summer time precipitation increment on the territory of Caucasus (Georgia) which improves seasonal distribution of simulated precipitation, but gives over estimation in annual total precipitation

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In [G. Jaiani, Piezoelectric Viscoelastic Kelvin-Voigt Cusped Prismatic Shells. Lecture Notes of TICMI, 19, 2019] transversely isotropic elastic piezoelectric nonhomogeneous bodies in the case when the poling axis coincides with one of the material symmetry axises is considered. The present talk is devoted to the dynamical problem such materials when the constitutive coefficients depending on the body projection (i.e., on a domain lying in the plane of interest) variables may vanish either on a part or on the entire boundary of the projection.

The current work examines two localization problems, which have the following physical meaning: on the middle point of the segment lying inside a body parallel to the border half plane in first case a point force is applied, and we must find such value of the normal stress along the section of the border half plane, which will cause this point force (stresses localization), while in the second case, there is given a vertical narrow deep trench outgoing of this point, and we must find such value of the normal stress along the section of the border half plane, which will result in such a pit (displacements localization). Finally, testable examples are given, which shows us, that what the normal stress supposed to be applied on part of the boundary, that at the midpoint of the segment lying inside the body to obtain a pre-given localized stress or displacement. Represented the numerical results, appropriate graphics, mechanical and physical interpretation of this problems.

An initial-boundary value problem is considered for the Timoshenko type nonlinear integro-differential equation. In particular consider an initial-boundary value problem for the J.Ball integro-differential equation, which describes the dynamic state of a beam (see, [1]). The solution is approximated with respect to by the Galerkin method, stabile symmetrical difference scheme and Jacobi iteration method (see,[2]-[4]). The algorithm has been approved on tests. First of all, we considered test examples that give the error of only the difference method and then we consider a test example in which, along with the error of the difference method, the error of the Galerkin method is taken into account. The results of recounts are represented in tables and graphics. References 1.Ball J.M. , Stability theory for an extensible beam, J. Differential Equations 14 (1973), 399-418. 2. Papukashvili G., On a numerical algorithm for a Timoshenko type beam nonlinear integro-differential equation, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics. Tbilisi. v.31 (2017), 115-118. 3. Papukashvili A., Papukashvili G., Sharikadze M. Numerical calculations of the J.Ball nonlinear dynamic beam. Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics. Tbilisi, v.32(2018). p. 47-50. 4. Papukashvili G., Peradze J., Tsiklauri Z., On a stage of a numerical algorithm for Timoshenko type nonlinear equation, Proc. A. Razmadze Math. Inst. 158 (2012), 67-77.

The Present talk is devoted to an up-dated exploratory survey of results obtained by various authors by using Vekua's dimension reduction method to investigation of hierarchical models of elastic, thermo-elastic, thermo-viscoelastic Kelvin-Voigt, piezoelectric, etc. (also with voids and microtemperatures) standard and prismatic shells, plates, and bars, including cusped ones. It contains the results of the speaker as well.

n the present paper is presented an explicit solution of the Dirichlet BVP for isotropic circle with diffusion, microtemperatures and microconcentrations. For this we constructed the regular solution of the system of equations for isotropic materials with diffusion, microtemperatures and microconcentrations by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The obtained solution of the Dirichlet BVP is represented as absolutely and uniformly convergent series.

The present talk is devoted to construction of hierarchical models for elastic shells with voids. It is considered to deriving the governing relations and systems of the N=0 approximation (hierarchical model) for elastic plates with voids. It is constructed the general solutions and solved some boundary problems.

he problem of pressure of rigid punch upon a viscous half-plane is considered. As is known, building and composition materials possess the property if visco-elasticity and its affect is reflected in the Hook's law. Unlike the elastic bond, the stresses for visco-elastic bodies are proportional to deformations and to their time derivatives. The goal of the present work is to extend the well-known Kolosov- Muskhelishvili's method elaborated for the problem of pressure of a rigid punch in the case of the classical theory of plane elasticity to the theory of linear visco-elasticity based of the Kelvin-Voigt model.

We consider the anisotropic inhomogenous not only elastic long (conventionally)pipe, when solid part's cross-sections represent either circular or elliptical rings. The stress-strain state is defined by the nonlinear strong elliptical system of differential equations with boundary conditions. We construct and investigate the corresponding numerical schemes. The process of approximation of boundary conditions on networks is not used for the rough transfer, in contrast to classical methods.

We investigate multi-field mixed problems for complex elastic aniso¬t¬ro¬pic structures when in different adjacent components of the com¬posed body different refined models of elasticity theory are considered. In particular, we analyse the case when we have the generalized thermo-elec¬t¬ro-magneto elasticity model (GTEME model) in one region of the com¬po¬sed body and the generalized thermo-elasticity model (GTE model) in another adjacent region. Both models are associated with Green-Lind¬say's model. This type of mechanical problem mathematically is described by systems of partial differential equations with appropriate transmission and boundary conditions. In the GTEME model part we have six dimensional unknown physical field (three components of the displacement vector, electric potential function, magnetic potential function, and temperature distribution function), while in the GTE model part we have four dimensional unknown physical field (three components of the displacement vector and temperature distribution function). The diversity in dimensions of the interacting physical fields complicates mathematical formulation and analysis of the corresponding boundary-transmission problems. We apply the potential method and the theory of pseudodifferential equations and prove uniqueness and existence theorems of solutions to different type basic and mixed boundary-transmission problems in appropriate Sobolev spaces. We analyse the smoothness and singularity properties of solutions to mixed and interfacial crack type problems. This is a joint work with Maia Mrevlishvili, Otar Chkadua and Tengiz Buchukuri.

Access control is a fundamental security requirement for computing environments: It controls the ability of a subject to use an object in some specific manner. Attribute-based access control (ABAC) is a logical access control with great flexibility to specify access control policies as rules which get evaluated against the attributes of participating entities (user/subject or subject/object), operations, and the environment relevant to a request. The access control policies that can be implemented in ABAC are limited only by the computational language and the richness of the available attributes. Considerable work has been done and a number of formal models have been proposed recently for ABAC, with minimal sets of features that are sufficient to implement many desirable capabilities. In this talk, we discuss the popular access control models ABACα and ABACβ, and propose to study and analyse them in ρLog, a rule-based framework developed by us. We specify the logical and operational semantics of their policies in our framework, and show how to use the ρLog system to decide some properties of interest.

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Access control is a fundamental security requirement for computing environments: It controls the ability of a subject to use an object in some specific manner. Attribute-based access control (ABAC) is a logical access control with great flexibility to specify access control policies as rules which get evaluated against the attributes of participating entities (user/subject or subject/object), operations, and the environment relevant to a request. The access control policies that can be implemented in ABAC are limited only by the computational language and the richness of the available attributes. Considerable work has been done and a number of formal models have been proposed recently for ABAC, with minimal sets of features that are sufficient to implement many desirable capabilities. In this talk, we discuss the popular access control models ABACα and ABACβ, and propose to study and analyse them in ρLog, a rule-based framework developed by us. We specify the logical and operational semantics of their policies in our framework, and show how to use the ρLog system to decide some properties of interest.

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ρ Log-prox is a calculus for rule-based programming with strategies, which supports both exact and approximate computations. Rules are represented as conditional transformations of sequences of expressions, which are built from variadic function symbols and four kinds of variables: for terms, hedges, function symbols, and contexts. ρLog-prox extends ρLog by permitting in its programs fuzzy proximity relations, which are reflexive and symmetric, but not transitive. We introduce syntax and operational semantics of ρLog-prox, illustrate its work by examples, and present a terminating, sound, and complete algorithm for the ρLog-prox expression matching problem.

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The lecture course is dedicated to the theoretical investigation of basic, mixed and crack type three-dimensional initial-boundary value problems of the generalized thermoelectro-magneto-elasticity 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 analyse dynamical initial-boundary value problems and the corresponding boundary value problems of pseudo-oscillations, which are obtained from the dynamical problems by the Laplace transform. The dynamical system of partial differential equations generate a nonstandard six dimensional matrix differential operator of second order, while the system of partial differential equations of pseudo-oscillations generates a second order strongly elliptic formally non-selfadjoint six dimensional matrix differential operator depending on a complex parameter. First, we prove uniqueness theorems of dynamical initial-boundary value problems under reasonable restrictions on material parameters and afterwards we apply the Laplace transform technique to investigate the existence of solutions. This approach reduces the dynamical problems to the corresponding elliptic problems for pseudo-oscillation equations. The fundamental matrix of the differential operator of pseudo-oscillations is constructed explicitly by the Fourier transform technique, and its properties near the origin and at infinity are established. By the potential method, the corresponding three-dimensional basic, mixed and crack type boundary value problems and the transmission problems for composite elastic structures are reduced to the equivalent systems of boundary pseudodifferential equations. The solvability of the resulting boundary pseudodifferential equations are analysed in appropriate Sobolev-Slobodetskii, Bessel potential, and Besov spaces and the corresponding uniqueness and existence theorems of solutions to the boundary value problems under consideration are proved. 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. It is shown that the smoothness and stress singularity exponents essentially depend on the material parameters and an efficient method for their computation is described. By the inverse Laplace transform, the solutions of the original dynamical initial-boundary value problems are constructed and their smoothness and asymptotic properties are analysed in detail.

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The Cauchy problem for the second order nonlinear evolution equation in the Hilbert space is investigated. This equation represents generalization of J. Ball non-linear integrodifferential equation. Approximate solution of the stated problem is searched using threelayer semi-discrete scheme. Convergence of the proposed scheme is proved and approximation error of the solution is obtained.

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At present the climate change problem is associated with a growing crisis in food production and health safety owing to environment pollution, more frequent heavy precipitations, hails, floods and droughts with a growing desertification processes in many regions of the earth. Climate change process is in progress in the South Caucasus region and in Georgia too where increased trends in mean annual temperature with heavy precipitations, hails, floods and droughts are more frequent. So for prevention of accidents to take more efficient steps in provision with scientific information (regional and local scale extreme weather prediction, climate change tendencies) against the freaks of nature is an urgent issue on the territory of Georgia. For regional weather forecasting and climate trends prediction one of the most essential means represents creation an ensemble of the regional climate forecasting system (WRF Chem, WRF Climate, RegCM) for the Caucasus (Georgia) territory for the purpose of studding the different climate change scenarios and for future climate projections. In this article a high resolution modeling ensemble system is developed for studying extreme natural phenomenon, climate variability and the human impact on climate on the territory of Georgia. The climate ensemble system is constructed by existing global and regional climate earth-system models which are based on numerical solution of nonlinear hydro-thermodynamics system of equations using high performance computation technologies. For achievement of these approaches the statistical and dynamical downscaling methods and Regional Climate Model Evolution System (RCMES) are used. Some results of numerical calculations are analyzed and presented

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The limit distribution of the Integral square deviation of a nonparametric estimator of the Bernoulli regression function for one sample and two independent samples is investigated.

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A lot of researchers developed a theory for the propagation of electromagnetic waves in the ionospheric plasma (1-5) . However , they made certain assumptions such as that the ambient magnetic field is vertical and the magnetic declination is zero, which is unrealistic in mid-latitude ionosphere. In this study , the general forms of conductivity tensor, refractive index and polarisation coefficients for ionospheric plasma have been obtained with included the magnetic declination and dip angle by solving relevant motion equation. Calculations show that polarization coefficients become complex numbers when magnetic declination is included. While they are pure imaginary in the absence of declination. The declinational effect on the real parts of the polarization coefficients is more pronounced around plasma frequency that at other frequencies. However , the imaginary parts and refractive index are not affected at the plasma frequency [1, 2, 3, 4].

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The polarization of the characteristic wave for the cold ionosphere plasma in the northern hemisphere is theoretically investigated. These wave is composed of two parts both real and imaginary as the ionosphere becomes double-refractive. As a result the characteristic wave has elliptic polarization due to the earth's magnetic field as theoretical [1, 2, 3, 4].

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In order to construct an approximate solution of initial-boundary value problem for one system of non-linear two-dimensional equations, two different approaches are used. In particular, decomposition methods based on variable directions difference scheme and on the averaged model are investigated. Algorithms which are necessary for realization are described for both methods. Software code is developed and calculations are made for different test cases. The number of operations is defined for both methods. The time necessary for the realization of algorithms and the accuracy of the numerical experiments are compared. Obtained results are analyzed and relevant conclusions are made.

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The present lecture course is devoted to construction of differential hierarchical models for piezoelectric nonhomogeneous porous elastic and viscoelastic Kelvin-Voigt prismatic shells on the basis of linear theories. Using I. Vekua's dimension reduction method, governing systems are derived and in the Nth approximation of hierarchical models boundary value problems (BVPs) and initial boundary value problems (IBVPs) are set. In the N=0 approximation, considering, e.g., elastic, plates of a constant thickness, governing systems mathematically coincide with the governing systems of the plane strain corresponding to the basic three-dimensional (3D) linear theory up to a separate equation for the out of plane component of the displacement vector. The ways of investigation of BVPs and IBVPs, including the case of cusped prismatic shells, are indicated and some preliminary results are presented. Antiplane deformation of piezoelectric nonhomogeneous materials in the three-dimensional formulation and in N=0 approximation is analyzed. Well-posedness of Dirichlet and Keldysh type problems (BVP) are studied in the N=0 order approximation of hierarchical models for cusped prismatic shells. Some BVPs are solved in explicit forms in concrete cases.

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The present talk is devoted to construction and investigation of hierarchical models for piezoelectric termo-viscoelastic Kelvin-Voigt bars with rectangular cross-sections. In particular, in (0, 0) approximation static and oscillation problems are discussed. A special attention is given to analysis of peculiarities of nonclassical setting boundary conditions (BCs) in the case of cusped bars. Namely, the criteria are established for piezoelectric transversely isotropic cusped bars when on one end or on both ends of the bar no data need to be prescribed. Weighted BCs are set as well. On the face surfaces of the bar under consideration stress vectors and outward normal components of the electric displacement vectors are prescribed, while at the ends of the bar all the admissible (in sense of well posedness of boundary value problems) BCs, including mixed ones, with respect to weighted (0, 0) moments of the components of the mechanical displacement vectors and electric potential, and (0, 0) moments of the components of the stress and electric displacement vectors are prescribed.

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For the controlled functional differential equation with delay parameters the necessary conditions of optimality is obtained.

The report contains two parts. I. One of the most principal objects in development of mechanics and mathematics is a system of nonlinear differential equations for an elastic isotropic plate constructed by von K´arm´an. In 1978 Truesdell expressed a doubt: “Physical Soundness” of von K´arm´an system. This circumstance generated the problem of justification of von K´arm´an system. Afterwards this problem has been studied by many authors, but with most attention it was investigated by Ciarlet. In particular, he wrote: “The von K´arm´an equations may be given a full justification by means of the leading term of a formal asymptotic expansion” ([1], p. 368). This result obviously is not sufficent for a justification of “Physical Soundness” of this system, because representations by asymptotic expansions is dissimilar and leading terms are only coefficients of power series without any “Physical Soundness.” Based on our works, the method of constructing such anisotropic nonhomogeneous 2D nonlinear models of von K´arm´an-Mindlin-Reissner (KMR) type for binary mixtures; (poro/visco/piezoelectric/electrically conductive)elastic thin-walled structures with variable thickness is given, by means of which the terms become physically sound. The corresponding variables are quantities with certain physical meaning: averaged components of the displacement vector, bending and twisting moments, shearing forces, rotation of normals, surface efforts. The given method differs from the classical one by the fact that according to the classical method, one of the equations of von K´arm´an system represents one of Saint-Venant’s compatibility conditions, i.e. it‘s obtained on the basis of geometry and not taking into account the equilibrium equations. II. In the second one if we consider the problems connected with an xtension(enlarge) of initial data for constructing by evident scheme to finding the approximate solution of evolutionary equations by high order of accuracy than Resolvent methods (or semi group operators theory) [see, for example, 2] or Courant, von Neumann, Lax direct methods for approximate solution some problems of mathematical physics [see, for example, 3]. As it’s well known for Resolvent methods for solving by high order of accuracy lies in the best approximation of corresponding kerners while for Difference methods difficulties represent incorrectness of multipointing (high order of accuracy) schemes. In the report we construct the explicit schemes giving the approximate solution of some initial-boundary value problems by arbitrary order of accuracy depending only on order of smoothness of the desired solution.

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Controlled natural languages (CNLs) are engineered languages that are based on natural language, but have their vocabulary, syntax, and/or semantics restricted [9]. The motivation is to have a language that, on one hand, looks as natural as possible and, on the other hand, is simple and unambiguous

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In this talk we present a combination of nominal and unranked languages, extending nominal languages by unranked symbols and studying the fundamental computational mechanism for them: unification. However, unlike the unranked languages, where sequences are introduced in the meta-level, nominal syntax allows us to introduce their analogs in the object level. This is done by generalizing already existing syntactic constructs, pairs, to arbitrary tuples. They should be flat, which is achieved by imposing a special α-equivalence rule for them.

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The limit distribution of the integral square measure of deviation of one nonparametric estimator of the Bernoulli regression function is investigated.

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The full cycle of fog- and cloud formation in the mesoscale boundary layer of the atmosphere (MBLA) is simulated by numerical methods. Aerosol diffusion in MBLA from a point source is also simulated. It is possible to simulate smog formation by ‘superposition’ models of humidity processes and aerosol diffusion in MBLA. All three tasks are stated in both (x, –z) and (r, z) vertical planes

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The present work is direct continuation of the following articles [2–4], in which considered the construction of algorithms and corresponding numerical computation for the approximate solution of non–linear integro–differential equation of the Kirchhoff and the Timoshenko types. In particular, we consider an initial–boundary value problem for the J. Ball integro–differential equation, which describes the dynamic state of a beam [1]. look for the approximate solution of the stated problem by applying the Galerkin method, the symmetric stable scheme and the Jacobi iterative method. The algorithm has been approved for tests. The results of computation are represented by tables and graphics. References [1] J. M. Ball, Stability theory for an extensible beam. J. Differential Equations 14 (1973), 399–418. [2] G. Berikelashvili, A. Papukashvili, G. Papukashvili, J. Peradze, Iterative solution of a nonlinear static beam equation. arXiv preprint arXiv:1709.08687, 2017; https://arxiv.org/pdf/1709.08687v1.pdf.7. [3] A. Papukashvili, G. Papukashvili, B. Dzagania, Numerical calculations of the Kirchhoff nonlinear dynamic beam. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 24 (2010), 103–107. [4] A. Papukashvili, J. Peradze, J. Rogava, An approximate algorithm for a Kirchhoff nonlinear dynamic beam equation. Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 23 (2009), 84–86.

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The main causes of gas line constipation (emergency shutdown) are the formation of hydrates, freezing of water jams, pollution, and so on. In order to take timely measures against the formation of hydrates, it is necessary to know better the distribution of humidity, pressure and gas temperature in the pipeline. It is well known that a favorable condition for the formation of hydrates in the main pipeline is the place where the dew point occurs (depending on pressure and humidity). Indeed, the dew point is the temperature below which liquid droplets begin to condense, and dew can form. In this paper, we study the problem of predicting the possible points of hydrate occurrence in main pipelines, taking into account unsteady gas flow and heat exchange with the medium. To solve the problem, a system of partial differential equations has been investigated that controls unsteady gas flow in the gas pipeline. The problem solution for adiabatic gas flow is presented.

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At present climate change problem is associated with increased anthropogenic environment pollution, more frequent heavy precipitations, hails, floods and droughts with a growing desertification processes in the territory of Georgia. Namely for the last four decades the number of the natural hazards has increased about three times in comparison with 60 years period of the last century on the territory of Georgia. So the question of studding formation of hazardous precipitations on the background of modern climate change is an urgent issue for Georgia. In this article a comparison study of the results of numerical calculations of three cumulus parameterization and five micro physics schemes of the Weather Research Forecast (WRF) v.3.6 model and the Real-time Environmental Applications and Display System (READY), against the radar’s observational data, on the background of four exceptional local scale precipitation events occurred in the capital city of Georgia Tbilisi during summertime of 2015 and 2016 years is presented. Also for evaluation of summer time short term, local scale, heavy showers prediction in Tbilisi area READY System is used. Aeorological diagrams of READY system for discussed cases precisely showed instability of atmosphere on local territory despite of the fact that in all four cases we had different level of instability. Predicted accumulated total precipitations (24 h) are evaluated by carefulnexamination of meteorological radar and radio zoned data against the WRF simulated fields. Some results of the numerical calculations executed for warm season convective events are presented and analyzed.

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Automated reasoning is one of the most important research area in logic and computer science. It is also considered as a sub-field of artificial intelligence. It studies different aspects of reasoning. The most important tools of automated reasoning are different calculi for classical logic. We study TSR-logic based methods, that can be used in automated theorem proving and develop TSR solver. The main targe to solver is to be used in real life applications. One of such applications is enviromental contamination and weather forecast. The expected results will have as practical as well theoretical character.

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The paper considers normal contact problems formulated as follows: an indenter with negligible weight presses the surface half-space with a certain force, i.e., normal stress acts on the contact surface and tangential stress is zero. In particular, we consider two types of distributed load that correspond to the following cases: when half-space is subjected to frictionless flat rigid indenter, and when half-space is subject to frictionless cylindrical rigid indenter. The article considers plane deformation. Problems are solved by boundary element methods (BEM), which are based on singular solutions of Flamant (BEMF) and Boussinesq (BEMB) problems. The stress-strain state of the half-plane, particularly the constructed contours (isolines) of stresses in half-plane, was studied. The results obtained by BEMF and BEMB are discussed and compared.

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In this talk we present one approach to model MaTRU-based cryptographic protocols in Maude-NPA. We discuss limitations and possible extension of the system to formally analyze MaTRU-based cryptosystems.

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Access control is a security technique that specifies which users can access particular resources in a computing environment. Formal description of access control is extremely important, since it should be defined, unambiguously, how rules regulate what action can be performed by an entity on the resource, how to guarantee that each request gets an authorization decision, how to ensure consistency, etc. It is also important that such a formal description is at the same time declaratively clear and executable, to avoid an additional layer between specification and implementation. In this talk we discuss an Attribute-Based Access Control model, called ABACβ, which is an extension of ABACα model with context, contextual attributes and meta-attributes. We present the current capabilities of PρLog system to formalize ABACβ operational model and discuss possible extensions of the system in this purpose.

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In the present paper the mathematical model of quantum properties of different nanostructures is studied. Nanostructures properties are significantly different from their similar materials at macro scale level.The influence of the surface atoms becomes important, the thermal, optical and electrical properties change dramatically at the nanoscale dimension, the dimensions are comparable with the wavelength of electrons which causes quantum confinement of electrons and quantization of their energy. We have studied the energy levels of electrons in the cubic and prismatic (with the hexagonal cross-section) nanostructures from the non-relativistic viewpoint. We considered the stationary Schr¨odinger Equation for the wavefunction of the electron with the homogeneous boundary conditions. The eigenfunctions and the corresponding eigenvalues are obtained and consequently the possible numerical values of the energy levels of electrons are estimated.

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The estimate for the Bernoulli regression function is constructed using the Bernstein polynomial. The question of its consistency and asymptotic normality is studied. Testing hypothesis is constructed on the form of the Bernoulli regression function. Also, the test is constructed for the hypothesis on the equality Bernoulli functions. The question of consistency of the constructed tests is studied

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We will provide brief introduction to classical Fourier analysis. The following topics will be covered: • The conjugate mapping. Integral representation of the conjugate operator. • The truncated Hilbert transform on L_2. • The Calderon-Zygmund interval decomposition. The Calderon-Zygmund decomposition. • The Hardy-Littlewood maximal function. • The Lebesgue differentiation theorem. • Existence of the Hilbert transform of integrable functions. • The Hilbert transform on L_p. • The Schwartz class and tempered distribution. • The Fourier transform on L_p. • Littlewood-Paley theory.

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In this talk we present Maude-NRL, which is a tool and language for formal analysis of cryptographic protocols. We will discuss its possibilities to analyze quantum protocols.

we consider solution spaces for some class of singular elliptic systems on Riemann surfaces and boundary-value problems for solution spaces of such systems. We also discuss some relations for the kernels of the Carleman–Vekua equation. In particular, representations of these kernels in the form of generalized power functions are completely analogous to the classical Cauchy kernel expansion. The obtained results are applied to some problems of the theory of generalized analytic functions.

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In our talk summable series with respect to the systems $\Phi=\{\varphi_{n}(t)\}_{n=0}^{\infty}$ of finite and measurable functions defined on [0,1] by positive, regular, triangular Λ matrixes are considered. It is introduced the notion of Cantor’s Λ functional for Λ summable series. This notion generalizes, in particular, any trigonometric integral in the sense of the reconstruction of coefficients of the series. Reconstruction of coefficients of multiple function series by iterated using of Cantor’s Λ functional is established

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The talk is devoted to the homogeneous Dirichlet problem for the vibration problem of cusped viscoelastic Kelvin-Voigt prismatic shells in case of the zero approximation of the hierarchical models. The classical and weak setting of the problem are formulated. The spacial weighted functional spaces are introduced, which are crucial in our analysis. The coerciveness of the corresponding bilinear form is shown and uniqueness and existence results for the variational problem are proved. We describe in detail the structure of this spaces and establish their connection with weighted Sobolev spaces.

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In this study we was investigated the response of the ionospheric conductivities on the middle latitudes to the southward turnings of the BZ component of the interplanetary magnetic field (IMF). For this purpose, parallel conductivity , Pedersen conductivity , Hall conductivity data calculated for four different geographic coordinates in the middle latitude region of the northern hemisphere during the 22nd solar cycle (1986-1996) were examined.The effect on the conductivities of the changes in the BZ component occurs before about 18 hours from the event moment, and this effect disappears after 24 hours from the event moment. Furthermore , all of ionospheric conductivities less react to changes in the BZ component as latitude increase.

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In this paper we study matching in equational theories that specify counterparts of associativity and commutativity for variadic function symbols. We design a procedure to solve a system of matching equations and prove its soundness and completeness. The complete set of incomparable matchers for such a system can be infinite. From the practical side, we identify two finitary cases and impose restrictions on the procedure to get an incomplete terminating algorithm, which, in our opinion, describes the semantics for associative and commutative matching implemented in the symbolic computation system Mathematica.

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The present paper deals with the construction and investigation of hierarchical models for piezoelectric viscoelastic Kelvin-Voigt prismatic shells (including the case of cusped prismatic shells ) with voids on the basis of the linear theories. Using I. Vekua’s dimension reduction method, governing systems are derived and in the Nth approximation boundary value problems are set. In the N=0 approximation, considering plates of a constant thickness, governing systems mathematically coincide with the governing systems of the plane strain corres-ponding to the basic three-dimensional linear theories up to a separate equation arising for the out of plane component of the displacement vector in the case under consideration. It is proved that initial conditions for weighted displacements, weighted electric potential, and a weighted volume fraction are always classical, while boundary conditions for weighted displacements, weighted electric potential, and a weighted volume fraction are non-classical in the case of cusped prismatic shells, namely, we are not always able to prescribe them at cusped edges. E.g., if the thickness looks like the power function, vanishing at the edge of the prismatic shell, then in the N=0 approximation for piezoelectric viscoelastic Kelvin-Voigt prismatic shells with voids the displacements, electric potential, and volume fraction we may prescribe at the cusped edge if the power is less than 1, while it is not allowed if the power is greater than or equal to 1.

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Access control is a security technique that specifies which users can access particular resources in a computing environment. Over the years, numerous access control models have been developed to address various aspects of computer security. In this paper, we focus on a modern approach, attribute-based access control (ABAC), which has been proposed in order to overcome limitations of traditional models: discretionary access control (DAC), mandatory access control (MAC) and role-based access control (RBAC). The work on integrating access control mechanisms in semantic web technologies is developing into two directions: (1) to use semantic web technologies for modeling and analyzing access control policies and (2) to protect knowledge encoded in an ontology. In this paper we focus on the first issue and investigate how ABAC can be integrated into ontology languages.

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Characterizing present climate conditions and providing future climate projections at a regional scale is an extremely difficult task as it involves additional uncertainties while reducing a spatial scale of Global Climate Models (GCMs) simulated climate parameters. Decreasing in spatial accuracy of GCMs simulated climate variables occurs from continental to local scale using statistical downscaling (SD) or dynamical downscaling (DD) techniques. A Statistical downscaling methods by itself is divided into three groups: multiple linear regression, nonlinear regression and stochastic weather enerators, which mostly are used in different sectoral impact studies. In this study monthly maximums and minimums of 2-meter air temperature from three GCMs of CMIP5 database has been statistically downscaled using RCMES package, with four different methods for 27 selected meteorological stations on the territory of Georgia. The extreme values downscaling methods have been trained for the 1961-1985 and validated for the 1986-2010 period. Some statistical parameters have been calculated by applying R statistics environment to compare observed and simulated time series and to evaluate temporal and spatial goodness of each method. Downscaling model driven by the validation study was used for future extremes (Tmin and Tmax) time series construction for the 2021-2070 period under RCP4.5 and RCP8.5 scenarios. Temperatures time series have been constructed from a multi-model ensemble, with mean and spread. The Future change tendencies have been assessed in comparison to 1986–2010 period. Validation of statistical downscaling methods shows that all of the methods have some advantages and disad vantages on the temporal and spatial scale. The metrics used for model performance evaluation varies from station to station, year to year, and season to season. Keywords: GCM, statistical downscaling, bias correction, future projection

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The present talk is devoted to an updated exploratory survey of results obtained by various authors by using Vekua’s dimension reduction method to investigation of hierarchical models of elastic standard and prismatic shells including cusped ones. In the last case, their thicknesses may vanish on their projection boundaries. Such bodies, considered as 3D ones, are occupying 3D domains with non-Lipschitz boundaries, in general. The problem mathematically leads to the question of well posing and solving of BVPs (IBVPs) for even order equations and systems of elliptic (hyperbolic) type with the order degeneration in the static (dynamical) case which are non-classical, i.e., Keldysh type and weighted ones, in general. The talk also contains some new and unpublished results, concerning thermoelastic piezoelectric cusped prismatic shells

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The strength of a sufficiently long thick-walled homogeneous isotropic circular tube (cylinder) under the action of external forces is studied using the problems of elasticity statics. In particular, there are established the minimum thickness of pipes with different materials and with different diameters, for which do not exceed the permissible stresses values. Cylinder is in state of plane deformation, therefore are considered a two-dimensional boundary value problems for circular ring. Represented tables and graphs of minimum thickness of a circular ring when a) the normal constant stresses act at internal border, while the outer boundary is free of stresses and b) the normal constant stresses act at external border, while the inner boundary is free of stresses. To the numerical realization above mentioned problems are used solutions obtained by two means: the analytical solution obtained by method of separation of variables, and Lame's solution.

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Constrained Bayesian method (CBM) and the concept of false discovery rates (FDR) for testing directional hypotheses is considered in the paper. There is shown that the direct application of CBM allows us to control FDR on the desired level for both one set of directional hypotheses and multiple case when we consider () sets of directional hypotheses. When guaranteeing restriction level on the desired level, the sequential method of Bayesian type must be applied, the stopping rules of which is proper and the sequential scheme of making decision strongly controls mixed directional false discovery rate. Computation results of concrete examples confirm the correctness of theoretical outcomes.

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ABACα is a foundational model for attribute-based access control with a minimal set of capabilities to configure many access control models of interest, including the dominant traditional ones: discretionary (DAC), mandatory (MAC), and role-based (RBAC). A fundamental security problem in the design of ABAC is to ensure safety, that is, to guarantee that a certain subject can never gain certain permissions to access certain object(s). We propose a rule-based specification of ABACα and of its configurations, and the semantic framework of ρLog to turn this specification into executable code for the operational model of ABACα. Next, we identify some important properties of the operational model which allow us to define a rule-based algorithm for the safety problem, and to execute it with ρLog. The outcome is a practical tool to check safety of ABACα configurations. ρLog is a system for rule-based programming with strategies and built-in support for constraint logic programming (CLP). We argue that ρLog is an adequate framework for the specification and verification of safety of ABACα configurations. In particular, the authorization policies of ABACα can be interpreted properly by the CLP component of ρLog, and the operations of its functional specification can be described by five strategies defined by conditional rewrite rules.

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Planetary EM 5 ULFW appears as a result of interaction of the ionospheric medium with the spatially inhomogeneous geomagnetic field. The shear flow driven wave perturbations effectively extract energy of the shear flow increasing own amplitude and energy. These perturbations undergo self organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation’s front. Depending on the features 10 of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices, or dipole vortex, or vortex streets and vortex chains.

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The solutions of the problem of linear conjugation in a closed form, solving a system of singular integral equations in quadratures, determining the splitting type of holomorphic vector bundle on the Riemann sphere is reduced on the computation of the partial indices of matrix-functions of Holder class on the unit circle. From the solutions of one of listed problems the solution of others follows. On the other hand, the linear conjugation problem for piecewise constant matrix functions reduces to monodromic Riemann-Hilbert problem for the class of regular systems of differential equations. Recently, an algorithm for solving the linear conjugation problem for piecewise constant boundary matrix functions and an algorithm for calculation splitting type of holomorphic vector bundle induced from Fuchsian system of differential equations on the Riemann sphere has been proposed. In the talk, we give an algorithm that permit for any set of marked points on the Riemann sphere and for any quadratic matrices constructed a rational matrix function, the set of singular points of which is contained in the set of marked points and the left partial indices of this matrix function coincide with splitting type of the holomorphic vector bundle induced from corresponding Fuchsian system. This work was supported, in part, by the Shota Rustaveli National Science Foundation under Grant N 17-96

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The presented talk is devoted to some almost invariant sets and their applications.

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In this talk we present preliminary work that is done under the joint project funded by Shota Rustaveli National Science Foundation of Georgia and TUBITAK. In the first part we discuss several quantum NTRU-based key exchange protocols, which are developed by our colleagues from Turkey. In the second part we speak about Maude-NPA, which is a tool for formal analysis of cryptographic protocols. We will discuss its possibilities to analyze quantum protocols.

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Application of CBM to the testing of the intersection of a sub-set of basic hypotheses against an alternative one is considered. Optimal decision rule allows us to restrict the Type-I and Type-II errors rates on the desired levels.

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In the presented talk we consider various families of the measures and their characterizations in the sense of the set theory. In particular, we investigate the cardinality of some classes of measures.

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Analytical and numerical simulations’ data show that the transport of energetic particles in the presence of magnetic turbulence can be superdiffusive. The so-called anomalous transport has gained growing attention during the last two decades in many fields including laboratory plasma physics, and recently in astrophysics and space physics. Here the examples, both from laboratory and from astrophysical plasmas are shown, where superdiffusive transport has been identified, with a focus on what could be the main influence of superdiffusion on fundamental processes like diffusive shock acceleration and heliospheric energetic particle propagation. The use of fractional derivatives in the diffusion equation is also discussed, and directions of future investigations are indicated.

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The dynamics problems for a porous nonelastic circle with double voids are considered. These problems with the help of Laplace transformation with respect to time are reduced to problems of so called “pseudooscillation” the solutions of which are obtained in the form of series.

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We investigate multi-field problems for complex elastic anisotropic structures when in different adjacent components of the composed body different refined models of elasticity theory are considered. In particular, we analyse the case when we have the generalized thermo-electro-magneto elasticity model (GTEME model) in one region of the composed body and the generalized thermo-elasticity model (GTE model) in another adjacent region. Both models are associated with Green-Lindsay's model.This type of mechanical problem is mathematically described by systems of partial differential equations with appropriate transmission and boundary conditions.In the GTEME model part we have six dimensional unknown physical fields (three components of the displacement vector, electric potential function, magnetic potential function, and temperature distribution function), while in the GTE model part we have four dimensional unknown physical fields (three components of the displacement vector and temperature distribution function). The diversity in dimensions of the interacting physical fields complicates mathematical formulation and analysis of the corresponding boundary-transmission problems. We apply the potential method and the theory of pseudodifferential equations and prove uniqueness and existence theorems of solutions to different type basic and mixed boundary-transmission problems in appropriate Sobolev spaces. We analyse the smoothness and singularity properties of solutions to mixed and interfacial crack type problems.

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Applying schemes for proofs for "symmetric" theorems about the closure and self-adjointness stability for linear operators, from the widely-known book "Perturbation Theory for Linear Operators " by T. Kato, we prove the following: Let A and B (linearity is not necessary) be densely defined operators in Banach space X. Let D(A) is a subset of D(B) and let the following inequalities hold for any vector u and v in D(A): ||Bu-Bv||≤c||Au-Av|| and ||Bu-Bv||≤q||Au-Av||+||(A+B)u-(A+B)v||, where c and q are positive constants and q<1. The operator A is bijection if and only if when the operator (A+B) is also a bijection. In this case, the operator A^(-1) is continuous (A is homeomorphism) if and only if when (A+B)^(-1) is continuous ((A+B) is homeomorphism).

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Investigating the linear mathematical model of equilibrium of the plane non-homogeneous elastic body by means of complex analysis methods, the immediate functional dependence between Poisson's ratio and Young's modulus was detected in one special case of non-homogeneity.

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The full cycle of fog- and cloud formation in the mesoscale boundary layer of the atmosphere (MBLA) is simulated by numerical methods. Aerosol diffusion in MBLA from a point source is also simulated. It is possible to simulate smog formation by ‘superposition’ models of humidity processes and aerosol diffusion in MBLA.

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We consider a problem of the estimation of density of a random value that is the an initial value of some dynamics. The dynamics is determined by differential equation whose solution is observable at the end of an interval. By using a method of transformation of a measure along an integral curve in combination with kernel estimates, we present a procedure of the estimation of density.

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The problem of determining the shape of a uniform-strength contour in the case of axsial stretching of a rectangular plate with partial unknown boundary is considered.The analysis of the results obtained for different values of external forces is given.

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The Riemann-Hilbert type boundary value problem for some classes of generalized analytic vectors is presented.

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In the present paper we consider the elastic infinite strip with voids. General representations of a regular solution of a system of equations for a homogeneous isotropic medium with voids are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. Using the Fourier method, the basic BVPs are solved effectively (in quadratures) for an infinite strip.

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The basic boundary value problems for the infinite plane with a circular hole with voids are solved.

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The special case of the linear conjugation nonhomogeneous boundary value problem for Carleman-Vekua irregular equation when the boundary function has zeros and poles at some points of the boundary curve is considered. The coefficients of this equation belong to sufficiently wide classes of functions which are the generalizations of Vekua classical space. The formula of the general solution and the necessary and sufficient solvability conditions of this problem are established.

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In this paper by means of I. Vekua’s method the system of differential equations for the nonlinear theory of non-shallow shells is obtained. Using the method of a small parameter and complex variable functions approximate solutions are constructed for any N approximation.

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Let X is any point-set in Euclidean space R_n . We say that a line segment l is admissible for X if its end-points are singletons in X and there exists an edge of X containing l. Theorem. Let X be a finite quasi-Diophantine subset in the R_n . Then the length of each admissible line segment for X is a rational number.

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The presented work is a direct continuation of the articles, which consider the construction of algorithms and the corresponding numerical computations for the approximate solution of nonlinear integro-differential equations of Kirchhoff and Timoshenko types. In particular consider an initial-boundary value problem for the J.Ball integro-differential equation, which describes the dynamic state of a beam. The solution is approximated by using the Galerkin method, symmetrical stable difference scheme and the Jacobi iteration method. The algorithm has been approved for tests. The results of computations are represented by tables and graphics.

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In basic approximation of hierarchical models of piezoelectric transversely isotropic cusped bars static and oscillation problems are studied by means of variation formulation.

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The algorithm to construct the eigenfunctions of the characteristic equation of the multi-velocity transport theory by the Legendre polynomials is presented

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Global atmospheric models based on numerical integration of the full system of hydrothermodynamics and describing the weather rocesses fair predicted the general character of the weather but can’t catch the smaller scale processes, especially for the territories with compound topography. Really, small-scale processes such as convection cannot be explicitly represented in models with grid size more than 10 km. A much finer grid is required to properly simulate frontal structures and represent cumulus convection. In this article two particulate cases of unexpected heavy showers were studied. Numerical simulations were performed by three sets of domains with horizontal grid-point resolutions of 19.8 km, 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 Caucasus territory was tested. Some results of the numerical calculations performed by WRFC model are presented.

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The strength of a sufficiently long thick-walled homogeneous isotropic circular tube (cylinder) under the action of external forces is studied using the problems of elasticity statics. In particular, there are established the minimum thickness of pipes with different materials and with different diameters, for which do not exceed the permissible stresses values. Cylinder is in state of plane deformation, therefore are considered a two-dimensional boundary value problems for circular ring. Represented tables and graphs of minimum thickness of a circular ring when a) the normal constant stresses act at internal border, while the outer boundary is free of stresses and b) the normal constant stresses act at external border, while the inner boundary is free of stresses. To the numerical realization above mentioned problems are used solutions obtained by two means: the analytical solution obtained by method of separation of variables, and Lame's solution

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The limit distribution of integral square deviation of Walverton-Vagner type estimate is found and its properties are studied.

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The report deals with the linear stability of the steady state solution and numerical approximation of one nonlinear partial differential system. The algorithm of the approximate solution and the results of numerical experiments are given.

The perturbation algorithm for the initial-boundary value problem for spatial one and two dimensional variable parabolic equation is considered. We have obtained the numerical realization of the algorithm, analysis of the numerical reasalts and grafics images.

In the present paper the mathematical model of quantum properties of different nanostructures is studied. Nanostructures properties are significantly different from their similar materials at macro scale level.The influence of the surface atoms becomes important, the thermal, optical and electrical properties change dramatically at the nanoscale dimension, the dimensions are comparable with the wavelength of electrons which causes quantum confinement of electrons and quantization of their energy. We have studied the energy levels of electrons in the cubic and prismatic (with the hexagonal cross-section) nanostructures from the non-relativistic viewpoint. We considered the stationary Schr¨odinger Equation for the wavefunction of the electron with the homogeneous boundary conditions. The eigenfunctions and the corresponding eigenvalues are obtained and consequently the possible numerical values of the energy levels of electrons are estimated.

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The report deals with the investigation and numerical solutionof the nonlinear integro-differential equation of parabolic type. Asymptotic behavior of solution of the initial-boundary value problem and convergence of the finite-difference scheme is studied.

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A.B. Kharazishvili proved the existence of $rt$-sets of cardinality $(d+1)$ in $R^d$ space, for every natural $d\geq 3$. Also, a characterization of all $rt-$sets in $R^d$ space is established in the same work. In our talk a new proof of the above-mentioned theorem is presented. In addition, several theorems related to the so-called $at-$, $rt-$, $ot-$ and $-at-$, $-rt-$, $-ot-$ sets are considered

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In (0,0) approximation of hierarchical models of piezoelectric transversely isotropic cusped bars we consider static and oscillation problems. We analyze peculiarities of nonclassical setting boundary conditions.

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For the delay controlled functional differential equation with the discontinuous initial condition the Necessary Conditions of Optimality is obtained.

Variation Formulas of Solutions for Controlled Delay Differential Equation with the Continuous Initial Condition is considered and their Application in the Optimization Problems

An applications of the mathematical theory of general systems can play a major role in the important areas, such that differential equations, dynamical systems, deference equation, numerical or algebraic algebras. etc. The starting point for the entire development is the concept of a system dened on the set-theoretic level. Among the standard axioms of set theory, the Axiom of Choice (AC) is a powerful set-theoretical assertion which implies many extraordinary and interesting consequences. Moreover, further additional set-theoretical axioms (e.g. the Continuum Hypothesis (CH), Martin’s axiom (MA), the existence of large cardinals, and others) allow to obtain new deep results in real analysis and measure theory. It's known that, the existence of the global response function is equivalent to the set-theoretical propositions, for example to the existence of the function of choice

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The problem of the ongoing climate change resulting from natural and growing anthropogenic factors acquires a particular importance for the territory of the Caucasus. Dust aerosol represent one of the main pollutants on the territory of Georgia and impact on regional climate. In this study, the WRF Chemistry model with dust module is used to study transportation of dust to the territory of the South Caucasus from the Sahara and Sahel in Africa, Arabian and ar-Rub’ al-Khali deserts located in the Middle East, Kyzylkum, Karakum in the Central Asia. The results of calculations have shown the WRF model was able to simulate dust aerosols transportation to the Caucasus reliably in conditions of the complex topography and that dust aerosol is an important factor in the climate system of the South Caucasus.

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Constrained Bayesian method (CBM) and the concept of false discovery rates (FDR) for testing directional hypotheses is considered in the paper. There is shown that the direct application of CBM allows us to control FDR on the desired level. Theoretically is proved that mixed directional false discovery rates (mdFDR) are restricted on the desired levels at the suitable choice of restriction levels at different statements of CBM. These facts are demonstrated by computation of concrete examples for different statements of CBM.

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For the nonlinear optimal problem with the discontinuous initial condition and several constant delays in the phase coordinates and controls, with the general boundary con¬diti-ons and functional, the necessary conditions of optimality are obtained: in the form of equa-lity and inequality for the initial and final moments, for delays containing in the phase coordinates and initial vector; in the form of the integral maximum principle for the initial function and control.

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The transversely isotropic elastic piezoelectric nonhomogeneous bodies in the case when the poling axis coincides with one of the material symmetry axises is considered, The present talk is devoted to the oscillation problem such materials when the constitutive coefficients depending on the body projection (i.e., on a domain lying in the plane of interest) variables may vanish either on a part or on the entire boundary of the projection.

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Mazurkiewicz set has a difficult and interesting descriptive structure and the study their measurability properties is very actual. In general, one can- not assert that a Mazurkiewicz set is necessarily nonmeasurable with respect to λ2-measure. Moreover, there exists Mazurkiewicz set X in the euclidian plane R2, which is measurable with respect λ2 and λ2(X) = 0. Slightly changing the argument of Mazurkiewicz, we can show that there exists a Mazurkiewicz set Y , which is λ2-thick. Here we can remark, that there exists a Mazurkiewicz set, which is relatively measurable with respect to the class M (R2). Finally we describe Mazurkiewicz sets in the context of negligible sets with respect M (R2): • all Mazurkiewicz sets are negligible with respect M (R2); • there exists Mazurkiewicz set which is absolutely negligible with respect M (R2); • there exists Mazurkiewicz set which is not-absolutely negligible with re- spect M (R2);

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Every holomorphic vector bundle    on Riemann sphere  splits into the direct sum of line bundles and the total Chern number of this vector bundle  is equal to sum of Chern numbers of line bundles. The integer-valued vector with components Chern number of line bundles is called splitting type of holomorphic vector bundle and is analytic invariant of complex vector bundles.  
There exists a one-to-one correspondence between the H\"older continues matrix function and the holomorphic vector bundles described above, wherein the splitting type of vector bundles coincides with partial indices of matrix functions. It is known that every holomorphic vector bundle equipped with meromorphic (in general) connection  with logarithmic singularities at finite set of marked points and corresponding meromorphic 1-from  have first order poles in marked points and removable singularity at infinity.  
The Fucshian system of equations induced from this 1-form gives the monodromy representation of the fundamental group of Riemann sphere without marked points. The monodromy representation induces trivial holomorphic vector bundles  with connection. The extension of the pair (\texttt{bundle, connection}) on the Riemann sphere is not unique and defines a family of holomorphically nontrivial vector bundles.    
In the talk we present about the following statements:     
1. From the solvability condition (in the sense Galois differential theory) of the Fuchsian    system  follows formula for computation of partial indices of piecewise constant matrix function.     
2. All extensions of  vector bundle on noncompact Riemann surface correspond to    rational matrix functions  algorithmically computable by monodromy matrices of Fucshian system.

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Attribute-Based Access Control (ABAC) has been proposed as a highly flexible method for providing access based on the evaluation of attributes (user attributes, resource attributes, environment attribute, etc.). Attributes characterize anything that may be defined and to which a value may be assigned. ABAC generalizes other access control models, and is considered to be more flexible, scalable, and secure in dynamic environments where the number of users is very high. The work about access control is growing into two paralell directions: developing access control models and developing policy languages to support the models. There was attempts to find common point of these two directions. Several integrations of other models into otology languages. On the other hand, there is a general ontology model proposed for access control management. In this talk we try to combine these two approaches to show how ABAC model can be integrated into ontology languages.

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The paper discusses the application of constrained Bayesian method (CBM) of testing the directional hypotheses. It is proved that all decision rules developed on the basis of CBM restricts the mixed directional false discovery rate (mdFDR) and total Type III error rate as well. When alternatives are skewed directional false discovery rate (DFDR) or mixed directional false discovery rate (mdFDR) are used. The optimal procedures controlling DFDR (or mdFDR) use two-tailed procedures assuming that directional alternatives are symmetrically distributed. Therefore, decision rule is symmetric in relation with the parameter’s value defined by basic hypothesis. For the experiments where the distribution of the alternative hypotheses is skewed, the asymmetric decision rule is preferable. There theoretically is proved, in the offered work that all possible statements of CBM guarantying restrictions of the abovementioned criteria on the desired level and therefore are optimal in this sense. The theoretical results are confirmed by simulation study.

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In the talk theorems related to AT-, RT- and OT- sets and to the existence of modified AT-, RT- and OT- sets are discussed.

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http://www.rmi.ge/eng/QUALITDE-2018/Shavadze_workshop_2018.pdf

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By the iteration method an Inverse Problem is solved for the Linear Controlled Neutral Differential Equation.

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The present talk is devoted to the nonlinear multi-dimensional integro-differential equation of parabolic type. The well-posedness of the initial-boundary value problem with first kind boundary condition and convergence of additive averaged semi-discrete scheme with respect to time variable are studied.

Vibration problem for porous elastic cusped prismatic shells is considered. The talk deals with the existence, uniqueness, and regularity properties with thickness of general form in case of N=0 of approximation.

In this work internal boundary value problem of elastic equilibrium of the homogeneous isotropic body bounded by coordinate lines of the parabolic coordinate system is considered, when on parabolic border normal stress is given. Analytical solution is obtained by the method of separation of variables. Using the MATLAB software numerical results and constructed graphs of the mentioned boundary value problem are obtained.

In the first part of the work the problems for both dynamic beam are studied. The complex nonlinear problem of Timoshenko type for dynamic beam, which vas solved using by us algorithm. The algorithm we constructed gives an approximation for both spatial and temporal variables. An algorithm for the resulting system of discrete equations is constructed, considering the nonlinear, namely, cubic structure of the model. In order to simplify the iterative process in this part of the algorithm, we used Cardano's formulas, which allowed us to optimize the algorithm in certain sense, and positively affected the number of iterations. In the second part of the thesis the problem of approximate solution of the nonlinear integro-differential equation for a static beam of Kirchhoff type is studied. We used an approach, which reduces the problem to a nonlinear integral equation, using Green's functions, and for its solution we use the Picard's iterative method. The condition of convergence of considered method is established and the accuracy is estimated. The theoretical results related to the convergence of approximate solutions are confirmed by the numerical experiments.

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The talk is devoted to two type piezoelectric prismatic shells consisting of cusped layers. The double-layer prismatic shells of one type are themselves cusped ones and the double-layer prismatic shells of another type are of a constant thickness but consist of cusped layers. Well-posedness of Dirichlet and Keldysh type problems (BVP) is studied in the zeroth order approximation of hierarchical models constructed by means of I.Vekua’s dimension reduction method. Some BVPs are solved in explicit forms in concrete cases. For I.Vekua’s dimension reduction method and cusped prismatic shells see

The boundary problems of thermoelastostatics for a porous circular ring with voids are considered. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of two analytic functions of a complex variable and one solution of Helmholtz equation. The first and second fundamental boundary value problems are solved explicitly for the concentric circular ring.

The problem of finding of the equally strong contour of the plane theory of elasticity for a rectangular plate weakened by a hole and notches at the vertices. The problem consist in finding analytical forms of boundaries of equally strong contour under the condition that the tangential normal stress takes on the contour value is a constant. Using complex analysis methods the complex potentials and the equation of the equally strong contour are constructed effectively (in analytical form). The case of cyclic symmetry (square) is studied and investigated in detail.

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In the present work we consider the elastic sphere with voids and microtemperatures. The general solution of the equations for a homogeneous isotropic thermoelastic medium with voids and microtemperatures is constructed. The Neumann type boundary value problem for the sphere is explicitly solved. The obtained solution is represented as absolutely and uniformly convergent series.

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This report represents the necessity part of the cycle of works dedicated to the problems connected with approximate solution of Cauchy problem for evolutionary equation by arbitrary (in the define sense) order of accuracy with respect of mesh width. We consider the cases when the objects are 2 dim strong elliptic systems of partial differential equations with classical boundary conditions, domains are square or circle. As examples there are considering Poisson, Helmholtz, the selfadjoint second order (with small parameter) differential equations, the problem of the definition of elastic generalized stress-strained (Filon) state and Mindlin-Reissner-Naghdi type refined theories to taking into account temperature field and porosity phenomenon with variable thickness, the stable hierachical models corresponding to thermodynamic elastic thin walled structures As mathematical apparatus will be used elaborated by us the continuous of Douglas-Rachford alternative direction method and multipoint finite-difference methodologies with effective (in sense of optimal order of necessity arithmetic operations) schemes.

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We investigate regularity properties of solutions to mixed boundary value problems for the system of partial differential equations of dynamics associated with the thermo-electro-magneto elasticity theory. We consider piecewise homogeneous anisotropic elastic solid structures with interior and interfacial cracks, in particular, smart materials and structures. Using the potential method and theory of pseudodifferential equations we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the thermo-mechanical and electro-magnetic fields are analyzed near the crack edges and near the curves where different types of boundary conditions collide. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are analyzed in detail as well.

The talk is dedicated to the theoretical investigation of basic, mixed and crack type three-dimensional initial-boundary value problems of the generalized thermo-electro-magneto-elasticity 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 analyze dynamical initial-boundary value problems and the corresponding boundary value problems of pseudo-oscillations which are obtained from the dynamical problems by the Laplace transform. The dynamical system of partial differential equations generate a nonstandard \linebreak$6\times 6$ matrix differential operator of second order, while the system of partial differential equations of pseudo-oscillations generates a second order strongly elliptic formally non-selfadjoint $6\times 6$ matrix differential operator depending on a complex parameter. First we prove uniqueness theorems of dynamical initial-boundary value problems under reasonable restrictions on material parameters and afterwards we apply the Laplace transform technique to investigate the existence of solutions. This approach reduces the dynamical problems to the corresponding elliptic problems for pseudo-oscillation equations. The fundamental matrix of the differential operator of pseudo-oscillations is constructed explicitly by the Fourier transform technique, and its properties near the origin and at infinity are established. By the potential method the corresponding three-dimensional basic, mixed and crack type boundary value problems, and the transmission problems for composite elastic structures are reduced to the equivalent systems of boundary pseudodifferential equations. The solvability of the resulting boundary pseudodifferential equations are analyzed in appropriate Sobolev-Slobodetskii ($W^{s}_p$), Bessel potential ($H^{s}_p$), and Besov ($B^{s}_{p,q}$) spaces and the corresponding uniqueness and existence theorems of solutions to the boundary value problems under consideration are proved. 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. It is shown that the smoothness and stress singularity exponents essentially depend on the material parameters and an efficient method for their computation is described. By the inverse Laplace transform the solutions of the original dynamical initial-boundary value problems are constructed and their smoothness and asymptotic properties are analyzed in detail.

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The paper presents new decomposition formulas for cosine operator function based on known trigonometric formulas. The validity of the constructed formula is proved when argument of cosine operator function is a sum of two bounded operators. High order decomposition formula is constructed, in case when there is a square root of the main operator in the argument of cosine operator function and the number of addends equals two. The decomposition formula is constructed using resolvents of the summand operators. There is also proposed an algorithm that allows to construct any order accuracy decomposition formula for cosine operator function. More precisely, the algorithm allows to construct 2p+2 accuracy order decomposition formula based on 2p order one.

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In the first part of the work the problems for both dynamic beam are studied. The complex nonlinear problem of Timoshenko type for dynamic beam, which vas solved using by us algorithm. The algorithm we constructed gives an approximation for both spatial and temporal variables. An algorithm for the resulting system of discrete equations is constructed, considering the nonlinear, namely, cubic structure of the model. In order to simplify the iterative process in this part of the algorithm, we used Cardano's formulas, which allowed us to optimize the algorithm in certain sense, and positively affected the number of iterations. In the second part of the thesis the problem of approximate solution of the nonlinear integro-differential equation for a static beam of Kirchhoff type is studied. We used an approach, which reduces the problem to a nonlinear integral equation, using Green's functions, and for its solution we use the Picard's iterative method. The condition of convergence of considered method is established and the accuracy is estimated. The theoretical results related to the convergence of approximate solutions are confirmed by the numerical experiments. The author express hearing thanks to Prof. J.Peradze for his active help in problem statement and solved.

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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 topographic structure of the terrain. The territory of Georgia is a good example for that. Indeed, about 85% of the total land area of Georgia is mountain ranges with compound topographic sections which play an important role for spatial-temporal distribution of meteorological fields. 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 is a mesoscale numerical weather prediction system, designed for forecasting needs. WRF consists of several solvers and it is quite flexible to be extended for different needs. One of such example is the Polar WRF. One of our goals is to combine WRF model with the solver developed by us, to get better prediction of temperature, wind velocity, showers and hails for different set of physical options in the regions characterized with the complex topography.

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In the present paper determine the location and amount of accidental gas escape from the main gas (oil) pipe-line has been studied. For solving the problem it has been discussed early-made method, reason is that the exact analytical method has not been existed. We have created quite general test, the manner of the solution has been known in advance. Comparison has shown us the affectivity of the suggested method

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On the basis of the numerical model developed by us, a number of the following processes are simulated: the simultaneous existence of a stratus cloud and radiation fog; cluster of clouds and fog. Along with wet processes, aerosol propagation from an instantaneous point source is simulated

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The talk concerns 50 years long history of Ilia Vekua Institute of Applied Mathematics of Ivane Javaxishvili Tbilisi State University. The Institute was founded by a Georgian mathematician and mechanist Ilia Vekua on October 29, 1968. The aim of the Institute was to carry out research on important problems of applied mathematics, to involve University professors, teachers and students in research activities on topical problems of applied mathematics in order to integrate mathematics into the educational processes and research, and to implement mathematical methodologies and computing technology in the non-mathematical fields of the University. In 1978, the Institute was named after its founder and first director Ilia Vekua. In December, 2006 - May, 2009 the Institute was functioning at the Faculty of the Exact and Natural Sciences. In June, 2009 - September, 2016 the Institute was directly subordinated to the University Administration. Since the end of September, 2016 the Institute has a status of the Independent Scientific-Research Institute. At present, the Institute successfully continues and develops activities launched by its founder in the following four main scientific directions: • Mathematical problems of mechanics of continua and related problems of analysis; • Mathematical modelling and numerical mathematics; • Discrete mathematics and theory of algorithms; • Probability Theory and mathematical Statistics. The institute sees its mission as threefold: • Carrying out fundamental and practical scientific research in applied mathematics, mathematical and technical mechanics, industrial mathematics and informatics, undertaking state and private sector contracts to provide expert services; • Offering the university a high-level computer technology base for University professors and teachers, research employees and students undertaking their scientific research activities; • Supporting PhD and post-graduate students to attain scientific grants, as well as through employment within the Institute and participation in scientific conferences.

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The paper discusses the application of constrained Bayesian method from the method of solving a practical problem to a new philosophy of statistical hypotheses testing.

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In this talk the 2D fully coupled theory of steady vibrations of poroelasticity for materials with double porosity is considered. The fundamental and singular matrices of solutions are obtained in terms of elementary functions. The single and double layer potentials are constructed and the basic properties of these potentials are established.

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Based on mathematical models of describing the multidimensional soliton-type structures in complex media (ionospherean atmosphere, hydrosphere, ionospheric and magnetospheric plasma) the nonlinear dynamics of electromagnetic solitary vortices and the wave structures have been studied. Nonlinear wave structures can be a purely monopoly vortex, a transverse vortex chain, and/or a longitudinal vortex path against the background of an inhomogeneous zonal wind, depending on the shear flow velocity profile. The accumulation of such vortices in the ionospheric medium can generate a strongly turbulent state. The interaction of soliton type multidimensional structures in the complex media, described by DNSL class of equations taking into account of dispersive and dissipative effects are studied numerically and interesting results are obtained.

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A goodness-of-fit test is constructed by using a Wolverton–Wagner distribution density estimate. The question as to its consistency is studied. The power asymptotics of the constructed goodness-of-fit test is also studied for certain types of close alternatives.

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We discuss some convergence and divergence properties of subsequences of logarithmic means of Walsh-Fourier series. We give necessary and sufficient conditions for the convergence regarding logarithmic variation of numbers.

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The linear mechanism of generation, intensification and further nonlinear dynamics of internal gravity waves (IGW) in stably stratified dissipative ionosphere with non-uniform zonal wind (shear flow) is studied. In the ionosphere with the shear flow, a wide range of wave disturbances are produced by the linear effects, when the nonlinear and turbulent ones are absent.

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The nonlinear Scr¨odinger equation describes wide range of physical phenomena. In the Euclidian space R3 we consider the cubic nonlinear Scr¨odinger equation (cNLS) By using previous results of the author (N. Khatiashvili et al, On effective solutions of the nonlinear Schrodinger equation, J.Phys; (2014)) we have obtained the approximate novel nonsmooth solitonic solutions of NLS.

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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 topographic structure of the terrain. The territory of Georgia is a good example for that. Indeed, about 85% of the total land area of Georgia is mountain ranges with compound topographic sections which play an important role for spatial-temporal distribution of meteorological fields. 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 is a mesoscale numerical weather prediction system, designed for forecasting needs. WRF consists of several solvers and it is quite flexible to be extended for different needs. One of such example is the Polar WRF. One of our goals is to combine WRF model with the solver developed by us, to get better prediction of temperature, wind velocity, showers and hails for different set of physical options in the regions characterized with the complex topography.

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Georgia Natural gas distribution networks are complex systems with hundreds or thousands of kilometers of pipes, compression stations and many other devices for the natural gas transportation and distribution service. In the gas transmission pipelines to achieve the power consumption points with the required conditions is the main and the most difficult issue. For solving this problem properly determination of the gas pressure and flow rate distribution along the pipeline is necessary step. Searching of the gas flow pressure and flow rate distribution along the inclined and branched pipeline network is the more difficult issue. For this reason development of the mathematical models describing the non-stationary processes in the branched, inclined pipeline systems are actual. The purpose of this study is determination of gas pressure and flow rate special and temporally distribution along the inclined and branched pipeline. A simplified mathematical model (based on the hypothesis that the boundary conditions do not change quickly and the capacity of gas duct is relatively large) derived from the nonlinear system of one-dimensional partial differential equations governing the dynamics of gas non-stationary flow in the inclined, branched pipeline is obtained. In this case gas pressure special and temporally distribution along the branched pipeline is presented. Some results of numerical calculations of gas flow in the inclined branched pipelines are presented

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In the first part of the work the problems for both dynamic beam are studied. The complex nonlinear problem of Timoshenko type for dynamic beam, which vas solved using by us algorithm. The algorithm we constructed gives an approximation for both spatial and temporal variables. An algorithm for the resulting system of discrete equations is constructed, considering the nonlinear, namely, cubic structure of the model. In order to simplify the iterative process in this part of the algorithm, we used Cardano's formulas, which allowed us to optimize the algorithm in certain sense, and positively affected the number of iterations. In the second part of the thesis the problem of approximate solution of the nonlinear integro-differential equation for a static beam of Kirchhoff type is studied. We used an approach, which reduces the problem to a nonlinear integral equation, using Green's functions, and for its solution we use the Picard's iterative method. The condition of convergence of considered method is established and the accuracy is estimated. The theoretical results related to the convergence of approximate solutions are confirmed by the numerical experiments. The author express hearing thanks to Prof. J.Peradze for his active help in problem statement and solved.

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In the present study with the view of finding out the details of the dust aerosols influence on the Georgian climate change some numerical experiments were performed by WRF and RegCM models. Toward this purpose we have executed as short term 104 Abstracts of Participants’ Talks Batumi–Tbilisi, September 3–8, 2018 (WRF/Chem/dust) as well long term (RegCMv.4.7) calculations. Namely sets of 30 years simulations (1985-–2014) with and without dust effects has been executed by RegCM 4.7 model with 16.7 km resolution (over the Caucasus domain) and with 50 km resolution (encompassing most of the Sahara, the Middle East, the Great Caucasus with adjacent regions). Results of calculations have shown that dust aerosol is an inter-active player in the climate system of Georgia. Numerical calculations have shown that mineral dust aerosol influenced on temperature and precipitations (magnitudes) spatial and temporally inhomogeneous distribution on the territory of Georgia and obtained results generally agreed with MODIS satellite data.

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A two-dimensional boundary value problem of elastic equilibrium of a plane-deformed infinite body with a circular opening is studied. A part of the opening is fixed and from some points of the unfixed part of the cylindrical boundary there come radial finite cracks. The problem is to find conditions for the fixed parts of the opening so that the damage caused by the crack, i.e. stresses on its surface, should be minimal. We should note that the crack ends inside the body are curved. The curve radii vary similar to boundary conditions. The problem is solved by the boundary element method

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The new test for homogeneity for p ≥ 2 independent samples based on Parzen’s type estimators of distribution density is constructed. The limiting power of the constructed tests is found for Pitman’s type “close” alternatives. Also is considered the comparison of constructed tests with Pearson’s chi-square test for two samples. For this is found the limiting power of chi-square homogeneity test for above-mentioned alternatives. It is established the limiting power of constructed test is grater then the limiting power of chi-square homogeneity test.

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In recent years, regular hedge languages [1] became very popular in programming languages, due to their expressive power. Languages supporting programming with regular hedge expressions are useful for Web-related applications. As examples, CDuce, PρLog, XDuce and XHaskell can be mentioned. Regular hedge expressions have been extensively used in search engines, rewriting, program verification, software engineering, lexical analysis, etc. Because of space limitation, we can not give an exhaustive overview of regular hedge expressions applications. The theory of hedge languages generalizes the theory of word languages. Therefore, it is not surprising that regular hedge languages provide more expressive and powerful platform for semistructured data manipulation than regular word languages. Solving regular word language equations with various restrictions have been intensively studied in the last decade. Solving regular word language equation systems without restrictions is hard and the class of smallest solutions of such systems corresponds to recursively-enumerable sets [2]. It should be noted that much less attention has been devoted to solving regular hedge language equations. In this talk we propose a solving algorithm for one side ground regular hedge language equations. The solving algorithm is based on factorization of regular hedge languages, which generalizes factorization of regular word languages given in [3]. We show that, the algorithm computes maximal solutions and is sound and complete.

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Access control is a security technique that specifies which users can access particular resources in a computing environment. Formal description of access control is extremely important, since it should be defined, unambiguously, how rules regulate what action can be performed by an entity on the resource, how to guarantee that each request gets an authorization decision, how to ensure consistency, etc. It is also important that such a formal description is at the same time declaratively clear and executable, to avoid an additional layer between specification and implementation. Over the years, numerous access control models have been developed to address various aspects of computer security. In this talk, we describe traditional models: discretionary access control (DAC), mandatory access control (MAC) and role-based access control (RBAC). Despite successful practical applications of these traditional models, they have certain disadvantages, which was the reason why new approaches emerged. We will focus on one modern approach, attribute-based access control, which has been proposed in order to overcome limitations of traditional models.

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The problem of investigating the measurability of sets and functions with respect to a concrete measure m on a base (ground) set E, to turn attention to the more general question of investigating the measurability of sets and functions with respect to a given class M of measures on E. We study the measurability properties of sets and real-valued functions with respect to various classes M of measures on the base set E. We say that a function f is relatively measurable with respect to the class M if there exists at least one measure μ ∈ M such that f is measurable with respect to μ. Let (E1, S1, μ1) and (E2, S2, μ2) be measurable spaces equipped with sigma-finite mea- sures. We Recall that a graph Γ ⊂ E1 × E2 is (μ1 × μ2)-thick in E1 × E2 if for each (μ1 × μ2)-measurable set Z ⊂ E1 × E2 with (μ1 × μ2)(Z) > 0, we have Γ ∩ Z ̸ = 0. Notice that, the thickness of graphs is pathological phenomenon for subsets of basic set. However, this feature plays an essential role in the problem of extensions of measures. Theorem 1. Let E1 be a set equipped with a sigma-finite measure μ and let f : E1 → E2 be a function satisfying the following condition: there exists a probability measure μ2 on ran(f ) such that the graph of f is a (μ1 × μ2)-thick of the product set E1 × ran(f ). Then there exists the measure μ′ such that: 1) μ′ is measure extending μ1; 2) f is relatively measurable with respect to μ

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Intensification and further dynamics of internal gravity waves (IGW) in the ionosphere with non-uniform zonal wind (shear flow) is studied. It is revealed that the transient amplification of IGW disturbances due time does not flow exponentially, but in algebraic - power law manner. The frequency and wave-number of the generated IGW modes are functions of time. Thus in the ionosphere with the shear flow, a wide range of wave disturbances are produced by the linear effects, when the nonlinear and turbulent ones are absent. The effectiveness of the linear amplification mechanism of IGW at interaction with non-uniform zonal wind is analyzed. It is shown that at initial linear stage of evolution IGW effectively temporarily draws energy from the shear flow significantly increasing (by order of magnitude) own amplitude and energy.

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The governing system of porous elastic prismatic shells in case of zero approximation is rewritten in case of harmonic vibration. The talk is devoted to the homogeneous Dirichlet and Keldysh problem for the obtained general system. The classical and weak setting of the problem are formulated. The special functional spaces is introduced. We show coerciveness of the corresponding bilinear form and prove uniqueness and existence results for the variational problem. We describe in detail the structure of the introduced spaces. Moreover, we give some sufficient conditions for a linear functional arising in the right-hand side of the variational equation to be bounded.

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Cusped prismatic shells considered as 3D bodies may have non-Lipschitz surfaces as the boundaries and their thicknesses may vanish at the edge. Using I. Vekua’s dimension reduction method, complexity of the 3D domain occupied by the body will be transformed in the degeneracy of the order of the 2D governing equations of the constructed hierarchy of 2D models on the boundary of the 2D projection of the 3D bodies under consideration. Consideration of boundary value problems (BVP) and initial boundary value problems (IBVP) for elastic cusped prismatic shells leads to investigation of nonclassical BVPs and IBVPs for the governing elliptic and hyperbolic systems of equations of the second order with order degeneracy on the boundary of the domain under consideration in the case of the two spatial variables. Initial conditions for the so called weighted mathematical moments of displacements remain classical, while the boundary conditions (BC) for them are nonclassical, in general. It means that in certain cases the Dirichlet BCs should be replaced by the Keldysh BCs (i.e. some parts of the boundary, where the order of the equations degenerate, should be freed from the BCs) and in certain cases weighted BCs should be set. The present talk deals with hierarchical models of cusped piezo-electric prismatic shells. It is proved that like the weighted mathematical moments of the displacements the BCs for the mathematical moments of the electric potential are nonclassical. Modifications of methods developed for the degenerate equations and systems of equations are used.

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A goodness-of-fit test is constructed by using a Wolverton–Wagner distribution density estimate. The question as to its consistency is studied. The power asymptotics of the constructed goodness-of-fit test is also studied for certain types of close alternatives.

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We discuss equilibrium configurations of Coulomb potential of point charges in convex domains of the plane and three-dimensional Euclidean space. For a triple of points, we give an analytic criterion of the existence of point charges for which the given triple is an equilibrium configuration. Using this criterion, rather comprehensive results are obtained for three charges in the circle and ellipse. Several related problems and possible generalizations are also indicated

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In the present paper some hydrological specifications of Georgian water resources on the background of regional climate change are presented. Some results of extreme precipitation numerical calculations and Georgian’s glaciers melting are given. The specific properties of regional climate warming process in the eastern Georgia is studied by statistical methods. Water resources alteration on the background of climate change is presented. The effect of the eastern Georgian climate change upon water resources is investigated

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At present climate change process is in progress in the South Caucasus region and this process has not avoided the territory of Georgia too. Namely glaciers melting, increasing the frequency of droughts and heavy showers are good indicators of ongoing climate change process on the territory of Georgia. The climate on the territory of Georgia alters through the parallels, from subtropical at the Black Sea coast to the arid continental in Eastern Georgia. Georgia's arid and semiarid regions(south-eastern part of Georgia) are especially sensitive to desertification process where during the last eight years four times (2010, 2012, 2014,2017) were reiterated catastrophic droughts which has reduced volume of water resources not only for hydro powers as well drinking water supplies in the dry regions of Georgia. Simultaneously for the last four decades the number of the extreme weather events and natural hazards (floods, landslides, storms, heavy showers, hails) has significantly increased on the territory of Georgia. Indeed for that period the number of the natural hazards has increased about three times in comparison with 60 years period of the last century. So the problem of the forthcoming climate change resulting from natural and growing anthropogenic factors acquires a particular importance for Georgia. In the present study some experiments correspond to RegCM4.7 model physics options that have been used to study both regional climate and dust effect over the territory of Georgia are presented. The regional climatic impact of dust was evaluated by means of two numerical experiments in which the first was executed without dust and second one was simulated through interactive dust inclusion. For each experiment, the dust climatic impact has been defined as the difference between the dust and without dust simulations for the variables of interest (average monthly, seasonal, annual temperatures and precipitations). Our experiment's target is finding out a possible dust impact on climate within 30-year simulation with activating dust coupler in RegCM model, with ERA-interim boundaries for the mentioned region. The three decadal (1985-2014) simulation was executed by the nested version of RegCM4.7 with the horizontal resolutions of 50 and 16.7 km for the coarse and nested meshes respectively. Some results of numerical calculations are presented.

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http://www.coia-conf.org/upload/editor/files/COIA_2018_V1.pdf

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For the delay differential equation with the mixed initial condition an analytic relation between solutions of equations with the original and perturbed initial data is established. Continuous of the minimum of integral functional is proved with respect to perturbation of the initial date.

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For the nonlinear controlled functional differential equations with several constant delays in the phase coordinates the local variation formulas of solutions are proved, in which the effects of the discontinuous initial condition and perturbations of delays and the initial moment are detected.

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Planetary EM ULFW appears as a result of interaction of the ionospheric medium with the spatially inhomogeneous geomagnetic field. An effective linear mechanism responsible for the generation and transient intensification of large scale EM ULF waves in the shear flow is found. It has been shown that the shear flow driven wave perturbations effectively extract energy of the shear flow and temporally algebraic increasing own amplitude and energy (by several orders). With amplitude growth the nonlinear mechanism of self-localization is turned on and these perturbations undergo self organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation’s front. Depending on the features of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices, or dipole vortex, or vortex streets and vortex chains.

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We discuss equilibrium configurations of point charges with Coulomb interaction on the circle, line segment and system of three concentric circles, and obtain characterization of stable electrostatic configurations with a few points (see [1]). In the case of the circle, we show that any configuration consisting of odd number of points on the circle can be realized as an equilibrium configuration of certain non-zero point charges and give a simple criterion for existence of positive charges with this property [2]. Several related problems to Fuchsian differential equations on complex plane and possible generalizations are also discussed.

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The effect of the dust forcing on the Caucasus (Georgia) regional climate change with a regional climate model interactively coupled to a dust model is studied. Toward this purpose the sets of 30 years simulations (1985–2014) with and without dust effects executed by RegCM 4.7 model with 16.7 km resolution over the Caucasus domain and with 50 km resolution encompassing most of the Sahara, the Middle East, the Great Caucasus with adjacent regions was investigated. Results of calculations have shown that dust aerosol is an inter-active player in the climate system of Georgia. Mineral dust aerosol influences on temperature and precipitations (magnitudes) spatial and temporally inhomogeneous distribution on the territory of Georgia and generally has been agreed with MODIS satellite data. According to results of comparisons of the simulated dust aerosol optical depth seasonal distributions against to the observed ones gave a good agreement. Also dust radiative forcing inclusion has improved simulated summer time temperature, as it was tend to warm bias mostly on continental territories of Georgia. Also there was observed summer time precipitation increment on the territory of Caucasus (Georgia) which improves seasonal distribution of simulated precipitation, but gives over estimation in annual total precipitation

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In this work local and global formulas of variation are proved. There is considered three cases when the variation occurs from left, or from right, or from both side.

In this article the problem of hydrates’ possible origin area prediction in the main pipelines taking into consideration gas non-stationary flow and heat exchange with medium is studied. For solving the problem the system of partial differential equations governing gas non-stationary flow in main gas pipe-line is investigated. Solution of the problem for gas adiabatic flow is presented. The results of some numerical calculations are presented

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An initial-boundary value problem is posed for the J. Ball integro-differential equation, which describes the dynamic state of a beam. The solution is approximated with respect to a spatial and a time variables by the Galerkin method and stabile symmetrical difference scheme, which requires carrying out of iteration process. The algorithm has been approved on tests and the results of recounts are represented in tables.

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A vibration problem is consider for hierarchical models of nonhomogeneous thermoelastic prismatic shells with porous.

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Exact (analytical) solution of 2D problems of elasticity are constructed in the elliptical coordinates in domain bounded by the hyperbola. Here we represent internal boundary value problems of elastic equilibrium of the homogeneous isotropic body bounded by coordinate lines of the elliptic coordinate system, when on hyperbolic border non homogeneous (nonzero) boundary conditions are given. Exact solutions are obtained using the method of separation of variables.

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In the talk questions related to geometric realizations of abstractly given families of sets by families of point sets of different types are discussed.

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The talk is devoted to construction of hierarchical models for piezoelastic nonhomogeneous viscoelastic Kelvin-Voigt prismatic shells with voids. Using I. Vekuas dimension reduction method, the governing systems of partial differential equations are derived. In the case of cusped prismatic shells the peculiarities of nonclasical setting of boundary conditions is discussed.

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We study TSR-logic based methods, that can be used in automated theorem proving and develop TSR solver. The main targe to solver is to be used in real life applications. One of such applications is enviromental contamination and weather forecast. The expected results will have as practical as well theoretical character.

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The new nonparametric kernel type estimate of the Bernoulli regression function is created. Its asymptotic properties are studied.

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Weisz proved-among others – that for f ∈ LlogL the Fejér means σ~n(t,m,u) of conjugate transform of two-parameter Walsh-Fourier series a. e. converges to f(t,u) . The main aim of this paper is to prove that for any Orlicz space, which is not a subspace of LlogL, the set of functions for which Walsh-Fejér Means of two parameter Conjugate Transforms converge in measure is of first Baire category.

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The non-linear elliptic equation is considered in the infinite area. By the trigonometric representations the approximate solutions of the high accuracy are obtained. Those solutions are bounded at infinity and represent peaked solitary waves.

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The stochastic integral for random measures with smoosh distributions has been constructed. Some properties of generalized stochastic integral are given.

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The aim of this paper is to construct the continuous solution of the nonhomogeneous linear equation corresponding to the characteristic equation of the multivelocity transport theory in the isotropic case.

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In this talk we consider problem of cryptographic protocol verification and analysis. Cryptographic protocol is used for secure communication over the network by two or more agents. Cryptographic protocol verification is a task, that determines whether the protocol is secure and can be broken by different kind of attacks, like men in the middle, etc. We try to model a cryptographic protocol in the Pρlog system and show whether it is vulnerable for attacks. We would like to mention, that it is easier task (decidable) to find out whether a protocol is vulnerable for attacks, than to find an attack that breaks the protocol (not decidable in general).

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The linear mechanism of generation, intensification and further nonlinear dynamics of internal gravity waves (IGW) in stably stratified dissipative ionosphere with non-uniform zonal wind (shear flow) is studied. In the ionosphere with the shear flow, a wide range of wave disturbances are produced by the linear effects, when the nonlinear and turbulent ones are absent. Observation data of gravity waves are analysis is carried out. Spectral features, recurrent quantitative and qualitative characters of obtained signals are studied. Special properties of their dynamics are revealed.

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The boundary problems of elastostatics for a porous circular ring with voids are considered.. Explicit solutions of problems are obtained in the form of series. Conditions are established that ensure absolute and uniform convergence of these series.

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A special case of a heterogeneous linear heterogeneity problem for the regular Carlmann-Vekua equation is investigated when the function G (t) participating in the boundary condition has zeros and poles at some points in the boundary circle.

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For solving the problems of study, analysis and quality management of the environment there is ne­ce­ssary operatively to treat great amount of measuring information on physical, chemical and biological parameters characteristic for them. To do it in a proper way, in conformity to the modern requirements, is possible only by wide use of modern mathematical methods and computers. For this purpose it is necessary to develop automated systems and universal program packages with developed mathematical methods consisting of self-learning algorithms requiring whenever it is possible minimum a prior information and having capability of adaptation to the most unexpected changes of the character of the investigated objects.

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In the work is proved a theorem about of the necessary condition of criticality of the continuous mapping defined on the quasiconvex filter. This theorem is a central result of R. Gamkrelidze and G. Kharatishvili extremal problems theory. By this theory, as rule, investigation of the optimal problems are reduced on the finding necessary condition of criticality. From the necessary condition of criticality follows necessary optimality conditions for the optimal problem.

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The tension-compression static problem is investigated in the zero approximation of governing system for Kelvin-Voigt plates with variable thickness using I. Vekua’s dimension reduction method.

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The estimate of an odds-ratio based on the kernel estimate of the regression function is constructed. The consistency, asymptotic normality and uniform convergence of the constructed estimate are proved.

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The historical development of Georgian nation was complicated and hard. During the long existence there were frequent internal wars and invasions from outside, which caused permanent struggle for survival and defending nationality. Despite such conditions Georgians were trying to develop national economy and culture, arts, literature and sciences. But very often the only way to perform this activity lay through cooperation with foreign countries more developed technically and culturally. There are no written direct sources about creative work in and/or teaching of mathematics and mechanics in Georgia in old times. But high level of architecture of a lot of old monuments (churches, castles, etc.) could not be reached without certain mathematical and mechanical knowledge. Ancient Georgian manuscripts tell us that Georgians used their alphabet to elaborate an original system of numeration, they have had the own chronology. In early centuries there have been translations of books devoted to astronomy. Beginning from medieval times Georgians have translated or compiled mathematical textbooks, composed encyclopedic dictionaries. Mathematical knowledge is contained also in manuscripts involving a heritage of some Georgian philosophers. The second part of the talk is devoted to state of mechanics and mechanical education in Georgia in XX and XXI centuries.

There is discussed Steinhaus property and their applications in the study of the invariant extensions of the Lebesgue measure.

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Fuchsian systems on a complex manifold with nontrivial topology are investigated and Hamiltonians, whose dynamic equations reduce to a Fuchs type differential equation, are given. These Hamiltonians and equations correspond to realistic physical models encountered in the literature.

The talk is devoted to the correctness of the initial-boundary value problems for two nonlinear multi-dimensional integro-differential equations of parabolic type. Construction and study of the additive averaged semi-discrete schemes with respect to time variable are also given.

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http://www.rmi.ge/eng/QUALITDE-2017/Shavadze_workshop_2017.pdf

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The paper discusses the generalization of constrained Bayesian method (CBM) for arbitrary loss functions and its application for testing the directional hypotheses. The problem is stated in terms of false and true discovery rates. One more criterion of estimation of directional hypotheses tests quality, the Type III errors rate is considered. The ratio among discovery rates and the Type III errors rate in CBM is considered. The advantage of CBM in comparison with Bayes and frequentist methods is theoretically proved and clearly demonstrated by a concrete computed example. It is shown that CBM surpasses the Bayes and frequentist methods with guaranteed reliability of decisions made. .

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In this article, we have discussed the logical method of program analysis. We formulated theorems whose analogues we proved in the TSR theory

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In the present article an anti-plane problem of the elasticity theory for the composite (piece-wise homogeneous) orthotropic body weakened by cracks, intersecting the interface (problem 1) or reaching it (problem 2) at the right angle is studied. The studied problem is reduced to the singular integral equation (when the crack reaches the interface) and system (pair) of singular integral equation (when the crack intersects the interface) containing an immovable singularity with respect to the unknown characteristic functions of the cracks disclosure. Behavior of solutions in the neighborhood of the crack endpoints is studied by the method of discrete singularity with uniform division of an interval by knots. In both cases (crack intersects and crack reaches to the interface) the question of behavior of approximate solutions are investigated. The corresponding algorithms are composed and realized. The results of numerical investigations are presented.

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The talk is devoted to a survey of scientific, pedagogical, and educational activities of the Georgian mathematician David Gordeziani.

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In this talk we consider problem of cryptographic protocol verification and analysis. Cryptographic protocol is used for secure communication over the network by two or more agents. Cryptographic protocol verification is a task, that determines whether the protocol is secure and can be broken by different kind of attacks, like men in the middle, etc. We try to model a cryptographic protocol in the Pρlog system and show that it is not vulnerable for attacks. We would like to mention, that it is easier task (decidable) to find out whether a protocol is vulnerable for attacks, than to find an attack that breaks the protocol (not decidable in general).

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We consider the time-harmonic acoustic wave scattering by a bounded layered anisotropic inhomogeneity embedded in an unbounded anisotropic homogeneous medium. The material parameters and the refractive index are assumed to be discontinuous across the interfaces between inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problem is formulated as a boundary-transmission problem for a second order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. We show that with the help of localized potentials constructed by a harmonic parametrix the boundary-transmission problem can be reformulated as a system of localized boundary-domain pseudodifferential equations (LBDIE) and prove that the corresponding localized boundary-domain pseudodifferential operator is invertible in appropriate function spaces. This leads to the unique solvability result for the original acoustic wave scattering problem with arbitrary frequency parameter.

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We consider an abstract hyperbolic equation with a Lipschitz continuous operator, where the main operator is a sum of finite number self-adjoint and positive definite operators. Semi-discrete implicit difference schemes corresponding to the summand operators are solved independently (parallelly) on each local interval. It is proved that weighted sum of the solutions of the semi-discrete implicit difference schemes converges to the exact solution of the given abstract hyperbolic equation.

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The nonlinear integro-differential equations and their systems describe various processes in physics, economics, chemistry, technology and so on. It is doubtless that the study of qualitative and structural properties of the solutions of initial-boundary value problems for those equations and systems, construction and investigation of discrete analogues and the study of numerical algorithms are very important. One type of integro-differential systems arise, for instance, in mathematical modeling of the process of penetrating of magnetic field into a substance. It is known that the mentioned process is described by the system of Maxwell equations. One nonlinear partial integro-differential model is considered in the pre sent study. The model is obtained by reducing of the above-mentioned Maxwell equations to the integro-differential form. Initial-boundary value problem with Dirichlet boundary conditions is considered. Asymptotic behavior as $t\to\infty$ of solutions is studied. Rates of stabilization are given. Stabilization and convergence of discrete analogs are proven. Wider classes of nonlinearity are investigated than the ones studied earlier. Various numerical experiments are carried out. Results of numerical experiments with the corresponding graphical illustrations are given and compared to the theoretical ones.

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Several combinatorial questions and facts connected with certain types of mutual positions of finitely many hyperplanes in a finite-dimensional affine space are considered. An application of one of such facts to a multi-dimensional version of the well-known Sylvester theorem is presented.

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In the talk theorems related to finite independent families of convex bodies of various types are considered. Among mentioned theorems are statements reveal a certain interrelation between the cardinality and the structure of a constituent of a finite independent family of convex bodies.

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ULF electromagnetic planetary waves can self-organize into vortex structures (monopole, dipole or into vortex chains). They are often detected in the plasma media, for instance in the magneto sheath, in the magnetotail and in the ionosphere. Large scale vortices may correspond to the injection scale of turbulence, so that understanding their origin is important for understanding the energy transfer processes in the geospace environment. In a recent work, the THEMIS mission has detected vortices in the magnetotail in association with the strong velocity shear of a substorm plasma flow which has conjugate vortices in the ionosphere. By analyzing the THEMIS data for that event, we found that several vortices can be detected together with the main one, and that the vortices indeed constitute a vortex chain. The study is carried out by analyzing both the velocity and the magnetic field measurements for spacecraft C and D, and by obtaining the corresponding holograms. It is found that both monopolar and bipolar vortices may be present in the magnetotail. The comparison of observations with numerical simulations of vortex formation in sheared flows is also discussed

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Investigation of differential and integro-differential models describing applied processes represent the actual sphere of modern mathematics. It is doubtless that construction of algorithms for approximate solutions, computer realization and analysis of the numerical results of corresponding initial-boundary value problems are very important. Mathematical models of real processes often lead to partial integro-differential equations of parabolic type. It is known that most of those problems have nonlinear character which essentially complicates of their study. Investigation and numerical solution of nonlinear integro-differential equations, appear for instance, as a result of significant process of mathematical modeling of electromagnetic field propagation in the medium. Various initial and initial-boundary value problems for those models and for their generalization are studied by many authors. Making certain physical assumptions in mathematical description of the same process of electromagnetic field propagation into a substance, Prof. G.I. Laptev proposed one of such generalization which he called as an averaged integro-differential model and pointed out their importance and complexity of their investigation. One should note that in this direction rather extensive bibliographical overview is given in the recently published monograph - T. Jangveladze, Z. Kiguradze, B. Neta, Numerical Solution of Three Classes of Nonlinear Parabolic Integro-Differential Equations. Elsevier, ACADEMIC PRESS, 2016. Investigations for the above-mentioned averaged models are conducted for some special cases so far. In particular, narrow class of nonlinearities are studied by scientists. Our goal is the investigation and numerical resolution of system of nonlinear integro-differential equations above with source terms. In particular, class of nonlinearity is widened and more general diffusion coefficients are considered. Uniqueness and large time behavior of solutions of the initial-boundary value problem for that model are fixed. Corresponding finite difference scheme is constructed and investigated. Stability and convergence of that scheme is proven. Results of numerical experiments with appropriate tables and graphical illustrations are given. Results of numerical experiments fully agree with theoretical findings.

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In mathematical modeling of many natural processes systems of nonlinear partial differential equations arise very often. Investigation and approximate solution of the initial-boundary value problems posed for these systems are the actual sphere of contemporary mathematical physics and numerical analysis. Economical finite-difference scheme for one system of nonlinear multi-dimensional partial differential equations is constructed. In particular case model can be used as a mathematical simulation of process of vein formation in meristematic tissues of young leaves. Stability and convergence of developed scheme are proven. Numerical experiments verifying theoretical findings for three-dimensional case are carried out. The appropriate graphical illustrations are given. Second important model is obtained at mathematical modeling of processes of electro-magnetic field penetration in the substance. In the quasi-stationary approximation, this diffusion process, taking into account of Joule law is described by nonlinear system of Maxwell equations. For more thorough description of electromagnetic field propagation in the medium, it is desirable to take into consideration different physical effects, first of all – heat conductivity of the medium. Special attention is paid to construction of discrete analogs, corresponding to one-dimensional models as well as to construction and analysis of decomposition algorithms with respect to physical processes. The above-mentioned decomposition is defined by splitting this model in two parts: in the first part the Joule heat release is taken into account and in the second – part the heat conductivity of the medium is considered.

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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 degenerate 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.

The talk is devoted to a concise survey of scientific, pedagogical, and educational activities of the outstanding Georgian mathematician and mechanist Ilia Vekua. Biographical data are also given.

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Natural gas distribution networks are complex systems with hundreds or thousands of kilometers of pipes, compression stations and many other devices for the natural gas transportation and distribution service. Achievement in the power consumption points with the required conditions is the main practical aspect and the most difficult issue in the gas transmission pipeline system. Determination of gas pressure and flow rate distribution along the pipelines is a necessary step for solving the above mentioned question. Searching of gas flow pressure and flow rate along the inclined and branched pipeline network has much more practical value but represents more difficult issue. For this purpose development of the mathematical models describing gas non-stationary flow in the branched and inclined pipeline systems are actual. The purpose of this study is determination of gas pressure and flow rate special and temporally distribution along the pipeline based on simplified one-dimensional partial differential equations governing the gas non-stationary flow in the inclined and branched pipeline. The simplification is established on the hypothesis that the boundary conditions do not change quickly and the capacity of gas duct is relatively large. Analytical solution of the simplified one-dimensional partial differential equations governing the gas non-stationary flow in the inclined and branched pipeline is obtained. Some results of numerical calculations of gas flow in the inclined and branched pipelines are presented.

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In this talk the 3D quasi-static theory of elasticity for materials with voids is considered. The representation of regular solution of the system of equations in the considered theory is obtained. There the fundamental and some other matrixes of singular solutions are constructed in terms of elementary functions.

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We consider the geometrically nonlinear and non-shallow spherical shells for I. N. Vekua N=3 approximation. The concrete problems, using complex variable functions and the method of the small parameter has been solved.

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The surfaces with non-zero Gaussian curvature are the Riemann’s diversity of 2-dimension, which are investment in the 3-D Euclidean space. Therefore, for these varieties of properties it is possible to construct quite clear representations. Further, it is shown that any regular surface can be put in the Riemann’s 3-dimensional diversity.

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The problem of the plane theory of elasticity with a partially unknown boundary (the problem of finding an equally strong contour) for a rectangular plate weakened by an equally strong contour (the unknown part of the boundary) is considered. It is assumed that the linear segments of the boundary are under the action of normal contractive forces with the given principal vectors and the unknown part of the boundary is free from external forces. The condition for the unknown contour to be equally strong is that the tangential normal stresses are stable on it. For solving the problem, the methods of complex analysis are used; the sought complex potentials and equations of an unknown contour are constructed effectively (in the analytical form).

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The talk is devoted to the updated survey of problems with the some elastic cusped structure-incompressible fluids interaction problems, when in the solid part either the Kirchhoff-Love plate or Vekua’s prismatic shell in the lower order approximations are considered. Application of I. Vekua’s dimensional reduction method to the viscous Newtonian fluid occupying thin prismatic domains will be also presented.

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In the present paper we consider approximate solution issues for following two problems: 1.Nonlinear boundary value problem for the Kirchhoff type static beam (see, for example [1], [2]). The problem is reducted by means of Green’s function to a nonlinear integral equation. To solve this problem we use the the Picard type iterative method. 2. Nonlinear initial-boundary value problem for the J.Ball dynamic beam (see, for example [3], [4]). Solution of problem is founded by means of an algorithm, the constituent parts of which are the Galerkin method, a symmetric difference scheme and Jacobi iterative method. For both of these problems are constructed the new algorithms of approximate solutions and numerical experiments are executed. The results of calculations are presented by tables and diagrams. REFERENCES 1.J.Peradze, A numerical algorithm for a Kirchhoff – type nonlinear static beam. J. Appl. Math. 2009, Art.ID 818269, 12pp. 2. T.F.Ma, Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Discrete Contin. Dyn. Syst. 2007, 694-703. 3. J. M. Ball, Stability Theory for an Extensible Beam, J. Diff. Eq., 14 (1973), 399-418. 4.G.Papukashvili, J.Peradze, Z.Tsiklauri, On a stage of a numerical algorithm for Timoshenko type nonlinear equation, Proc. A.Razmadze Math.Inst., Tbilisi, 158, (2012), 67-77.

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The main purpose of this work is mathematical modeling and study of stress-strain state of spongy bone of the jaw with implant. Spongy bone may be considered as a multiporous medium where fractures and intervening porous blocks are the most obvious components of the dual-porosity system. Spongy bone consists of the solid and liquid phases. The paper, in the solid phase, presents the equations, which describe of the effect of fluid pressure on the solid deformation within each individual component and in the fluid phase, a separate equation written for each component of distinct porosity or permeability. This paper studies the stress-strain state of spongy bone of the jaw near the implant in the case of occlusal load. Mathematical model of this problem represents a contact problem of elasticity between the implant and the body of the jaw. Boundary element methods, which are based on the solutions to the problems Flamant (BEMF) and Boussinesq (BEMB), are used to obtain numerical values of stresses in the bone tissue under the occlusal load on the implant.

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In the talk I will discuss about the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of the functions from Llog L.

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In this talk we present an extended traditional tableaux inference system to work with formulas built over unranked terms. Unranked unification is used in tableaux as a mechanism that decides whether a path can be closed. It selects terms for replacement in quantification rules. We show, that the calculus is sound and complete, thus non-terminating in general. As unranked unification is not finitary in general, that is another reason of nontermination of the given algorithm. Finally, we illustrate the potential of the extended calculus in Web-related applications. In such applications unification problem is reduced to matching and is thus finitary.

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An initial-boundary value problem is posed for the J. Ball integro-differential equation, which describes the dynamic state of a beam. The solution is approximated utilizing the Galerkin method, stable symmetrical difference scheme and the Jacobi iteration method. For approximate solving boundary value problem the some of programs in algorithm language Maple is composed and many numerical experiments are carried out.

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It consists in reducing the problem of the Kirchhoff type Nonlinear static beam equation by means of Green's function to a nonlinear integral equation, to solve with we use the iterative process. The condition for the convergence of the method is established and numerical realization is obtained. The algorithm has been approved tests and the results of recounts are represented in graphics.

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In the present work, we consider the classical nonlinear Kirchhoff string equation and study its two-dimensional generalization.

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The paper set a non-classical problems, which formulate in the following way: what normal stress supposed to be applied on part of the boundary, that at the segment lying inside the homogeneous isotropic elastic half-plane to obtain a pre-given conditions. The problems solved by Boundary element method . Testable examples are given, which shows us, that what the normal stress supposed to be applied on part of the boundary, that at the segment lying inside the body to obtain a pre-given stress or displacement. Represented the numerical results, appropriate graphics, mechanical and physical inter-pretation this problems.

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In this presentation we consider various coverings of the Euclidean plane and Euclidean space which are produced by using a set-theoretical approach. Special attention is paid to so-called homogenous coverings which are constructed with the aid of Axiom of Choice and method of transfinite induction. In this context, it should be noted that in 1985 A. B. Kharazishvili solved a well-known problem concerning homogenous coverings of the Euclidean plane and three-dimensional Euclidean space with congruent circumferences (see [1]). We apply the method similar to that of [1]. In our joint paper (see [2]), some questions related to indicated problems are highlighted too. We also consider several decompositions of certain subsets of Euclidean space and demonstrate their close connections with independent families of sets. Some applications of special decompositions in mathematical analysis are also discussed. References [1] A. B. Kharazishvili, Partition of a three-dimensional space into congruent circles. (Russian) Soobshch. Akad. Nauk Gruzin. SSR 119 (1985), no. 1, 57–60. [2] A. B. Kharazishvili, T. Sh. Tetunashvili, On some coverings of the Euclidean plane with pairwise congruent circles. Amer. Math. Monthly 117 (2010), no. 5, 414–423

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We use theory extended with contracted operators to study Bourbaki fundamental theory. We have defined unranked invariance and monotonicity notions and proved corresponding theorems. In particular, we have proved following theorem: Assume type of derivable unranked operators is I, II, II1 . If any operators belonging before derivable operators are invariant, then operator is also invariant. Result 1. Unranked operators and are invariant operators. Result 2. Unranked operators and are monotonic operators. Result 3. Unranked operators and are monotonic with respect to last operand. Acknowledgment. This work was supported by the Shota Rustaveli National Science Foundation under the grants # FR/508/4-120/14. References [1] B. Dundua, M. Rukhaia, Kh. Rukhaia, L. Tibua, Pρ Log for access control. J. Technical Sci. Technol. 5 (2015), no. 2, 41–44. [2] Sh. S. Phakadze, Some questions of notation theory. (Russian) Izdat. Tbilis. Univ., Tbilisi, 1977

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Dangerous meteorological events have significantly increased on the territory of Georgia over the last decades. In accordance with Georgia’s complex Orography, the main part of these events has local character and they mainly are associated with the air mass convective movements. Therefore, in the agenda there is an urgent question about timely prediction of possible convection processes in the non-uniform areas of Georgia. For this, it is necessary to study the thermodynamic condition of the atmosphere in the particular areas and therefore determine the degree of instability of the atmosphere. The presented article examines several cases of strong convective meteorological processes processing on the territory of Georgia based as on per the data of the synoptic maps, as well as on the basis of the Sighnaghi meteorological radar data. Aerological diagrams have been built for the each dangerous meteorological event taken part in the region. The diagrams were designed to measure and assess the thermodynamic state of the atmosphere and the rate of uncertainty of the atmosphere based on the particle method. It has been confirmed that the degree of volatility of the atmosphere for the four days was accurate in conjunction with data received from meteorological radar and synoptic mapping. This fact allows us to predict the atmospheric thermodynamic condition in the specific region on the basis of forecasted aerogical data obtained through the model and evaluate the quality of the convective processes in the local area.

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Regional scale summer time precipitations, among others (wind, temperature, sea level pressure, geopotential height etc.), still remain more difficult parameter for prediction by weather research forecast (WRF) model. This inconvenience mainly is stipulated by insufficient parameterization the resolved and subgrid-scale precipitation processes in the WRF model and by the lack in setting reliable initial and boundary conditions at nested grids of the WRF model. Furthermore, in comparison with cold-season precipitations, warm-season convective events and precipitations are much more difficult for prediction and especially for the territories having complex orography. In this study the problem of micro physics and cumulus rameterization schemes options for several warm-season convective events predictions above the Caucasus territory is studied. With the purpose of investigating impact of detailed orography on summer time heavy showers prediction three set of domains with horizontal grid-point resolutions of 19.8, 6.6 and 2.2 km have been used. Computations were performed by Grid system with working nodes (16 cores+, 32GB RAM on each) situated at GE-01-GRENA. Some results of the numerical calculations performed by WRFv.3.6.1 model with different and convective scheme components are presented. Acknowledgment. The research leading to these results has been co-funded by the European Commission under the H2020 Research Infrastructures contract no. 675121(project VI-SEEM).

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The tests are constructed for the hypotheses that p ⩾ 2 independent samples have the same distribution density (homogeneity hypothesis) or have the same well-defined distribution density (goodness-of-fit test). The limiting power of the constructed tests is found for some local “close” alternatives.

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Applying the classical Banach theorem, we have the following proposition. Theorem. There exists a Lebesgue measurable subset X ⊂ R_n which does not possess the uniqueness property in the class of all πn-volumes.

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An almost linear differential equation is considered (such equations were first discussed by us) and sufficient conditions are established for it to have a so-called property A or B.

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For the delay differential equation variation formulas of solution are given. For the optimization problems the necessary conditions of optimality are obtained. The linear representation of the first order sensitivity coefficient is obtained.

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The existence theorems of an optimal initial data and necessary optimality conditions for initial data are proved.

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Estimation of the increment of solution with respect to small parameter is obtained . There are considered two cases when variation of the initial moment take place from the left side or from the right side. The increment value of solution at the initial moment is calculated.

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The ტალკ is devoted to construction of hierarchical models for viscoelastic Kelvin-Voigt prismatic shells with voids on the basis of the linear theory. Using I. Vekua’s dimension reduction method, governing systems are derived and in the Nth approximation boundary value problems are set. In the N=0 approximation, considering plates of a constant thickness, governing systems mathematically coincide with the governing systems of the plane strain corresponding to the basic three-dimensional linear theory up to a separate equation for the out of plane component of the displacement vector in our case. The ways of investigation of boundary value problems and initial boundary value problems, including the case of cusped prismatic shells, are indicated and some preliminary results are presented. Initial conditions for weighted displacements and a weighted volume fraction are always classical. Boundary conditions for weighted displacements and a weighted volume fraction are non-classical in the case of cusped prismatic shells. Namely, we are not always able to prescribe them at cusped edges. If the thickness looks like the power function, vanishing at the edge of the prismatic shell, then in the N=0 approximation for viscoelastic Kelvin-Voigt prismatic shells with voids we can prescribe the displacements and volume fraction at the cusped edge if the power is less than 1, while we cannot do it if the power is not less than 1.

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The main goal of this paper is investigation of some singularities and specific features of atmosphere flows above the complex terrain of the Georgian territory, for prediction a regional scale dangerous events (heavy rains, hails) formation by different cumulus parameterization (CPSs) and micro physics (MP) schemes. To achieve the specified goal, we have used 3-D non-hydrostatic, non-stationary Whether Research Forecast - Advanced Researcher Weather (WRF-ARW) version 3.6 model. We have configured the WRF-ARW nested grid model for Caucasus region considering geographical-landscape character, topography height, land use, soil type and temperature in deep layers, vegetation monthly distribution, albedo and others. Investigations required High Performance Computer systems. That is way we have ported the WRF-ARW application to the GRID site GE-01-GRENA in Georgia which is located at Georgian Research and Educational Networks Association (GRENA). As GRENA connected in European GRID infrastructure so it was a good opportunity for running model on larger number of CPUs and storing large amount of data on the grid storage elements. The ability of the WRF model in prediction precipitations with different set of these MP and CPSs was examined using two precipitation events occur on the territory of eastern Georgia for warm season of 2015. Two set of domains with horizontal grid-point resolutions of 6.6 and 2.2 km are chosen to represent complex topography in current research WRF v.3.6 model. Accumulated total (24 h) precipitations are evaluated by careful examination of meteorological radar and radio zoned data and simulated fields. Some results of the numerical calculations performed by WRF model are presented

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The nonlinear singular integral equation associated with the Stokes gravity waves in the incompressible Euler fluid is studied. The existence of the solution is proved and the approximate solution is constructed by means of Maple.

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A new approach to the statistical hypotheses testing, called Constrained Bayesian Methods (CBM), was developed by (Kachiashvili (1989, 2003), Kachiashvili et al. (2012a, b), Kachiashvili and Mueed (2013)). This method differs from the traditional Bayesian approach with a risk function split into two parts, reflecting risks for incorrect rejection and incorrect acceptance of hypotheses and stating the risk minimization problem as a constrained optimization problem when one of the risk components is restricted and the another one is minimized (Kachiashvili (2011), Kachiashvili et al. (2012b)). Application of this method to different types of hypotheses (two and many simple, composite and multiple hypotheses, directional hypotheses) with parallel and sequential experiments showed the advantage and uniqueness of the method in comparison with existing ones (Kachiashvili (2014a, b), Kachiashvili (2015), Kachiashvili (2016), Kachiashvili and Bansal (unpublished)). The uniqueness of the method consists in the emergence of the regions of impossibility of making a simple or any decision alongside with the regions of acceptance of tested hypotheses, which allows us based on this approach to develop both parallel and sequential method without any additional efforts. The advantage of the method is the optimality of made decisions with guaranteed reliability and minimality of necessary observations for given reliability. CBM uses not only loss functions and a priori probabilities for making decisions as the classical Bayesian rule does, but also a significance level as the frequentist method does. The combination of these opportunities improves the quality of made decisions in CBM in comparison with other methods.

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The main goal of the present work is to find the solution of the contact problems for a homogeneous isotropic half-space by boundary element methods based on two different singular solutions (one is the solution of Flamant problem (BEMF), and another is the solution of Boussinesq's problem (BEMB)), and then to compare the obtained results.

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Some applications of Fuchsian system are listed.

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In the present paper unranked tableaux calculus is discussed, which extends the classical first order tableaux calculus for formulas over unranked terms. The correctness and completeness theorems of the calculus are proved and its expressive power in Web-related applications are illustrated.

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In this talk we discuss logic programming formalism extended with second-order conditional rewriting rules. We show expressive power of the extended formalism and demonstrate its applications.

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For the delay differential equation variation formulas of solution are given. For the optimization problems the necessary conditions of optimality are obtained. The linear representation of the first order sensitivity coefficient is obtained.

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Considered properties of weak distribution of the Hilbert space. Defined the necessary conditions for existence of measure.

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The nonlinear singular integral equation connected with the gravity waves propagation in the Eulerian fluids is studied. By the method of small parameter the equation is linearized and the solutions are obtained by means of the step-wise approximation method.

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The tests are constructed for the hypotheses that p ⩾ 2 independent samples have the same distribution density (homogeneity hypothesis) or have the same well-defined distribution density (goodness-of-fit test). The limiting power of the constructed tests is found for some local “close” alternatives.

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Based on investigations [1]. we discuss the following results: In the nonlinear dynamic equations of von Kármán type term a member describing wave propagation in the longitudinal direction usually is absent. The influence of this term can be proved to be very important at the description of behaviour of wings and tail parts of aircraft construction. Analogous phenomenon holds in the static problems too. Introduction of the corresponding needed terms eliminates the well-known problem of “Physical Soundness”in Truesdell’s sense. The corrections, introduced according to the proposed theory, in the average boundary conditions, consist in a refinement of the influence of boundary layer. It can cause significant changes in the neighbourhood of cuts (porthole, doors and etc.). Introduction of this term also explains and resolves set of paradoxes usually characteristic of existing refined theories (e.g. Kirchhoff, von Kármán, Mindlin, Reissner and all others).

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In this paper we prove that, in the case of some unbounded. Vilenkin groups, the Riesz logarithmic means converges.

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This talk is concerned with the linear theory of thermoelasticity with microtemperatures for homogeneous and isotropic solids.

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We analyse some new aspects concerning application of the fundamental solution method to the basic three-dimensional boundary value problems, mixed transmission problems, and also interior and interfacial crack type problems for steady state oscillation equations of the elasticity theory. First we present existence and uniqueness theorems of weak solutions and derive the corresponding norm estimates in appropriate function spaces. Afterwards, by means of the columns of Kupradze’s fundamental solution matrix special systems of vector functions are constructed explicitly. The linear independence and completeness of these systems are proved in appropriate Sobolev–Slobodetskii and Besov function spaces. It is shown that the problem of construction of approximate solutions to the basic and mixed boundary value problems and to the interior and interfacial crack problems can be reduced to the problems of approximation of the given boundary vector functions by elements of the linear spans of the corresponding complete systems constructed by the fundamental solution vectors. By this approach the approximate solutions of the boundary value and transmission problems are represented in the form of linear combinations of the columns of the fundamental solution matrix with appropriately chosen poles distributed outside the domain under consideration. The unknown coefficients of the linear combinations are defined by the approximation conditions of the corresponding boundary and transmission data.

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A special case of a homogeneous linear homogeneity problem for the regular Karlmann-Vekua equation is investigated, when the function G (t) participating in the boundary condition has zeros and poles at some points in the boundary circle.

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The paper presents the general procedure of solving of the equations of linear multivelocity neutron transport theory in plane geometry. Elementary solutions are found and then it is proved that the general solutions can be formed by the superposition of elementary solutions. As an application the Green’s function for a uniform infinite medium is constructed.

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In this talk we will speak about Calculus of Structures (CoS for short), which is a formalism introduced by Alessio Guglielmi. The structures are intermediate expressions between formulae and sequents. The inferences are to-down symmetric and can be applied deeply inside the expressions. Any deductive system can be presented in CoS. Here we will speak about linear logic in CoS and show its cut-elimination procedure.

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In the talk we discuss some projective sets and their measurability properties.

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Boundary problems of elastostatics for a porous circle with voids are considered. The uniqueness theorems for solutions of these problems are proved. Explicit solutions of problems in the form of series are obtained. Conditions are established that ensure the absolute and uniform convergence of these series.

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The paper deals with a boundary value problem for the Kirchhoff type static beam nonlinear integro-differential equation. The problem is reduced by Green function to an integral equation which is solved using the Picard iteration method. The convergence of the iteration process is established and numerical realization is obtained.

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The present work consider the normal contact problems, which are formulated as follows: the block, weight which can be neglected, to press with a certain force on the surface of half-space, i.e. at the contact surface is given normal stress (is given contractive stress). In particular, we examine two types distributed load, which correspond to the following cases: a) when contact surface is flat and b) when the contact surface is parabolic shape. The work consider plane deformation state. There is studied the stress-strain state of a half plane, namely, are obtained contours (isoline) of the maximum values of stresses and displacements in the half plane. The problems are solved by the boundary element method, which is based on the solutions of the problems of Flamant and Boussinesq's.

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The talk is devoted to a survey of scientific, pedagogical, and educational activities of the Georgian mathematician and mechanist Ilia Vekua. Biographical data are also given.

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Studying behavior of gas and liquid substances flow in horizontal and especially inclined branched pipelines became topical problem of today and had attracted attention of a number of scientists [1-8]. The present work is devoted to the one mathematical model describing a movement of gas flow in the inclined branched pipeline. A quasi-nonlinear system of two-dimensional partial differential equations describing gas non-stationary flow in pipe is considered. Gas pressure and flow rate distribution along the branched pipeline is investigated. Some results of numerical calculations of gas flow in horizontal and inclined branched pipelines are presented. Preliminary numerical calculations have shown efficiency of the suggested method

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In mathematical modeling of many natural processes systems of partial differential equations arise very often. Most of these models are nonlinear. This circumstance significantly complicate the study of such models. Investigation and approximate solution of the initial-boundary value problems posed for these systems are the actual sphere of contemporary mathematical physics and numerical analysis. One such type important model is obtained at mathematical modeling of processes of electro-magnetic field penetration in the substance. In the quasi-stationary approximation, this diffusion process, taking into account of Joule law is described by nonlinear system of Maxwell equations. For more thorough description of electromagnetic field propagation in the medium, it is desirable to take into consideration different physical effects, first of all - heat conductivity of the medium. In this case, with taking into account again the Joule law, the different type nonlinear system of partial differential equations is obtained. Many scientific papers are devoted to the investigation and approximate solution of abovementioned differential models. Many authors are studying convergence of semi-discrete analogs and finite-difference schemes for the models described here and for the problems similar to them. There are still many open questions in this direction. We study some properties of solutions of different kind of initial-boundary value problems for investigated systems, as well as numerical solution of those problems. We compare theoretical results to numerical ones. Special attention is paid to construction of discrete analogs, corresponding to one-dimensional models as well as to construction, analysis and computer realization of decomposition algorithms with respect to physical processes for the second system. The above-mentioned decomposition is defined by splitting this model in two parts: in the first part the Joule heat release is taken into account and in the second - part the heat conductivity of the medium is considered.

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Antiplane strain (shear) of an orthotropic non-homogeneous prismatic shell-like body is con- sidered when the shear modulus depending on the body projection (i.e., on a domain lying in the plane of interest) variables may vanish either on a part or on the entire boundary of the projection. The dependence of well-posedeness of boundary conditions on the character of vanishing the shear modulus is studied. Vibration problem is considered. The classical and weak setting of the boundary value problems are considered. Appropriate weighted functional spaces are introduced. Uniqueness and existence results for the variational problem are proved. The structure of the constructed weighted space is described and its connection with weighted Sobolev spaces is established.

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In this talk we present an extended traditional tableaux inference system to work with formulas built over unranked terms. Unranked unification is used in tableaux as a mechanism that decides whether a path can be closed. It selects terms for replacement in quantification rules. We show, that the calculus is sound and complete, thus non-terminating in general. As unranked unification is not finitary in general, that is another reason of nontermination of the given algorithm. Finally, we illustrate the potential of the extended calculus in Web-related applications. In such applications unification problem is reduced to matching and is thus finitary.

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In this talk we sketch a solving procedure for one-side ground hedge regular and context regular inequalities.

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Dangerous meteorological events in accordance with Georgia's complex orography mainly have local character and associated with the dusted air mass convection. Therefore an urgent question was to review the possible convection processes taken into account dust aerosols impact on forming regional climate of Georgia and to study the thermodynamic condition of the atmosphere and therefore determine the degree of instability of the atmosphere in the particular non-uniform areas of the Kakheti region

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http://conference.ens-2017.tsu.ge/uploads/5892254091a98Tea_Shavadze-2017.pdf

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The new criterion of testing hypothesis of equality distribution densities is created for grouped data.

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This paper describes PρLog a tool that combines Prolog with the ρLog calculus.

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For the delay functional differential equation with several delays and the discontinuous initial condition the variation formulas of solution are obtained. In formulas the effects of perturbations of the delays are discovered.

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The talk 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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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. 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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In the present paper pressure and gas flow rate distribution in the branched pipeline on the bases of one quasi-stationary nonlinear mathematical model is investigated. For realization of this purpose the system of partial differential equations describing gas quasi-stationary flow in the branched pipeline have been studied. Effective solutions of the quasi-stationary nonlinear partial differential equations (pressure and gas flow rate distribution in the branched pipeline) with the purpose of leak detection in the horizontal branched pipeline is presented. Some results of numerical calculations are presented

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A generalized Delaunay system was introduced and it was shown that most properties of the standard Delaunay systems remain valid for generalized Delaunay systems. At the same time, an example of a property of all standard Delaunay systems was pointed out, which fails to be true for some generalized Delaunay system.

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In our talk we consider some questions concerning geometrical realizations of families of sets. Namely, several geometrical realizations of families of sets in the finite-dimensional $R^n$ spaces are discussed with the corresponding algorithms. Also, some theorems establishing different properties of constituents of the above-mentioned realizations are presented.

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Regional climate formation above the territory of complex terrains is conditioned due to joint actions of large-scale synoptic and local atmospheric processes where the last one is basically stipulated by complex topography structure of the terrain. The territory of Caucasus and especially territory of Georgia are good examples for that. As known the global weather prediction models can well characterize the large scale atmospheric systems, but not enough the mesoscale processes which associated with regional complex terrain. With the purpose of modeling these smaller scale atmospheric phenomena and its characterizing features it is necessary to take into consideration main features of the local terrain, heterogeneity of land surfaces, influence of large scale atmosphere processes on the local scale processes etc. Recently the Weather Research and Forecasting (WRF) models represent a good opportunity for studding regional and mesoscale atmospheric processes such are: regional climate, extreme precipitations, hails, influence of orography on mesoscale atmosphere processes, sensitivity of WRF to physics options such are: microphysics, cumulus parameterizations, surface physics, planetary boundary layer physics and atmospheric radiation physics. Taking into account this broad availability of parameterizations it is not easy to define the right combination that better describes a meteorological phenomenon dominated above the investigated region. In the present article we have configured the nested grid WRF v.3.6 model for the Caucasus region, taking into consideration geographical-landscape character, topography heights, land use, soil type, temperature in deep layers, vegetation monthly distribution, albedo and others. Computations were performed by High Performance Computer systems. 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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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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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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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. 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 etc. 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. To achieve the specified goal, we have configured the WRF v.3.7 nested grid, wet model on the GRENA’s cluster which gave a good opportunity for running model on larger number of CPUs and storing large amount of data on the grid storage elements. Simulations were performed using a set of 2 domains where the coarser domain had a grid of 94x102 points which covers the South Caucasus region, while the nested inner domain has a grid size of 70x70 points mainly territory of Georgia. Both used the 51 vertical levels. We have studied some particulate cases of dangerous unexpected heavy showers which have taken place on the territory of Georgia and were accompanied with damage results. Some results of numerical calculations are presented

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In this paper we focus our attention on the nonlinear elliptic equation connected with the solitary waves. We consider the equation in the infinite area. By the trigonometric substitutions the equation is transformed to another nonlinear elliptic equation. The exact non-smooth solutions of this equation vanishing at infinity are obtained. Those solutions represent non-smooth solitary waves.

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The 2-D problem of the mezoscale boundary layer of the atmosphere (MBLA) in case of temperature nonhomogenouso underlying surface is set and solved. Such ecologically important processes as stratus clouds and radiation fogs are simulated (they are active "aggressors", accumulators of hazardous substances); space-time fields distributions of components of wind, temperature, pressure, specific humidity and water content are received. This task is considered by us also as distribution of a thermal wave in the atmosphere. Periodic decisions in the conditions of the daily course of temperature of the underlying surface are received. This task contains the properties of synergy characteristic processes...

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Functional series and interpolation algorithms for solving identification problems are used in the theory of nonlinear systems. It’s constructed interpolation formula of the Newton type and obtained evaluation of residual term in V. Makarov’s and V. Khlobistov’s works for nonlinear operators functional (see for example [1]). This approach is based on “continual” knots from interpolation conditions in the definition of kernels of functional (operator) polynomials. These “continual” knots represent linear combination of Heaviside functions. The abovementioned works have theoretical and practical importance in applied problems of the theory of operators’ approximation. Issues of realization of interpolation approximations on the electronic computers haven’t been discussed by the abovementioned authors. Calculating algorithms for approximate solution for boundary value problems of ordinary differential equations with non-constant coefficients are subscribed in the works [2], results of calculations of test problems are given, convergence issues are studied by the numerical-experimental way. Issues of approximate solutions for two-point boundary value problem with nonconstant coefficient by the use of operator interpolation polynomials of the Newton type are also discussed in the given work. Besides, the Green function of the differential equation of the boundary value problem) as a non-linear operator with respect to the nonconstant coefficient, is replaced by the known kernels of operator interpolation polynomial of the Newton type. Formulas of approximate solution of different type are constructed for finding the solution for two-point boundary value problem. Description of realization algorithm sand the calculation results of test problems are given. The convergence with respect to m parameter from the series of numerical experiments is exposed (m-degree of the operator interpolation polynomial of the Newton type). References [1] V. L. Makarov and V. V. Khlobystov, On the identification of nonlinear operators and its application (invited contribution). Boundary elements IX, Vol. 1 (Stuttgart, 1987), 43–58, Comput. Mech., Southampton, 1987. [2] A. R. Papukashvili, Approximate solution of a two-point boundary value problem with a variable coefficient using operator interpolation polynomials of Newton type. (Russian) Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 44 (1992), 45–74, 221.

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In the present work the boundary value problems of elasticity are considered in parabolic coordinate system. In the parabolic coordinates are written the equilibrium equation system and Hooke's law. Analytical (exact) solution of 2D problems of elasticity are constructed in the domain bounded by coordinate lines of the parabolic coordinate system. Here we represent external boundary value problems of elastic equilibrium of the homogeneous isotropic infinite body bounded by parabola, when on parabolic border normal or tangential stresses are given. Exact solutions are obtained using the method of separation of variables. Numerical results and corresponding graphs of above-mentioned problems are presented.

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Functional series and interpolation algorithms for solving identification problems are used in the theory of nonlinear systems. It’s constructed interpolation formula of the Newton type and obtained evaluation of residual term in V. Makarov’s and V. Khlobistov’s works for nonlinear operators functional (see for example [1]). This approach is based on “continual” knots from interpolation conditions in the definition of kernels of functional (operator) polynomials. These “continual” knots represent linear combination of Heaviside functions. The abovementioned works have theoretical and practical importance in applied problems of the theory of operators’ approximation. Issues of realization of interpolation approximations on the electronic computers haven’t been discussed by the abovementioned authors. Calculating algorithms for approximate solution for boundary value problems of ordinary differential equations with non-constant coefficients are subscribed in the works [2], results of calculations of test problems are given, convergence issues are studied by the numerical-experimental way. Issues of approximate solutions for two-point boundary value problem with nonconstant coefficient by the use of operator interpolation polynomials of the Newton type are also discussed in the given work. Besides, the Green function of the differential equation of the boundary value problem)as a non-linear operator with respect to the nonconstant coefficient, is replaced by the known kernels of operator interpolation polynomial of the Newton type. Formulas of approximate solution of different type are constructed for finding the solution for two-point boundary value problem. Description of realization algorithm sand the calculation results of test problems are given. The convergence with respect to m parameter from the series of numerical experiments is exposed (m-degree of the operator interpolation polynomial of the Newton type). References [1] V. L. Makarov and V. V. Khlobystov, On the identification of nonlinear operators and its application (invited contribution). Boundary elements IX, Vol. 1 (Stuttgart, 1987), 43–58, Comput. Mech., Southampton, 1987. [2] A. R. Papukashvili, Approximate solution of a two-point boundary value problem with a variable coefficient using operator interpolation polynomials of Newton type. (Russian) Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 44 (1992), 45–74, 221

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As pipelines become one of the main sources of liquid and gas substances transportation so studying behaviour of gas and liquid substances flow in horizontal and inclined branched pipelines became topical problem of today. Recently, many gas flow models have been developed and a number are using by the gas-liquid industry. In spite of the fact that most of those have been based on the result of gas-liquid flow experiments, accounting practices have shown none of them are universal, as yet they needs to be carefully analyzed, retreated, reworked and checked by the flow pattern. It has been shown in modern publications that the most complicated part in the practice especially are connected with branched pipeline networks and as a consequence mathematical modelsÁÀÈÖÌÉ, describing flow in the pipelines having outlets containing essential mistakes, which are owing significant simplification of the modelling environment and processes. For this reason development of the detailed numerical models adequate describing the real non-stationary not isothermal processes processing and progressing in the branched pipeline systems is necessary want. And as a consequence study of the problem by analytical methods from the mathematical point of view is prerequisite and represents a very actual problem. In the present paper gas pressure and flow rate distribution along the branched pipeline is investigated. The study is based on the analytical solution of the simplified nonlinear, non-stationary partial differential equations describing gas quasi-stationary flow in the branched pipeline. The effective solutions of the quasi-stationary nonlinear partial differential equations are presented. Preliminary numerical calculations have shown efficiency of the suggested method.

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Antiplane shear of an orthotropic non-homogeneous prismatic shell is considered when the shear modulus depending on the body projection (i.e., on a domain lying in the plane of interest) variables may vanish either on a part or on the entire boundary of the projection (problems of antiplane strain of isotropic non-homogeneous prismatic shell-like bodies are considered in [1], [2]). The dependence of well-posedeness of boundary conditions on the character of vanishing the shear modulus is studied. References [1] G. Jaiani, On the relation of boundary value problems for cusped plates and beams to three-dimensional problems. Semin. I. Vekua Inst. Appl. Math. Rep. 28 (2002), 40–51. [2] N. Chinchaladze, On some dynamical problems of the antiplane strain (shear) of isotropic non-homogeneous prismatic shell-like bodies. Bull. TICMI 19 (2015), no. 2, 55–65.

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Numerical solution of the non-classical problem, namely problem of localization of stresses, are obtained by the boundary element method. In a certain sense, the problem of localization of stresses in a body is the inverse problem to the delocalization problem. The localization problem is defined as follows: to change a sufficiently uniform stressed-deformed state of a body for a sharply expressed non-uniform stressed-deformed state (in conditions of constant external perturbations) by changing and appropriate selection of parameters of the medium. The problem can be set as follows: find on the part of border half plane distribution of normal stress so that is the same normal stress on the segment given length at a given distance from the boundary half plane is equal of given function (function describes a concentrated force). By the changes of elastic characteristics, the distance and the length of segment of the border will be select the normal stress optimal distribution at part of border half plane. Numerical results, corresponding graphs and mechanical and physical interpretation of above-mentioned problems are presented.

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In the parabolic coordinates equilibrium equations system and Hook’s law are writing. Exact solutions of 2D static boundary value problems of elasticity are constructed for the homogeneous isotropic bodies occupying domains bounded by coordinate lines of system parabolic coordinates. At parabolic border are given normal or tangential loads, and at xi = 0 are given symmetrical or anti-symmetrical conditions. The exact solutions are obtained by the separation variables method. In the work the numerical values of the components of stress tensor and displacement vector at some points of the body, and visualization and discussion of gained results are presented.

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New criteria of hypothesis for equality some distribution functions is constructed. Asymptotics of the test is investigated.

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The Cauchy type integral with the Weierstrass kernel taken over the smooth open line is studied. By means of the Muskhelishvili theory the behavior of this integral at the ends of this line is studied. The applications to hydrodynamics are given.

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In our talk geometrical realizations of families of sets and some algorithms for such realizations in finite dimensional $R^n$ spaces are presented and discussed. Special attention dictated by visibility aspects is paid to geometrical realizations of families of sets in $R^1$, $R^2$ and $R^3$ spaces. Several of our theorems describe properties of special important cases of geometrical realizations of families of sets and some of their applications are shown. Namely, different statements dealing with geometrical realizations of independent families of sets are presented.

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It is well known that the researches in the field of automatic theorem proving mainly have been conducted in two directions: a) the plain representation of the problem through the improvement on the logical language. b) search for the effective nethods and their realization. Kh. Rukhaia has contracted tau-logic, the language of which includes tau operator as one of its main symbols. Some properties of this operator are studied

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The present talk is a survey devoted to mathematical and engineering models of elastic multi-layered prismatic shell-like structures. In particular, it presents a model constructed by the speaker, based on modifications and a combination of the engineering method of equivalent single-layered model (see, e.g., [1]) and I. Vekua dimension reduction method [2] of constructing hierarchical models. The layers may be cusped prismatic shells as well. In the case of cusps peculiarities of setting boundary conditions (for such peculiarities see [3]) do not arise if the thickness of the structure does not vanish at the lateral boundary in spite of the fact that the structures may consist of cusped layers. References [1] H. Altenbach, J. Altenbach, W. Kissing, Mechanics of Composite Structural Elements. Springer-Verlag, Berlin-Heidelberg, 2004. [2] I. Vekua, Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston-London-Melbourne, 1985.

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The Weather Research and Forecasting (WRF) model version 3.6 represents a good opportunity for studding regional and mesoscale atmospheric processes such are: extreme precipitations, hails, sensitivity of WRF to physics options, influence of orography on mesoscale atmosphere processes, etc. In the present article the WRF model was applied to the selected weather events for predicting rainfall with numerous combinations of physics options. For fulfillment of this plan we have configured the WRF v.3.6 nested grid, wet model for Caucasus region (Georgian territory), considering geographical-landscape character, topography height, land use, soil type, temperature in deep layers, vegetation monthly distribution, albedo and others. The computations were performed by the Georgian Research and Educational Networking Association (GRENA) GRID system GE01-GRENA which is integrated in the European GRID infrastructure. Therefore it was a good opportunity for running model on larger number of CPUs and storing large amount of data on the GRID storage element. On the GRENA’s cluster WRF was compiled for both Open MP and MPI (Shared + Distributed memory) environment and WPS was compiled for serial environment using PGI (v7.1.6) on the platform Linux-CentOS. Simulations were performed using a set of 2 domains with horizontal grid-point resolutions of 6.6 km and 2.2 km, both defined as those currently being used for operational forecasts. The coarser domain is a grid of 94x102 points which covers the South Caucasus region, while the nested inner domain has a grid size of 70x70 points mainly territory of Georgia. Both use the default 54 vertical levels. We have studied some particulate bases of dangerous unexpected heavy showers which have taken place in warm seasons of 2015 in eastern part of the territory of Georgia and were accompanied with damage results consequences of the events were hard to foresee. The predicted rainfall by WRF model was compared with the observed rainfall data. In this study some comparisons between WRF forecasts was done in order to check the consistency and quality of WRF model with the heavy precipitations occur on the territory of Georgia. Some results of the numerical calculations performed by WRF model are presented. Acknowledgement. The research leading to these results has been co-funded by the European Commission under the H2020 Research Infrastructures contract no. 675121 (project VI-SEEM).

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In the report we consider three-dimensional elastic static equilibrium system of equations of bodies with double porosity. From this system of equations, using a reduction method of I. Vekua, we receive the equilibrium equations for the plates having double porosity. The systems of equations corresponding to approximations N=0 and N=1 are written down in a complex form and we express the general solutions of these systems through analytic functions of complex variable and solutions of the Helmholtz equation. The received general representations give the opportunity to solve analytically boundary value problems about elastic equilibrium of plates with double porosity.

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I. Vekua has constructed several versions of the refined linear theory of thin and shallow shells by means of his method of reduction of three-dimensional problems of elasticity to two-dimensional ones. This method for nonshallow shells in case of geometrical and physical nonlinear theory was generalized by T. Meunargia. In the present paper by means of the I. Vekua method the system of differential equations for the geometrically nonlinear spherical shells is obtained. Using the method of a small parameter,in the approximations of order N=2 the complex representations of the general solutions are obtained. Some concrete problems are solved.

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In this paper the 2D linear equilibrium theory of thermoelasticity with microtemperatures for isotropic microstretch solids is considered and the fundamental and singular matrices of solutions are constructed in terms of elementary functions. Representation of regular solution is obtained. Some basic results of the classical theories of elasticity and thermoelasticity are generalized.

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In the present paper we consider a plane problem of elasticity for a doubly-connected domain bounded by the convex 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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The structure of the solutions of sufficiently wide class of singular elliptic systems in the neighborhood of singular point are studied. On this basis the correct boundary value problems are posed and their complete (in some sense) analysis is given.

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In the talk we present proof search methods for first-order unranked logic. The unranked languages have unranked alphabet, where function and predicate symbols do not have a fixed arity. Such languages can model XML documents and operations over them, thus becoming more and more important in semantic web. We present a version of a 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 logic and proof layers of the semantic web stack.

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Descriptive Set theory is an important branch of set theory and plays an important role to solve many problems and questions in set theory ([2], [4]). When Luzin have constructed Projective sets hierarchy, he give to the mathematicians a new idea to develop set theory in new direction ([2], [3]). Descriptive set theory was applied in measure theory and was clear that, projective sets are very good objects in the sense of Lebesgue measure and are measurable sets. We discuss a modified version of the concept of measurability of sets and functions, in particular, we consider the measurability not only with respect to a concrete given measure, but also with respect to various classes of measures ([1], [3]). So, for a class M of measures, the measurability of sets and functions has the following three aspects: a. absolute measurability with respect to M; b. relative measurability with respect to M; c. absolute non-measurability with respect to M. Definable Sets of real line have a many interesting properties and we consider such sets in the sense measure extension problem. It is well known, that assuming Martin’s axiom it can be shown that there exists absolutely non-measurable functions. In particular, It is proved, that there exists absolutely non-measurable functions whose graph is projective subset of R.

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For the last decades intensive growth of pollution of the Black Sea by highly toxic substances of an anthropogenic origin and respective sharp deterioration of an ecological situation of the sea basin has made actual the study of spatial-temporal distribution of polluting substances and development of forecasting methods of pollution dispersion processes in the Black Sea. These toxic substances impact perniciously on the organic world of the seas and oceans. The formation and evolution of any substances of the anthropogenic or natural origin getting in the oceans and seas is a complex process and in general it is caused by simultaneous by actions of the following factors: physical (transition in other state of matter, radioactive disintegration, etc), chemical (chemical disintegration, chemical reaction with other substances), biological (accumulation, carrying over of substances by live organisms) and hydrodynamic ones

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Well-posedeness of boundary value (BVP) and initial boundary value problems for elastic cusped (tapered) prismatic shells (i.e. for shells with vanishing thickness at the boundary) claims non-classical setting of boundary conditions (BC). The present paper is devoted to a concise up-dated exploratory survey of results within the framework of various models of prismatic shells, in particular, plates of variable thickness (Kirchoff-love model, hierarchical models, multi-layer models, etc.). It also contains the non-published results of the author concerning micropolar and classical (with microtemperatures) elastic cusped prismatic shells.

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Antiplane strain (shear) of an orthotropic non-homogeneous prismatic shell-like body is considered when the shear modulus depending on the body projection (i.e., on a domain lying in the plane of interest) variables may vanish either on a part or on the entire boundary of the projection. The dependence of well-posedeness of boundary conditions (BCs) on the character of vanishing the shear modulus is studied

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Our report deals with the integrals with the Weierstraß kernel and their applications. In a complex z-plane we consider the non-linear integral equation containing the Weierstraß kernel. This equation is associated with the planar Stokes waves in the ideal fluid. Having found solutions of this equation the profile of the Stokes wave will be defined. By means of the con-formal mapping method the existence of solution of the integral equation is shown and the approximate solution is obtained.

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The variation formulas of solutions are proved in which the effects of the discontinuous initial condition and perturbations of delays and the initial moment are detected. There are considered three cases when variation of the initial moment take place from the left side, or from the right side or from the both side.

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Modern measure theory substantially relies onpowerful set-theoretical methods and, moreover, without using these methods the further development of this theory is doubtful. In our presentation, we discuss a modified version of the concept of measurability of sets and functions. In particular, we consider the measurability not only with respect to a concrete given measure, but also with respect to various classes of measures. From this general position, for any class M of measures, the measurability of sets and functions may be of three categories: a) absolute measurability with respect to M; b) relative measurability with respect to M; c) absolute non-measurability with respect to M. With the aid of some additional set-theoretical axioms, we specify the above-mentioned aspects of measurability. It is also investigated how the classes of absolutely measurable, relatively measurable and absolutely non-measurable functions (with respect to a fixed class M of measures) behave under action of standard operations, such as composition, addition, multiplication, limit operation, and so on.

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Summary The controlled differential equation with two constant delays in the phase coordinates is considered . Three types variation formulas are proved with respect to a new class of variation, which separately are corresponding to cases when variation of the initial moment take place from the left or from the right or from the both sides. The new class of variation means that the signs of variation of the initial moment and delays may be not same.

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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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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

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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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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The variation formulas of solution are derived , in which the effect of perturbations of the initial moment and the delay parameter, and also that of the discontinuous initial condition are detected. For initial data optimization problems the necessary conditions of optimality are obtained and the existence theorems are proved.

We consider also the problem of mathematical modelling of thermo-dynamic elastic nonshallow shells by 2D von Kảrmản-Koiter-Ciarlet type refined theories. The statical part of corresponding models represents boundary value problems for nonlinear systems of 2D partial integro-differential equations with Monge-Ampere operators and Poisson brackets. As a typical example we consider some details when shells present longitudinal TWS tube. In this problem, some schemes of the applied theory of analytical functions and a projective method will be given

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In this paper we consider the 2D linear equilibrium theory of elasticity for triple-porosity. We construct the fundamental and singular matrices of solutions for the system of equilibrium equations in terms of elementary functions. Some basic properties of the fundamental solution are also established. Representation of regular solution is obtained.

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The problem of consistency estimation of integral functional from distribution density and its derivatives is considered. For the density used estimation of Rosenblatt-Parzen and for the integral functional is used the “plug-in-estimator”. Theorems of Consistency and asymptotically normality are proved. Applications are given. The results extend the results from the paper [1]. References 1. Nadaraya E., Sokhadze G. On Integral Functional of a Density. Communications in statistics – Theory and Methods. 2016 (in press)

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The problem of atmospheric boundary layer is stated and solved taking into account humidity processes. Classification of foehn processes is made. On the basis of the present numerical model foehn process is simulated

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The report is dedicated to the set-theoretical invariants of algebraic and transcendental circles. Some features of algebraic circle care and evolution will also be presented.

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The new criterion of testing hypothesis of equality distribution densities is created.

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In this paper we study the a. e. strong summability of the cubic partial sums of the d-dimensional Walsh-Fourier series of the functions belonging to L( log⁺ L)d-1.

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Using the special exp-function method travelling wave exact solutions of 2D nonlinear Burgers’ equation are obtained. It is shown that such solutions have spatially isolated structural (soliton-like) forms.

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The linear conjugation boundary problem for the nonhomogeneous irregular CarlemanVekua equation is considered. The coefficients of the equation belong to sufficiently general classes of functions. The formula of general solution and necessary and sufficient conditions of solvability are obtained.

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We present a calculus for first-order unranked logics. Describe a proof construction algorithm under this calculus and show its implementation.

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We consider various paradoxical point sets which have an extra-ordinary descriptive structure. We study the measurability properties of such pathological sets in some set-theoretical models. In particular, we study measurability properties of Hamel bases and projective sets in models of ZFC and, especially, in Gödel’s Universe L ([1],[2],[3]). Moreover, we discuss projective hierarchy and large cardinals

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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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The talk 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 NeymanPearson 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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In our talk the problem of geometrical realization of any finite family of sets will be considered. Special attention will be paid to geometrical realizations of independent families of sets and to those figures which can be used for constructing such realizations. Some geometrical properties of constituents induced by the above-mentioned realizations will be presented

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We consider well-posedeness of two-dimensional BVPs within the framework of hierarchical models for cusped prismatic shells, plates, and bars on the basis of classical [1] and micropolar theories of elasticity, in addition with microtemperatures [2] and voids. We discusse peculiarities of setting boundary conditions for displacements, microrotations, couple stress vectors, tractions, temperatures, microtemperatures, and volume fraction fanctions. We carry out comparative analysis as well. References 1. Jaiani, G.: Cusped Shell-like Structures, SpringerBriefs in Applied Science and Technology, 84 p., Springer-Heidelberg-Dordrecht-London-New York (2011). 2. Jaiani, G.: Differential Hierarchical Models for Elastic Prismatic Shells with Microtemperatures. ZAMM - Z. Angew. Math. Mech., DOI 10.1002/zamm.201300016, 95 (1), 77-90, (2015).

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The present work is devoted to the analysis of some mathematical models describing a movement of subsoil waters (liquids) into the soil having the non-homogeneous multilayer structure in the vertical direction. The corresponding systems of two-dimensional differential equations in stationary and nonstationary 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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Analytical (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 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 are given normal or tangential stresses. Exact solutions are obtained using the method of separation of variables. Using the MATLAB software are obtained numerical results and constructed graphs of the mentioned boundary value problems.

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At present environment pollution by harmful substances having anthropogenic origination several times excised pollution caused by natural phenomena such are: volcanic eruptions, fires of forests, earthquakes, tornado, cyclones etc. Generally, harmful substances are emitted from power plants and vehicle engines, however in addition to ordinary events it is possible that non-ordinary situations like oil and gas pipelines and auto traffic and railway accidents arise (also terrorist attacks become more frequent). Activity of anthropogenic factors resulted in the considerable change of the area of underlying surface and water supplies in Georgia. Namely there are observed decreasing of the mowing, arable, unused lands, water resources, shrubs and forests, owing to increasing of the production and building. Transformation of one type structural unit into another one, naturally, results in local climate change. As research studies have shown, in recent decades, climate change process directly or indirectly impacts on different components of water resources on the territory of Georgia. In this study some hydrological specifications of Georgian water resources (surface, underground, thermal, mineral and drinking) and its potential pollutants are presented. Owing to climate change process in modern glaciers of Georgia dominate the processes of the retreat and melting, sizes of large glaciers come apart into smaller ones, the volume and length of glaciers are reduced. Some results of investigation of Georgian’s glaciers pollution and its melting process are given. In the last decades, study on regularities of space-temporal distribution of anthropogenic admixtures in the Black Sea have ecome extremely important and urgent, because of the sharp deterioration of an ecological situation of this unique sea basin. Among different pollutants, oil and oil products present the most widespread and dangerous kind of pollution for the Black Sea. Spreading of the oil pollutants in the Georgian Black Sea coastal zone on the basis of numerical model in case of different sea circulation regimes dominated for the four seasons in the Georgian Black Sea coastal zone is simulated. Some results of the river Rioni’s and the Georgian the largest rivers Kura’s conditions are given

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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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We consider a modified version of the concept of measurability of sets and functions, and analyze this version from the point of view of additional set-theoretical axioms. The main feature of such an approach is that the measurability is treated not only with respect to a concrete given measure, but also with respect to various classes of measures. So, for a class M of measures, the measurability of sets and functions has the following three aspects: a) absolute measurability with respect to M; b) relative measurability with respect to M; c) absolute non-measurability with respect to M. With the aid of additional set theoretical axioms, we specify the above-mentioned aspects of measurability. It is also investigated how the classes of absolutely measurable, relatively measurable and absolutely non-measurable functions (with respect to a fixed class M of measures) behave under action of standard operations, such as composition, addition, multiplication, limit operation, and so on. In particular, it is shown that: (1) Any function, which has a ????2 -massive graph, is relatively measurable with respect to the class of extensions of Lebsgue measure; (2) There exists a Bernstein set which is absolutely negligible with respect to the class of all nonzero sigma-finite translation invariant measures on R. (3) There exists a Bernstein set which is absolutely non-measurable with respect to the class of all nonzero sigma-finite translation invariant measures on R.

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In the first and zero approximations are considered Vekua hierarchicas models for prismatic shells then on the face surfaces tangential components of displacements and normal components stresses are given.

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Oscillation criteria for higher order functional differential equation is proved.

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It is well known, that set theory plays a fundamental role in all of mathematics branches. In particular, set-theoretical methods are successfully applied in various topics of real analysis and measure theory. Our research work is devoted to some applications of additional set-theoretical axioms in point set theory and in measure theory. We consider certain types of interesting and important point sets on the real line, such as Vitali sets, Bernstein sets, Hamel bases, Luzin sets, and Sierpinski sets. These point sets are pathological in the sense of measure theory or Baire property. We study the above-mentioned sets from the point of view of measure extension problem. Vitali, Bernstein and Hamel constructions are based on uncountable forms of the Axiom of Choice, while Luzin and Sierpinski constructionsrely on the Continuum Hypothesis. In the present work various combinations of such subsets are considered. In particular: (1) There exists a subset X of R which is simultaneously a Vitali set and a Bernstein set. (2) There exists a Hamel basis of R which simultaneously is a Bernstein set. (3) There exists no Hamel basis in R which simultaneously is a Vitali set. (4) There exists no Luzin (Sierpinski)subset (generalized Luzin(Sierpinski) subset) of R which simul- taneously is a Vitali set. (5) There exists no Luzin (Sierpinski) subset (generalized Luzin(Sierpinski) subset) of R which si- multaneously is a Bernstein set. (6) Under the Continuum Hypothesis (Martin Axiom) there exists Luzin (Sierpinski) subset (gener- alized Luzin(Sierpinski) subset) of R which simultaneously is a Hamel bases. We investigate a modified version of the concept of measurability of sets and functions, and analyze this version from the point of view of additional set-theoretical axioms. The main feature of such an approach is that the measurability is treated not only with respect to a concrete given measure, but also with respect to various classes of measures. So, for a class M of measures, the measurability of sets and functions has the following three aspects: (a) absolute measurability with respect to M; (b) relative measurability with respect to M; (c) absolute non-measurability with respect to M. With the aid of additional set theoretical axioms, we specify the above-mentioned aspects of measurability and use Marczewski method of extending measures for some pathological subsets of the real line

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The present talk is dedicated to the investigation and approximate resolution of the initial-boundary value problem with first type boundary conditions for one one-dimensional integro-differential parabolic model.

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For the nonlinear delay functional differential equations with the discontinuous initial condition linear representations of the first order sensitivity coefficients are proved.

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The article focuses on the discussion of the problem arising at testing composite hypothesis using classical Bayesian approach and known as Lindley’s paradox. Along with Bayesian approach with arbitrary choice of a priory probabilities here is considered offered by Bernardo the special choice of a priori probability with the purpose of avoiding of this paradox and the constrained Bayesian method (CBM) proposed by the author of this paper. The positive and negative sides of these approaches are considered. Theoretically and on the basis of computation of special examples (where arises Lindley’s paradox) is shown that CBM is free from the problems like the abovementioned owing to new, original properties of this method connected with the regions of no-making the decision about the validity of hypotheses along with the regions of making the decision with given levels of error probabilities Type I and Type II. Computed practical examples clearly demonstrate high quality and reliability of CBM. In critical situations, when other tests give opposite decisions, it gives the most logical decision.

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It is shown that, for any nonzero σ-finite translation invariant (translation quasi-invariant) measure μ on the real line R, the cardinality of the family of all translation invariant (translation quasi-invariant) measures on R extending μ is greater than or equal to 2ω1, where ω1 denotes the first uncountable cardinal number. Some related results are also considered

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We have constructed a τ-logical theory, in the language of which, among the main symbols is the operator τ. Viktor Mikhailovich Glushkov in his work “Some Problems in the Theory of Automata and artificial intelligence” considers it necessary to develop a practical formal a language for writing mathematical propositions and their proofs. However, this language must refer to the formal languages ​​of mathematical logic, and it must also contain a generic assignment operator - i.e. variables should be assigned not only numerical values, but also values ​​corresponding to various mathematical concepts. In our opinion, the τ-logical theory described above partially solves this problem.

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In the paper the fluid flow of large viscosity and low Reynolds number is considered in large reservoirs. It is assumed that viscosity is large and the linearized 2D Navier-Stokes equation (Stokes system) is studied in the rectangular area partly filled with the heavy fluid. The case of the solenoidal body force is considered. The solutions of the Stokes system are obtained with the appropriate initial-boundary conditions. It is proved that for the given pressure the solution is uniquely defined. The profiles of free surfaces are constructed for the different pressure.

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მოხსენებაზე წარმოგიდგენთ ავტომატურ დამამტკიცებელს, რომელიც ამოწმებს პროპოზიციული ურანგო ლოგიკის წინადადების მართებულობას. აღნიშნული დამამტკიცებლის ინოვაციურობა მდგომარეობს იმაში, რომ იგი დაწერილია SQL ენაზე. გრაფიკული გარსი კი აგებულია Visual Prolog-ის საშუალებით. ამჟამად მიმდინარეობს დამამტკიცებლის წარმადობის ტესტირება. მარტივ ამოცანებზე ჩატარებული ტესტირების შედეგები დამაკმაყოფილებელია. SQL ენზე დამამტკიცებლის შექმნის იდეა გამომდინარეობს იმ ფაქტიდან, რომ გვინდა სამომავლოდ შევქმნათ დამტკიცებათა მონაცემთა ბაზა, რომელიც დაეხმარება დამამტკიცებელს უფრო სწრაფად შეასრულოს დავალება.

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Iteration algorithm of computation of effective estimators of the shape parameters of beta distributions using unbiased estimators of the end points parameters of the random variable were obtained and investigated. For the cases when more accurate estimations of the parameters are required, one more step of computation, realized optimization of the obtained estimations, is necessary. The consistent, unbiased and efficient estimators of the parameters of a concrete case of Beta distribution, irregular right-angled triangular distribution, on the basis of maximum likelihood estimators were obtained and investigated. The computation results realized on the basis of the simulation of the appropriate random samples demonstrate theoretical outcomes. The Beta distribution optimally describes the probability character of biometric measurement information of a fingerprint reader for the people having no access to the controlled system.

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Considered boundary-value problem for the ellipse, when on the elliptic border and the segment between the foci of the ellipse are give tangential stresses, and free from normal stresses. This task is obtained from the corresponding problem for semi-ellipse , when conditions of uninterrupted continuation of the solution are given at and , therefore it is possible to bind the semi-ellipse as an elliptic ring, in which the tangential stresses are given on and the conditions of uninterrupted continuation of the solution not performed along this part, i.e. we have a crack on which the tangential stress acts. For solution this problem we apply a method by which the solution of complex problems of elasticity is reduced to the solution of simple problems of elasticity, namely, to solution internal and external problems of elasticity, which are easily solved by separation of variables. Using the MATLAB software we obtained numerical results and constructed 2D and 3D graphs of the distribution of displacements and stresses in body.

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At present impact of global climate change on the territory of Georgia is evident at least on the background of the Caucasus glaciers melting which during the last century have decreased to half their size. Glaciers are early indicators of ongoing global and regional climate change. Knowledge of the Caucasus glaciers fluctuation (melting) is an extremely necessary tool for planning hydro-electric stations and water reservoir, for development tourism and agriculture, for provision of population with drinking water and for prediction of water supplies in more arid regions of Georgia. Otherwise, the activity of anthropogenic factors has resulted in decreasing of the mowing, arable, unused lands, water resources, shrubs and forests, owing to increasing the production and building. Transformation of one type structural unit into another one has resulted in local climate change and its directly or indirectly impacts on different components of water resources on the territory of Georgia. In the present paper, some hydrological specifications of Georgian water resources and its potential pollutants on the background of regional climate change are presented. Some results of Georgian’s glaciers pollution and its melting process are given. The possibility of surface and subsurface water pollution owing to accidents at oil pipelines or railway routes are discussed. The specific properties of regional climate warming process in the eastern Georgia are studied by statistical methods. The effect of the eastern Georgian climate change upon water resources is investigated

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A biofilm is a complex gel-like aggregation of microorganisms like bacteria, cyanobacteria, algae, protozoa and fungi. They stick together, they attach to a surface and they embed themselves in a self-produced extracellular matrix of polymeric substances, called EPS. Even if a biofilm contains water, it is mainly in a solid phase. Biofilms can develop on surfaces, which are in permanent contact with water, i.e. on solid/liquid interfaces or on different types of interfaces such as air/solid, liquid/liquid or air/liquid. 1D and 2D problem for the biofilm occupying thin prismatic domain are considered. 2D problem is solved using Vekuas dimension reduction methods.

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In the present paper pressure and gas flow rate distribution in the branched pipeline on the basis of one quasi-stationary nonlinear mathematical model is investigated. For realization of this purpose the system of partial differential equations describing gas quasi-stationary flow in the branched pipeline was studied. We have found effective solutions of these quasi-stationary nonlinear partial differential equations (pressure and gas flow rate distribution in the branched pipeline) for leak detection in a horizontal branched pipeline. For studying the affectivity of the method quite a general test was created. Preliminary data of numerical calculations have shown efficiency of the suggested method. Some results of numerical calculations defining localization of gas escape for the inclined pipeline are presented. The results of calculations on the basis of observation data have shown that the performed simulations were much closer to the results of observation.

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In this work research of antiplane problems of the theory of elasticity by a method of the integral equations for composite (piece-wise homogeneous) orthotropic bodies weakened by cracks is studied. Two problems are considered. In case of the first problem the crack spreads to the interface (the crack extends until the interface) under a right angle, and in case of the second problem the crack crosses dividing border (intersect an interface) under a right angle. The first problem is reduced to the singular integral equation which containing an immovable singularity (contains motionless feature), and the second problem is reduced to system (pair) of the singular integral equations containing an immovable singularity, with respect to the tangent stress jumps (the characteristic function of crack expansion) (see [1]). In the above – stated problems our main objective was research of behavior of characteristic function of disclosing of cracks, calculation of stress intensity factors at the ends of a crack and forecasting of distribution of a crack. The basic integral equation and system of the integral equations has been solved by a collocation method, in particular, a method of discrete singularity (see [2]). orresponding systems of the linear algebraic equations have been obtained for uniform (using rectangular quadratic formulas) as well for non-uniformly distributed knots (with the use of quadratic formulas of higher accuracy constructed on the knots of Chebyshev polynomials). Solving the system of linear algebraic equations and graphic interpretation was carried out by means of programming system Matlab. In the presented work for the approached solution of the above – stated problems new numerical algorithms were constructed, corresponding numerical calculations were performed and the hypothetical forecast for a crack distribution in the body had been given.

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The present talk is, in a certain sense, the speaker’s account about his main activities at the I. Javakhishvili Tbilisi State University, Georgian Mathematical Union, and Georgian Mechanical Union; besides, it is devoted to his principal results obtained in the theory of Partial Differential Equations, mainly in the theory of degenerate ones with applications to cusped (tapered) elastic shells, plates, and bars. It contains also a concise survey of some two- and one-dimensional models constructed by him in the fields of elastic solids and fluid-elastic solid interaction problems

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We consider the time-harmonic acoustic wave scattering by a bounded layered anisotropic inhomogeneity embedded in an unbounded anisotropic homogeneous medium. The material parameters and the refractive index are assumed to be discontinuous across the interfaces between inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problems are formulated as boundary transmission problems for a second order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. We show that with the help of localized potentials the boundary-transmission problems can be reformulated as a localized boundary-domain integral equations (LBDIE) systems and prove that the corresponding localized boundary- domain integral operators are invertible.

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I. Vekua constructed the linear theory of 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 I. Vekua’s method, the system of differential equations for physically and geometrically non-linear theory of non-shallow shells is obtained and compared with other theories (Ressner, Koiter-Naghdi, Lurie,...).

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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. This method for non-shallow shells in case of the geometrical and physical non-linear theory was generalized by T. Meunargia. In this paper we consider non-shalow shells. By means of I. Vekua’s normed moments method we get the approximate expression of the stress tensor which is compatible with boundary data on face surfaces.

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The paper considers the problem of the plane theory of elasticity for a circular hole with a finite polygonal domain. For the solution of the problem the use is made of the methods of conformal mappings and of boundary value problems of analytic functions, and unknown complex potentials are constructed effectively (analytically). The estimates of the solution behavior at the vicinity of angle are given.

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The main goal of this paper is to consider the Dirichlet and the Neumann type boundary value problems (BVPs) of equilibrium theory of thermoelasticity for a sphere with double porosity. We construct explicitly the solutions of the Dirichlet and the Neumann type BVPs in the form of absolutely and uniformly convergent series.

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Structures of complicated geometry play very important role in the modern practical constructions. Thin and fine elastic parts of structures with angles and cusps such as cusped (tapered) shells, plates and bars belong to the complicate components of constructions. The lecture gives foundations of the theory of such bodies. Their consideration leads to non-classical problems in mathematical and mechanical points of view in the sense of setting boundary conditions in the corresponding boundary and initial-boundary value problems. The main attention will be paid to explanation of peculiarities of setting boundary conditions for cusped shells, plates, and bars

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In this paper we introduce λR: A foundational calculus for sequence processing with regular expression types. Its term language is the lambda calculus extended with sequences of terms and its types are regular expressions over simple types. We provide a flexible notion of subtyping based on the semantic notion of nominal interpretation of a type. Then we prove that types are preserved by reduction (subject reduction), and that there exist no infinite reduction sequences starting at typed terms (strong normalization).

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The aim of this talk is to present an efficient proof-search procedure, that will, for a given formula schema, obtain its proof schema. It is well known that naive proof-search in sequent calculus leads to a redundant search space. We use standard and well known approaches to reduce the search space.

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The estimate for the Bernoulli regression function is constructed using the Bernstein polynomial. The question of its consistency and asymptotic normality is studied. Testing hypothesis is constructed on the form of the Bernoulli regression function. Also, the test is constructed for the hypothesis on the equality Bernoulli functions. The question of consistency of the constructed tests is studied

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In the present article the problem for composite (piece-wise homogeneous) body weakened by crack when the crack intersect an interface or penetrate it at rectangular angle is studied. Antiplane problems of the elasticity theory for piece-wise homogeneous orthotropic plane is reduced to the singular integral equation (when crack spreads to the interface) and system (pair) of singular integral equations(when crack intersects the interface) with respect to the unknown characteristic function of disclosing of cracks containing an immovable singularity. First time behavior of solutions in the neighborhood of the crack endpoints is studied by a method of discrete singularity in the both uniform, and non-uniformly cases of the knots arrangement. The question of the one system (pair) of the singular integral equations approximate solution is investigated. A general scheme for the approximate solution of the task by collocation method is presented. The corresponding algorithms are composed and realized. The results of numerical investigations are presented

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Study of air flow spatial-temporal propagation in the atmosphere, which is generated by high-power pulse in a small time period, has both theoretical and practical values and is in urgent need of attention. As a rule, such kind of physical and chemical processes take place during a small period of time over a relatively small territory, but the result of this action is impressive and brings serious damages. Investigation of the air flow advective propagation, especially over non-homogenous territory, is very useful for different scientific areas. The whole point is that even low height hills slow down the velocity of flow motion and often change its movement, sometimes to the opposite direction. Exactly such kind of orographic peculiarities characterize some regions of Georgia, namely Tskhinvali and Sachkhere regions, where military actions took place in 2008. In the present paper, spatial-temporal propagation of air flow in the atmosphere over mountainous region generated by high-power pulse is investigated. Some results of theoretical and numerical investigations for the territory of Georgia are given.

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The problem of the forthcoming global climate change resulting from natural and growing anthropogenic factors (much more economical and technological development, overexploitation of land, water, oil and gas resources) acquires a particular importance for the territory of Georgia. The geographic location of Georgia, its complex orography and circulation conditions specify the diversity of its climate. Here form 11 types of climates from semi-desert to subtropics. For example there were observed: mountainous zone of Caucasus with constant snow and glacier, wet subtropical climate of the Black Sea, continental climate of Eastern Georgia and semi desert climate in some regions of Eastern Georgia. As well on the background of the global climate warming the statistical processing of the data of mean climatic temperature for the last ninety years exposed the versatility of climate over the territory of Georgia. Namely there were observed regularity of the climate cooling in the West Georgia, warming in the East Georgia and also there were elicited those micro-regions where mean climatic temperature had not changed in time...

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Two nonlinear partial integro-differential models are considered. Those models at mathematical modeling of process of electro-magnetic fiels penetration in the substance. In the quasistationary approximation this process is described by Maxwell’s system of nonlinear partial differential equations.

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Research work is devoted to some applications of additional set-theoretical axioms in the theory of real-valued functions and measure theory. It is well known that various studies in mathematical analysis, theory of real functions and measuretheory leads to various paradoxical point sets. We consider certain types of interesting and important point sets on the real line, such as Vitali sets, Bernstein sets, Hamel bases, Luzin sets, and Sierpinski sets.Vitali’s, Bernstein’s and Hamel’s constructions are based on uncountable forms of the Axiom of Choice, while Luzin’s and Sierpinski’s constructions rely on the Continuum Hypothesis. In the present work various combinations of such subsets are investigated. In particular: (1) There exists a subset X of R whichis simultaneously a Vitali set and a Bernstein set. (2) There exists a Hamel basis of R which simultaneously is a Bernstein set. (3) There exists no Hamel basis in R which simultaneously is a Vitali set. (4) There exists no Luzin (Sierpinski)subset (generalized Luzin (Sierpinski) subset) of R which simultaneously is a Vitali set. (5) There exists no Luzin (Sierpinski) subset (generalized Luzin (Sierpinski) subset) of R which simultaneously is a Bernstein set. (6) Under the Continuum Hypothesis (Martin’s Axiom) there exists Luzin (Sierpinski) subset (generalized Luzin (Sierpinski) subset) of R which simultaneously is a Hamel bases. The above-mentioned classical pathological subsets of the R are envisaged and their various structural properties are investigated from the measure-theoretical view-point. We are dealing with these sets in the context of function theory and measure theory, and establish some interrelations between them. In particular,it is shown that: (a) the existence of an additive function from R into ????2 , whose range contains three non- collinear points, implies (within ZF & DC theory) the existence of a nontrivial solution of Cauchy’s functional equation; (b) there exists a translation invariant measure μ on R extending the standard Lebesgue measure and such that all Sierpinski sets are of μ-measure zero; (c) there exists a Bernstein subset of R which is absolutely negligible with respect to the class of all sigma-finite translation invariant (quasi-invariant) measures on R.

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In this talk 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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In this thesis our discussion is devoted to certain exotic point sets of the real line R, in particular, to Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpinski sets and their properties with respect to invariant measures. The general measure extension problem is to extend a given measure μ onto a maximally large family of subsets of R. It is known that: (a) Vitali sets are not measurable with respect to any measure on R which extends the Lebesgue measure λ and is invariant under the group of all translations of R. (b) Any Hamel basis of R is absolutely negligible with respect to translation invariant (quasi-invariant) measures on R. (c) Luzin sets are very small from the point of view of measure theory since they have measure zero with respect to the completion of any σ-finite diffused Borel measure on R. Here we are dealing with the above-mentioned sets in light of function theory and measure theory, and establish some interrelations between them. In particular, it is shown that: 1. Some Bernstein set X is relatively measurable with respect to the class of all translation invariant measures on R extending λ, but X does not possess the uniqueness property with respect to the same class of measures; 2. There exists a translation invariant measure μ on R extending the standard Lebesgue measure and such that all Sierpinski sets are of μ-measure zero; 3. There exists some Bernstein set which is absolutely negligible with respect to the class of all σ-finite translation invariant (quasi-invariant) measures on R.

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For the time-delay controlled functional differential equation with the discontinuous initial condition linear representation of the first order sensitivity coefficient is obtained with respect to perturbations of initial data. For the initial data optimization problems the necessary conditions of optimality are obtained.

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It is well known that independent families of sets play an important role in different branches of contemporary mathematics, especially, in general topology and mathematical analysis. In [4] and [5] we investigated independent families of sets not only from set-theoretical view-point, but also from geometrical one. According to one of our results, there exists an uncountable independent family of convex compact figures in the Euclidean plane. Studies in this direction show that it is natural to answer the following question: how the structure of atomic components of any independent family of convex compact figures in the Euclidean plane depends on the cardinality of the family? We established the following two results. Theorem. If five or more convex compact figures in the Euclidean plane form an independent family of sets, then for any point A of any atomic component W of the family there exists a triangle $ABC$ with positive area, such that the following inclusion holds: $$ \Delta ABC\subset W $$ Theorem. There exist independent families $F_1$ and $F_2$ of convex compact figures in the Euclidean plane, such that $card(F_1) = card(F_2) = 4$ and one of the atomic components of the family $F_1$ is a singleton while one of the atomic components of the family $F_2$ is a line segment with positive length.

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We defined τSR-analog of Herbrand universe, Herbrand τSR-base,Herbrand τSRinterpretation and the following theorems are proved.Theorem 1. If an interpretation I over some domain D satisfies a formula A of τSRlogic, then any one of the τSR-interpretations I corresponding to I also satisfies a formula A of τSR-logic. Theorem 2. A formula A of τSR-logic is unsatisfiable if and only if A is false under all the τSR-interpretation. References [1] Ch. L. Chang, R. Ch. Lee, Symbolic Logic and Mechanical Theorem Proving. Academic Press New York, San Francisco London, 1973. [2] Kh. Rukhaia, L. Tibua, G. Chankvetadze, B. Dundua, One method of constructing a formal system. Appl. Math. Inform. Mech. 11 (2006), no. 2, 81–89, 92–93. [3] Sh. S. Phakadze, Some Questions of Notation Theory. (Russian) Izdat. Tbilis. Univ., Tbilisi, 1977.

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At present pipelines become one of the main sources of liquid and gas substances transportation and play a vital role in our daily lives. That is way study of gas and liquid substances flow behavior in horizontal and inclined branched pipelines became topical problem of today and had attracted attention of a number of scientists. Recently, many gas flow equations have been developed and a number are using by the gas-liquid industry but as accounting practices have shown none of them are universal. In spite of the fact that most of those have been based on the result of gas-liquid flow experiments as yet they needs to be carefully analyzed, retreated, reworked and checked by the flow pattern. It has been shown in many modern publications that the most complicated part describing practical methods of modeling especially are branched pipeline networks and mathematical models describing flow in the pipelines having outlets containing essential mistakes, which are owing significant simplification of the modeling environment and processes. For this reason development of the detailed numerical models adequate describing the real non-stationary not isothermal processes processing and progressing in the branched pipeline systems and study the problem by analytical methods are actual. In the present paper pressure and gas flow rate distribution in the branched pipeline based on the one quasi-stationary nonlinear mathematical model using analytical methods is investigated. For realization of that purposes the system of partial differential equations describing gas quasi-stationary flow in the branched pipeline was studied. We have found effective solutions of the quasistationary nonlinear mathematical model(pressure and gas flow rate distribution in the branched pipeline). For learning the affectivity of the method quite general test was created. Preliminary data of numerical calculations have shown efficiency of the suggested method

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New algorithms of the approached decision of antiplane problems of elasticity theory (Poisson’s equation) for a two-dimensional crosswise body by means of Schwartz iterative method [1] are considered. Let us solve a problem Dirichlet for the Poisson equation by an ∆u(x, y) = f(x, y), (x, y) ∈ Ω, (1) u(x, y) = g(x, y), (x, y) ∈ ∂Ω, (2) where u(x, y) ∈ C^2(Ω) is unknown, f(x, y) ∈ C(Ω), g(x, y) ∈ C(∂Ω) are given functions, Ω = Ω_1 ∪ Ω_2 is a given body, ∂Ω = Γ = Γ_1 ∪ Γ_2 is a boundary of the given body, Ω_1 = {(x, y) : −2 ≤ x ≤ 2, −1 ≤ y ≤ 1}, Ω_2 = {(x, y) : −2 ≤ y ≤ 2, −1 ≤ x ≤ 1}, Γ_1 = {(x, y) : y = ±1, −2 ≤ x ≤ −1 or 1 ≤ x ≤ 2; x = ±2, −1 ≤ y ≤ 1}, Γ_2 = {(x, y) : x = ±1, −2 ≤ y ≤ −1 or 1 ≤ y ≤ 2; y = ±2, −1 ≤ x ≤ 1}. The algorithm consists of two parts: the Schwartz method and the Galerkin method. Unknown function expands in row Fourior–Legendre. Differences of polynoms Legendre are used as basic functions. It is received the five-dot linear system of the algebraic equations concerning unknown coefficients (see [2]). A count process is stable, as corresponding matrix of algebraic equation system has diagonal dominating property relative to rows. It is created the program code (on the basis of Maple 16) for the approached decision of the consider problem (1), (2). The authors express hearty thanks to Prof. T. Vashakmadze for his active help in problem statement and solving. References [1] S. L. Sobolev, Schwartz algorithm in elasticity theory. (Russian) Dokl. Akd. Nauk. SSSR 4 (1936), 235–238. [2] A. Papukashvili, Y. F. Gulver, Z. Vashakidze, To numerical realizations and stability of calculating process of some problems of theory of elasticity for cross-shaped regions. Rep. Enlarged Sess. Semin. I. Vekua Inst. Appl. Math. 28 (2014), 94–97.

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The research in recent several decades has established the boundary element methods (BEM) as a powerful tool in computational mechanics. One of the remaining drawbacks is that the BEM as previously used is based on an explicit knowledge of fundamental solutions. In many engineering problems we do not know these fundamental solutions. The overcome this drawback, an alternative BEM is presented here the method developed by means of the spatial Fourier transform generalizes the boundary element method to the so-called Fourier BEM [1]. Recent approach is available for all cases as long as the differential operator is linear and has constant coefficients and possible for all variants of the BEM. The basis of Fourier BEM are two well known theorems of the Fourier transformation: the theorem of Parseval and the convolution theorem. Parseval’s theorem states the equivalence of energy or work terms in the original space and in the Fourier space, and the convolution theorem links a convolution in the original space to a simple multiplication in the transformed space. The idea is to avoid the inverse Fourier transform of the fundamental solution and to work directly with the Fourier transformed fundamental solution. The elements and shape functions also can be transformed to the Fourier domain. In this work, the method is presented and then applied to elasticity problem for demonstrated the equivalence between traditional BEM and Fourier BEM.

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The integral square deviation between kernel setimates of Bernoulli regression function is investigated. Its asymptotic proprties is studied for grouped data.

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The integral square deviation between kernel setimates of Bernoulli regression function is investigated. Its asymptotic proprties is studied for grouped data.

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It is well known that, under some additional set-theoretical axioms, many interesting and exotical objects on real line R can be constructed. In this thesis our discussion is devoted to certain paradoxical subsets of R, in particular, Luzin sets and Sierpiński sets. These sets have many applications in real analysis, measure theory, general topology, and modern set theory. Luzin sets were constructed by Luzin in 1914, and Sierpiński sets were constructed by Sierpiński in 1924. Both Luzin and Sierpiński worked under the assumption of the Continuum Hypothesis (CH). These sets are dual objects from the point of view of Lebesgue measure and Baire category (see, for instance, [1] and [3]). We consider the above-mentioned paradoxical subsets of R and analyze these sets from the point of view of measure and category. (a) There exists a translation invariant measure μ on R which extends the Lebesgue measure λ and has the property that all Sierpiński subsets of R are measurable with respect to μ; moreover, all of them are of μ-measure zero. (b) If X is a λ-thick Sierpiński subset of R and λX is the induced measure on X, then the completion of the product measure λX ⊗ λX is not isomorphic to λX . (c) Any Luzin set Y is universal measure zero but no uncountable subset of Y has the Baire property.

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The present talk is devoted to results obtained within the framework of the project #30/28 finance by SRNSF, namely, to hierarchical models of layered prismatic shells and of structures consisting of two and three rectangularly linked bars. Bars and layers of prismatic shells are elastic and may be cusped ones

In [1] I. Vekua constructed hierarchical models for elastic prismatic shells based on expansions in the Fourier-Legendre series with respect to the thickness variable of the stress and strain tensors, and displacement vector components within the framework of the linear theory of elasticity. Generalizing this idea and using the double Fourier-Legendre series of the above mentioned physical and geometrical quantities, in [2] we constructed hierarchical models for elastic prismatic bars with variable rectangular cross-sections. The survey of further developments of these topics one can find in [3]. The present paper is devoted to static and dynamical boundary value-contact problems for two and three linked elastic bars with longitudinal axes parallel to the coordinate axes within the framework of the (0,0) approximation of hierarchical models [3]. They can be linked either really (in this case they may have different elastic constants) or mentally (in the case when two bars represent an entire (undivided) body). The paper is organized as follows. In Introduction the elastic structures under consideration are described. Section 2 covers governing equations. In Section 3 boundary value-contact problems are formulated and solved in the explicit form. In Section 4 perpendicularly linked prismatic bars loaded by self-weight are treated. In Section 5 numerical examples are presented. The results obtained are summarized in Conclusion. [1] I.N. Vekua, Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston-London-Melbourne 1985. [2] G.V. Jaiani, Z. Angew. Math. Mech., 81, 147 (2001). [3] G. Jaiani, Cusped Shell-like Structures. Springer, Heidelberg-Dordrecht-London- New York 2011.

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An effective linear mechanism responsible for the generation and transient intensification of large scale EM ULF waves in the shear flow is found. It has been shown that the shear flow driven wave perturbations effectively extract energy of the shear flow and temporally algebraic increasing own amplitude and energy (by several orders). With amplitude growth the nonlinear mechanism of self-localization is turned on and these perturbations undergo self organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation’s front. Depending on the features of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices, or dipole vortex, or vortex streets and vortex chains. From analytical calculation and plots we note that the formation of stationary nonlinear vortex structure requires some threshold value of translation velocity for both non-dissipation and dissipation complex ionospheric plasma. The space and time attenuation specification of the vortices is studied. The characteristic time of vortex longevity in dissipative ionosphere is estimated. The long-lived vortex structures transfer the trapped particles of medium and also energy and heat.

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In this talk we study correspondence between proof schemata and unranked logics. Proof schemata is a new formalism, an alternative to inductive reasoning, where cut-elimination theorem holds. Unranked logics are very important formalisms used in knowledge representation and semantic web. We describe a transformation, how an unranked logic sentence can be encoded into a formula schema.

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We discuss equilibrium confgurations of Coulomb potential of point charges on convex planar contour. In particular, for given n points on the contour we give an analytic criterion for existence of point charges for which the given conguration is an equilibrium. Using this criterion, detailed results are obtained for three and four charges on a circle and ellipse. Perspectives of using computer algebra in more complicated analogous problems will also be outlined.

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The concept of formula and term is introduced for this theory and theorems are proved

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The work is a continuation of the work [1]. It defines [2] rankless existential and universal quantifiers and formulate some of their properties [3,4]. ∃n+1xA1 ...AnA — (τx ∧n+1 A1 ...AnA/x) ∧n+1 A1 ...AnA It reads: ”there is such x propertiesA1,...,An, which has property A, n = 0, 1,.... ∀n+1xA1 ...AnA — ¬∃n+1xA1 ...An¬A It reads: every x properties A1,...,An has a property A,n = 0, 1,.... Note that if the superscripts are removed from the above-defined operators, then we get rankless quantifiers (1) ∀xA1 ...AnA → (T /x)[[∧A1 ...An] → A] (2) ¬∀xA1 ...AnA ↔ ∃xA1 ...An¬A (3) ∀xA1 ...An¬A ↔ ¬∃xA1 ...AnA (4) ∀xA1 ...An[A ∧ B] ↔ [∀xA1 ...AnA ∧ ∀xA1 ...AnB] (5) ∃xA1 ...An[A ∨ B] ↔ [∃xA1 ...AnA ∨ ∃xA1 ...AnB] (6) ∀xA1 ...AnA ↔ ∀x[[A1 ...An] → A] (7) ∃xA1 ...AnA ↔ ∃x[[A1 ∧ ... ∧ An] ∧ A] (8) ∀xA1 ...An[A ∨ B] ↔ [A ∨ ∀xA1 ...AnB] (9) ∀xA1 ...An[A ∧ B] ↔ [A ∧ ∀xA1 ...AnB] Если ∀x[A ↔ B], тогда (10) ∀xA1 ...AnA ↔ ∀xA1 ...AnB (11) ∃xA1 ...AnA ↔ ∃xA1 ...An References [1] Рухая Х. М., Тибуа Л. М., Чанкветадзе Г. О., Миканадзе Г. М. Безранговая формальная математическая теория // Международная конференция Мальцевские чтения 2012, тезисы докладов. 2012. С. 32. [2] Пхакадзе Ш. С. Некоторые вопросы теории обозначений. Тбилиси: Изд. ТГУ, 1977. C. 195. [3] Бурбаки Н. Теория множеств. М.: Наука, 1965. C. 3–13. [4] Rukhaia Kh. М., Tibua L. М. One Method of constructing a formal system // Applied Mathematics, Informatics and Mechanics. 2006. Vol. 11, № 2. P. 3–15

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Humidity processes, in particular, a stratus cloud and radiation fog, originating in the mesoscale atmospheric boundary layer, were numerically modeled. The influence on the process under consideration of such fundamental parameters as atmospheric stratification, relative humidity, and underlying surface temperature has been studied. The corresponding conclusions are made.

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A representation of the solution of second-order ordinal differential equation with random coefficients and random right side is given.

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In the talk the existance of independent families of convex sets of various type is shown and estimations of cardinalities of these families are established.

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A multi-layered model is constructed and investigated. Layers can have both blunt and sharp sharpness

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The asymptotic properties of a general functional of the Gasser–Muller estimator are investigated in the Sobolev space. The convergence rate, consistency, and central limit theorem are established.

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In the paper the connection of the blood flow with the amount of the cancer cells and proteins in the blood is investigated at the small arteriole level. Cancer proteins change viscosity and density of blood. The oxygen consumption process is described by the Stokes system and depends on viscosity and density of blood plasma. It is shown that when viscosity and density grow, oxygen consumption rate decreases. The velocity profile of oxygen consumption is constructed by using Maple for the different parameters.

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In this talk the 2D fully coupled quasi-static theory of poroelasticity for materials with double porosity is considered. For these equations the fundamental and some other matrixes of singular solutions are constructed in terms of elementary functions. The properties of single and double layer potentials are studied

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The Liouville-type theorems for the Karleman-Vekua irregular systems of equations are proved, which is a generalization of these types of theorems for a single equation.

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In the present paper gas pressure and flow rate distribution along the branched pipeline is investigated. The study is based on the analytical solution of the simplified nonlinear, nonstationary partial differential equations describing gas quasi-stationary flow in the branched pipeline. The effective solutions of the quasi-stationary nonlinear partial differential equations are presented. Preliminary numerical calculations have shown efficiency of the suggested method

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For the solution of a boundary value problems and boundary-contact problems of elasticity in polar coordinates system is formulated the boundary element method, namely the fictitious load method, for domain limited with axes of system polar coordinates. Circular boundary is divided on the small size arcs and linear part is divided on the small size segments.

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The talk is devoted to the system of degenerate partial differential equations arise from the investigation of elastic two layered prismatic shells. Harmonic vibration of cusped double-layered plates is consider. The weak setting of the BVPs in the case of the zero approximation of hierarchical models is considered.

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The paper presents some results of numerical investigations of airflow in the troposphere above the territory of Georgia based on a 3-D hydrostatic non-stationary numerical model for the meso-scale atmospheric processes. The upper boundary of the calculated domain is simulated by the free surface and on the lower boundary the condition of air particles slipping is used. The problem numerically is solved by the two-step Lax-Wendroff method. Numerical experiments have been performed in case of both western and eastern air flow invasion on the territory of Georgia with modeled isolated obstacle and real relief of Caucasus. Some results of numerical calculations are presented and analyzed

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The multi-dimensional system of nonlinear partial differential equations is considered. In two-dimensional case this system describes process of vein formation in higher plants. Variable directions finite difference scheme is constructed. Absolute stability and convergence of this scheme are studied. Rate of convergence is given. Various numerical experiments are carried out. Comparison of numerical experiments with the results of the theoretical investigation is given too. The appropriate graphical illustrations are given.

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The model which is based on Maxwell system that describes the electromagnetic field diffusion process into a substance is investigated. Large time behavior of solutions of corresponding initial-boundary value problems as well as numerical solution of considered problems are studied.

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For difference and ordinary delayed argument differential equations, the conditions of the oscillation of the solutions, which do not occur in the case of the ordinary equation, are studied.

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In this paper we have elaborated and configured Weather Research Forecast - Advanced Researcher Weather (WRF-ARW) model in the GRID system for Caucasus region considering geographical-landscape character, topography height, land use, soil type and temperature in deep layers, vegetation monthly distribution, albedo and others. Porting of WRF-ARW application to the grid was a good opportunity for running model on a larger number of CPUs and storing large amount of data on the grid storage elements. The WRF was compiled on the platform Linux-x86. Simulations were performed using a set of 2 domains with horizontal grid-point resolutions of 15 and 5 km, both defined as those currently being used for operational forecasts The coarser domain is a grid of 94x102 points which covers the South Caucasus region, while the nested inner domain has a grid size of 70x70 points mainly territory of Georgia. Both use the default 31 vertical levels. We have studied the effect of thermal and advective-dynamic factors of atmosphere on the changes of the West Georgian climate. Some results of calculations of the interaction of airflow with complex orography of Caucasus

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Constraint logic programming is one of the most successful areas of logic programming, combining logical deduction with constraint solving. I The main technique used in constraint logic programming research is introducing a new constraint domain, designing an efficient satisfiability and solving procedure for it, and putting it in the general constraint logic programming framework. I The domain we studied is the domain of sequences and contexts. Constraint logic programming over this domain is denoted by CLP(SC).

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The integral square deviation between kernel setimates of Bernoulli regression function is investigated. Its asymptotic proprties is studied.

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In this paper some effects of thermal and advective-dynamic factors of atmosphere on the background of the west Georgian climate have been studied. Mainly on the basis of numerical integration of full system of hydrodynamic equations main features of the atmospheric currents changeability while air masses were transferred from the Black Sea to the land’s surface had been investigated. Some results of numerical simulation of the air flow dynamics in the troposphere in conditions of large-scale undisturbed background flow are presented. It was shown that non-proportional warming of the Black Sea and Colkhi lowland provokes the intensive strengthening of circulation processes and effect of climate change on the territory of western Georgia

The present talk is devoted to the system of degenerate partial differential equations that arise from the investigation of elastic two layered prismatic shells. The well-posedness of the boundary value problems (BVPs) under the reasonable boundary conditions at the cusped edge and given displacements at the non-cusped edge is studied. The classical and weak setting of the BVPs in the case of the zero approximation of hierarchical models is considered.

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The quantum billiard problem in the stripe of the hexagonal configuration is considered. The corresponding 2D Helmholtz equation is studied with the homogeneous condition. By means of the conformal mapping method the equation is reduced to the linear elliptic equation in the rectangle. By means of the method of small parameter the approximate solution is obtained. This solution represents the wave function of the electron in the hexagonal stripe.

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The article focuses on the discussion of basic approaches to hypotheses testing, which are Fisher, Jeffreys, Neyman, Berger approaches and a new one proposed by the author of this paper and called the constrained Bayesian method (CBM). Wald and Berger sequential tests and the test based on CBM are presented also. The positive and negative aspects of these approaches are considered. Namely, it is shown that CBM has all positive characteristics of the above-listed methods. It is a data-dependent measure like Fisher’s test for making a decision, uses a posteriori probabilities like the Jeffreys test and computes error probabilities Type I and Type II like the Neyman-Pearson’s approach does. Combination of these properties assigns new properties to the decision regions of the offered method. In CBM the observation space contains regions for making the decision and regions for no-making the decision. The regions for no-making the decision are separated into the regions of impossibility of making a decision and the regions of impossibility of making a unique decision. These properties bring the statistical hypotheses testing rule in CBM much closer to the everyday decision-making rule when, at shortage of necessary information, the acceptance of one of made suppositions is not compulsory. Computed practical examples clearly demonstrate high quality and reliability of CBM. In critical situations, when other tests give opposite decisions, it gives the most logical decision. Moreover, for any information on the basis of which the decision is made, the set of error probabilities is defined for which the decision with given reliability is possible.

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In mathematical modeling of many natural processes nonlinear nonstationary differential models are received very often. One such model is obtained at mathematical modeling of processes of electromagnetic field penetration in the substance. For thorough description of electromagnetic field propagation in the medium, it is desirable to take into consideration different physical effects, first of all heat conductivity of the medium has to be taken into consideration. In this talk difference schems for such systems are discussed.

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Sensitivity analysis of the differential equation consists in finding an analytic relation between solutions of the original and perturbed equations. It is an important tool for assessing properties of the mathematical models. For example, in an immune model , it allows one to determine dependence of viruses concentrations on the model parameters. In the present work linear representation of the first order sensitivity coefficient is obtained with respect to perturbations of the initial data.

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Languages ​​in which functional or predicate symbols do not have a fixed arity (locality), in In recent years, they have been the subject of intensive research on due to the rather wide scope of their applicability [1]. Definition of operators occurs within the rational rules of introduction derivative operators of Shalva Pkhakadze [2]. Based on them in rank-free egalitarian theory, analogues have been proved results obtained in the egalitarian theory of N. Bourbaki [3].

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Initial-boundary value problem with mixed boundary conditions is considered for one nonlinear integro- differential equation with source term. Considered model is based on Maxwell’s system describing the process of the penetration of a magnetic field into a substance. Semi-discrete and finite difference schemes are studied. Attention is paid to the investigation more wide cases of nonlinearity than already were studied. Existence, uniqueness and long-time behavior of solutions are given too.

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The purpose of this talk is to consider the three-dimensional version of the full coupled theory of elasticity for solids with double porosity and to solve explicitly the Dirichlet and Neumann BVPs of statics in the full coupled theory for a sphere and for an elastic space with spherical cavity. The explicit solutions of these BVPs are represented by means of absolutely and uniformly convergent series.

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In the present talk the solutions of the Kirsch’s type problems are considered by means of different refined theories (E. Reissner, A. Lourie, I. Vekua). The obtained results are compared with each other.

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The plane problem of elasticity for a polygonal domain with a rectilinear cut is considered under the condition that uniformly distributed stretching forces or normal displacements i. e.,under the conditions of the third modified problem of elasticity are prescribed on the external boundary of the domain, while the cut edges are free from external forces. For solving the problem, the methods of conformal mappings and those of the boundary value problems of analytic functions are used. Solutions are given effectively (analytically).

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In this talk we consider equations of equilibrium of the isotropic elastic plate. By means of Vekua’s reduction method, the system of differential equations for plates is obtained, when on upper and lower face surfaces displacements are assumed to be known. Then for finite systems of equations, namely for approximations N = 1 and N = 2 , the general solutions are found. The main problem has been solved.

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see http://www.viam.science.tsu.ge/others/gnctam/GeoMech5/abstracts.pdf pages 8-9

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As is known, definition of the stressed-deformed condition for designs having difficult geometry is an actual problem. Constuction, research and computer realization of corresponding algorithms of numerical calculations are also actual. In this paper some “bridge-form” multystructures studied having complicated geometry stress-deformed state. Particularly the boundary-contacted problem is considered. Two rectangle form membranes are united by a string; We consider classic linear boundary problems for membranes (Poissons equation), but for string nonlinear Kirchhoff type integro-differential equation (see, for example [1], [2]). Numerical methods (finite-difference methods) for studying the above – stated multistructural is stressed – deformed condition are used. Direct numerical methods are used for finding the function of a bend in central points, and the iterative method is applied to definition of numerical values of function of a bend of a string for the approached decision of nonlinear equation of Kirchhoff type. The account program in MATLAB is created and numerical experiments are made. Acknowlegment. The designated protect has been fulfilled by financial support of the Rustaveli Science Foundation (Grant project # 30/28). References 1. Kirchoff G. Vorlesungen uber Mechanik. Teuner, Leipzig, 1883. 2. Peradze J. A numerical algorithm for the nonlinear Kirchhoff string equation. Numer. Math. 2005, 102, pp. 311-342.

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Using an analytical (precise) solution of the interior and exterior boundary value problem of elasticity for an ellipse and its corresponding exterior problem a solution of a boundary value problemo for elasticity is constructed for a confosale lliptic ring and its parts.

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see http://www.viam.science.tsu.ge/others/gnctam/GeoMech5/abstracts.pdf pages 14-15

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We investigate regularity properties of solutions to mixed boundary value problems for the system of partial differential equations associated with the theory of thermo-piezoelectricity (thermo-electro-elasticity) of piecewise homogeneous anisotropic elastic solid structures with interior and interface cracks. Using the potential method and theory of pseudodifferential equations we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the thermo-mechanical and electric fields are analyzed near the crack edges and near the curves, where different types of boundary conditions collide. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are analyzed in detail as well.

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This work is devoted to investigation of nonlinear dynamics of planetary electromagnetic (EM) ultra-low-frequency wave (ULFW) structures in the rotating dissipative ionosphere in the presence of inhomogeneous zonal wind (shear flow). Planetary EM ULFW appears as a result of interaction of the ionospheric medium with the spatially inhomogeneous geomagnetic field. The shear flow driven wave perturbations effectively extract energy of the shear flow increasing own amplitude and energy. These perturbations undergo self organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation's front. Depending on the features of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices, or dipole vortex, or vortex streets and vortex chains.

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Nonlinear dynamics of Rossby–Khantadze electromagnetic planetary waves in the weakly ionized ionospheric E-layer is investigated. Along with the prevalent effect of Hall conductivity for these waves, the latitudinal inhomogeneity of both the Earth’s angular velocity and the geomagnetic field becomes essential. It is shown that such short wavelength turbulence of Rossby–Khantadze waves is unstable with respect to the excitation of low-frequency and large-scale perturbations of the zonal flow and magnetic field. The nonlinear mechanism of the instability is driven by the advection of vorticity, leading to the inverse energy cascade toward the longer wavelength. The growth rate of the corresponding instability is found. It is shown that the generation of the intense mean magnetic field is caused by the latitudinal gradient of the geomagnetic field.

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Methods for taking into account the phase transitions of moisture used in numerical models of the atmospheric mesoboundary layer are given in detail. Examples of modeling stratus clouds and radiation fogs are given.

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At the human body blood supply all cells by nutrients, but cancer cells grow faster. When the volume of the solid cancer reaches the critical size it becomes dangerous. At this stage cancer cells begin to circulate in the blood and the viscosity and density of blood flow changes dramatically. The normal cells within the tumor volume are in the permanent oxidative stress and begin to die. The death rate depends on the tumor volume. We derived the formula which connects normal cell death rate with the tumor volume. The cases of spherical and cylindrical tumor volume are considered.

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The integral square deviation between kernel setimates of Bernoulli regression function is investigated. Its asymptotic proprties is studied.

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n the discussed theory "Taus", quantifier of exsitence and generality, did not point the locality. They are unranked operators. Similar results to the results obtained in the quantatic theory [3] by Burbakadze, were approved in the unranked quantatic theory.

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Interest in the problems of liquid filtration into the soil especially increased in the 20-th century. Namely, the problems related to the soil and subsoil water pollution by oil products, polluted waters, industrial wastes and pesticides (problem of mineral dressing of the soil, draining of swamp places, topics of water drainage from water reservoirs and hydroelectric stations, etc.), having practical values, were widely discussed and are studied nowadays too. In the present article a mathematical model of the accidental spilled liquid’s infiltration into the non-homogeneous structured in the vertical direction soil is discussed. The mathematical model is based on the integration of the non-linear and non-stationary hydrodynamic equation. The numerical model is taking into consideration the spilled liquid’s evaporation process, the main characteristic parameters of soil and some physical-chemical processes characterizing non-stationary processes in the soil. Some results of numerical calculations are presented

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New algorithms of the approached decision of Poisson equation (Dirichlet boundary problem) for a two-dimensional crosswise body by means of Schwartz iterative method are considered. The unknown function expands into the Fourier-Legendre series. Differences of Legendre polynomial are used as basic functions. The five-dot linear system of the algebraic equations concerning unknown coefficients is received. The program code (on the basis of Matlab) for the approached decision of the considered problem is created; corresponding numerical experiments are made which revealed stability of the account process.

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We introduce summability methods with a variable order and prove theorems connected with these methods. Namely, one of our theorems shows close connection between summabilities of an orthogonal series by these methods and the convergence of this series. Using above-mentioned theorem and such well-known results as Menshov-Rademacher, Menshov and Tandori ones we prove theorems on divergence of an orthogonal series almost everywhere by the introduced methods

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We prove uniqueness theorems for Rademacher series. Namely, we obtain the formulae for reconstruction of coefficients of the Rademacher single rearranged series by the sum of the series. As a result we strengthen such well-known theorems as Stechkin-Ul’yanov and Bakhshetsyan ones. It should be noted that mentioned formulae generalize well-known Fourier-Rademacher formulae. Indeed, using our formulae one can reconstruct the coefficients of the series for such case too if the values of the series sumare known only at the countable union of the appropriate pairs of points. Of course, Fourier-Rademacher formulae cannot be used for such case. We also prove the uniqueness of a Rademacher d-multiple series (d ≥ 2) if the values of the series sum are known only at the countable union of sets of the appropriate $2^d$ points. As a result we strengthen known theorem on the uniqueness of multiple series in the Rademacher system obtained by Mushegyan and Tetunashvili

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As our practice shows while transportation of natural gas by pipelines over the territory of Georgia , gas pressure and temperature changes cause formation of a liquid phase owing to partial condensation of the gaseous medium. In this paper a new mathematical model of hydrates origination in the main gas pipeline taking into consideration gas non-stationary flow is studied. For solving the problem of hydrates origination in the gas main pipeline taking into consideration gas non-stationary flow the system of partial differential equations describing this process is investigated. For learning the affectivity of the method one general test was created. Numerical calculations have shown efficiency of the suggested method

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see http://solmech2014.ippt.gov.pl/SolMech-2014.pdf pages 219-220

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For the quasi-linear neutral functional differential equation with the discontinuous initial condition variation formulas variation of solution are proved. In the variation formulas, the effects of delay function perturbation and discontinuous initial condition are detected.

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The present paper is devoted to construction and investigation of dimensional reduction algorithm for three-dimensional static and dynamical problems for multi-structures. Multistructures are elastic bodies, which consist of several parts with different geometrical shapes. In this paper applying spectral method we construct and investigate algorithm of approximation of three-dimensional boundary and initial-boundary value problems for elastic multi-structures by a sequence of problems defined on a union of domains with two different dimensions. Note that the constructed approximations of the three-dimensional problems can be treated as hierarchical mathematical models of multi-structures. We consider a multistructure, which is a junction of three-dimensional body with general geometrical shape and multilayered substructure clamped in it. The multilayered substructure consists of several layers, which are shells with variable thicknesses. Along the interfaces between threedimensional body and multilayered substructure and between adjacent shells rigid contact conditions, i.e. continuity of the displacement and stress vector-functions, are given. The three-dimensional models of shells we approximate by two-dimensional ones and construct static and dynamical hierarchical models of multi-structure defined on the union of threedimensional and several two-dimensional domains. We investigate the constructed hierarchical model reduction algorithm for static and dynamical models of multi-structure. More precisely, we prove the existence and uniqueness of solutions of the boundary and initial-boundary value problems corresponding to the constructed hierarchical models in suitable Sobolev spaces. Moreover, we prove the convergence of the sequence of vectorfunctions of three space variables restored from the solutions of the obtained pluri-dimensional problems to the solution of the original three-dimensional problem in corresponding spaces and if it satisfies additional regularity conditions we estimate the rate of convergence.

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In this paper, some results of numerical simulation of the air flow dynamics in the troposphere over the Caucasus Mountains taking place in conditions of nonstationarity of large-scale undisturbed background flow are presented. Main features of the atmospheric currents changeability while air masses are transferred from the Black Sea to the land’s surface had been investigated. In addition, the effects of thermal and advective-dynamic factors of atmosphere on the changes of the West Georgian climate have been studied. It was shown that non-proportional warming of the Black Sea and Colkhi lowland provokes the intensive strengthening of circulation and effect of climate cooling in the western Georgia.

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see http://www.aimsconference.org/conferences/2014/abstracts-book-finalized-2014-06-10.pdf p.518 A vibration problem for a double-layered cusped prismatic shell in the zeroth approximation of Vekua’s hierarchical models for elastic laminated prismatic shells is considered. The problem mathematically leads to the question of setting and solving BVPs for even order equations and systems of elliptic type with order degeneration in the statical case and of IBVP for even order equations and systems of hyperbolic type with order degeneration in the dynamical case. The well-posedness of the BVPs under the reasonable boundary conditions at the cusped edge and given displacements at the noncusped edge is studied. The classical and weak setting of the BVPs is considered. Appropriate weighted functional spaces are introduced. Uniqueness and existence results for the variational problem are proved. The structure of the constructed weighted space is described, its connection with weighted Sobolev spaces is established.

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In the present paper the linear theory of thermoviscoelasticity for Kelvin-Voigt materials with voids is considered. The fundamental solutions of the systems of equations of steady vibrations and quasi-statics are constructed by means of elementary functions and their basic properties are established. The formulas of representations of the general solutions for the systems of equations are established. The completeness of these representations of solutions is proved. The Green’s formulas and integral representations of Somigliana type of regular vector and classical solution are obtained. The uniqueness theorems of the internal and external basic boundary value problems of are proved. The basic properties of thermoelastopotentials and singular integral operators are established. Finally, the existence theorems for classical solutions of the above mentioned boundary value problems are proved by using the boundary integral equation method and the theory of singular integral equations.

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This article discusses some of the features of the SEE-GRID-SCI infrastructure and the implementation of the WRF-ARW model in the SEE-GRID-SCI infrastructure. Namely, on the implementation of the WRF-ARW model for the Caucasus region, some results of calculations in the SEE-GRID-SCI infrastructure are presented.

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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 constructingmodels of laminated structures. UsingVekua’s dimension reductionmethod 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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Various versions of hierarchical models of layered prismatic shells are constructed using Vekua's reduction method.

Book cover International Symposium on Functional and Logic Programming FLOPS 2014: Functional and Logic Programming pp 285–301Cite as Constraint Logic Programming for Hedges: A Semantic Reconstruction Besik Dundua, Mário Florido, Temur Kutsia & Mircea Marin Conference paper 604 Accesses 4 Citations Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8475) Abstract We describe the semantics of CLP(H): constraint logic programming over 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 give algebraic semantics of CLP(H) programs, define a sound, terminating, and incomplete constraint solver, and describe some fragments of constraints for which the solver returns a complete set of solutions.

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In the talk a theorem reveals an interrelation between the cardinality of a finite independent family of sets and the structure of constituents of an Euler-Venn diagram of this family produced by convex compact subsets of the Euclidean plane is presented.

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Static and dynamical boundary-contact problems for two rectangularly (with respect to their longitudinal axes) linked elastic bars with variable rectangular cross-sections are considered within the framework of the (0, 0) approximation of hierarchical models. They may have a contact interface either really (in this case the bars may have different elastic constants) or mentally (in the case when two bars represent an entire (undivided) body).

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The theorem proving techniques are divided into two parts, goal-directed and refutational. In this talk we present a goal-directed proof-search algorithm, which is based on a sequent calculus. Usually sequent calculus inference rules can be applied freely, producing a redundant search space. The technique, called focusing, removes this nondeterminism and redundancy in proof-search. Although we do not present a focused calculus, our algorithm is obtained according to the principles of focusing, achieving similar effect.

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In the present article a mathematical model of the accidental spilled liquid’s penetration into the soil having non-homogeneous structure in the vertical direction is discosed.The mathematical model is based on the integration of the non-linear and non-stationary systems of the hydrodynamic equations. The numerical model is taking into cosidiration the spilled liquid’s evaporation process, the main characteristic parameters of soil and some physical-chemical processes characterizing non-stationary processes in the soil. Some results of numerical calculations are presented

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A two-dimensional boundary value problem of elastic equilibrium of a planedeformed infinite body with an elliptic opening is studied. A part of the opening is fixed and from some points of the unfixed part of the cylindrical boundary there come curvilinear finite cracks. The problem is to find conditions for the fixed parts of the opening so that the damage caused by the crack, i.e. stresses on its surface, should be minimal. We should note that the crack ends inside the body are curved. The curve radii vary similar to boundary conditions.

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The article gives a new classification of foehns, descending warm winds. Among them, the foehn is especially interesting, in which the air rises uphill along the wet adiabat, and after transshipment it descends along the dry adiabat. In this case, the air heats up significantly. This is relevant from the point of view of agriculture, ecology, tourism. Numerical modeling of foehns is also considered in the framework of a flat model of the mesometeorological boundary layer of the atmosphere.

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Random measures and connected with them questions of absolute continuity under nonlinear transformations in abstract Hilbert space are considered. Estimation of solution of first order Differential Equations of adiabatic process with random measures parameters are given.

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The integral square deviation between kernel setimates of Bernoulli regression function is investigated. Its asymptotic proprties is studied.

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The purpose of this paper is to consider two-dimensional version of the full coupled theory of elasticity for solids with double porosity and effective solve the Dirichlet and Neumann BVPs of statics in the full coupled theory for an elastic circle. The explicit solutions of this BVPs are constructed by means of absolutely and uniformly convergent series.

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The problem of the plane theory of elasticity for a rectangular plate with a straight line cut is considered. Complex potentials to be searched based on complex analysis are effectively constructed (in analytical form) and their estimates are given near the ends of the slit (it is equal to 1/2). In the present report 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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Liouville-type theorems for the system of irregular equations of Carleman-Vekua is obtained.

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In this report, boundary value problems for elliptic systems in angular areas are considered.

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In this talk some basic boundary value problems for hierarchical models are solved.

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Cusped plates, on the one hand, are very important details from the practical point of view, such plates and beams are often encountered in spatial structures with partly fixed edges, e.g., stadium ceilings, aircraft wings, submarine wings etc., in machine-tool design, as in cutting-machines, planning-machines, in astronautics, turbines, and in many other areas of engineering (e.g., dams); on the other hand, their theoretical analysis and calculation are mathematically connected with the study of the problems for degenerate partial differential equations which are not covered by the general theory for degenerate partial differential equations. Some satisfactory results are achieved in this direction in the case of Lipschitz domains but in the case of non-Lipschitz domains there are a lot of open problems. To investigate such problems are the main part of the the present paper. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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The Caucasus is located in a seismic zone of high activity. The present paper discusses one mathematical model, based on which the issue of possible contamination of the territory of Georgia with radioactive substances in the event of an accident at the Armenian nuclear power plant will be studied. Some results of numerical counting are presented

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A biofilm is a complex gel-like aggregation of microorganisms like bacteria, cyanobacteria, algae, protozoa and fungi, embedded in an extracellular matrix polymeric substances (EPS). EPS develops resistance to antibiotics, to our immune system, to disinfectants or cleaning fluids. Biofilms can develop on surfaces which are in permanent contact with water, i.e. solid/liquid interfaces, but the growth of microorganisms also occurs in different types of interfaces such as air/solid, liquid/liquid or air/liquid. Biofilms are found everywhere: in industrial process, on medical devices, but also on the surface of monuments. We are interested on the formation and evolution of biofilms on fountains walls, i.e.: on stone substrates and under a water layer. These biofilms cause much damage, such as unaesthetic biological patinas, decoesion and loss of substrate-material from the surface of monuments or degradation of the internal structure. Since the topic is huge and of great interest, some mathematical models have already been proposed. At the beginning, mathematical modeling of biofilm was mainly focused on predicting growth balance, sometimes with practical applications in mind, as inWanner et al. (2006) and Wanner and Gujer (1984, 1986). These are generally 1D models with reaction– diffusion equations for nutrients and other substrates, sometimes with a moving boundary. The present talk is devoted to a vibration problem within the framework of a linearised (Ozeen’s type) 2D model of the boifilm occupying a thin domain

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Necessary optimality conditions of the initial data are obtained for the quasi-linear neutral functional differential equation with the continuous initial condition.

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There are some effective algorithms for solving the multi-dimensional problems. These algorithms mainly belong to the methods of splitting-up or sum approximation according to their approximate properties. Scheme of the variable directions are constructed and studied for Mitchison’s type multi-dimensional models.

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It is shown, within ZF&DC theory, that the existence of a Lebesgue nonmeasurable subset of R implies the existence of a partition of R into continuum many thick sets.

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The talk is devoted to simulations on computer of stress-strain state of laminated prismatic shells consisting of two elastic plies within the framework of the laminated prismatic shell model suggested in [1]. References 1. G. Jaiani. Hierarchical Models for Laminated Prismatic Shells, Abstracts

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In the papers [1,2] a problem of the rectangular plate deflection by thin rigid inclusion are brought to the smooth heart first type integral equation. Solution of this equation is defined in the singularized not integrated function class. It is possible that Jump function has the following not integrated singularity. The obtained integral equation in the papers [1,2] was solved by the spectral (orthogonal polynomials) method, namely Jacobs’ orthogonal polynomials was used. In this paper for approximate solution of the above mentioned integral equation a new algorithm using collocation method is suggested. Namely in this paper in case of uniformly ranged net a discrete singularity method [3] is used. By analogy with [1.2] we assumed that plane border is articulate leaned. By numerical calculations influence of the thin rigid inclusion’s length and mass on the values of the deflection function was investigated. Acknowlegment. This work was supported by Shota Rustaveli National Science Foundation (Grant AR/320/5-109/12). References 1. Онищук О.В., Попов Г.Я. О некоторых задачах изгива пластин с трещинами и тонкими включениями. –Изд. АН СССР. МТТ. 1980. 4. с. 141-150. 2. Попов Г.Я. Концентрация упругих наприажений возле штампов, разрезов. тонких включений и подкреплений. М. : Наука, 1982. 342 с. 3. Белоцерковский С.М., Лифанов И.К. Численные методы в сингулярных интегральных уравнениях. – М.: Наука, 1985. 256 с.

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The paper is devoted to an analytical solution construction of some applied thermoelasticity problems for multilayer cylindrical bodies when some of the layers may consist of an incompressible elastic material. Analytical solutions are obtained for the mentioned problems and on their basis an appropriate program is developed. For a body -layer along the radial coordinate and bounded by coordinate surfaces of a circular cylindrical system of coordinates static thermoelastic equilibrium is considered. On the flat boundaries of the cylindrical body boundary conditions of either symmetrical or antisymmetrical continuous extension of the solution are imposed. Between the layers contact conditions of rigid, sliding or other contact type may be defined. On the upper and lower cylindrical boundary surfaces arbitrary boundary conditions are given. The stated problems are analytically solved by the method of separation of variables. A general expression of the solution by means of harmonic functions is used. The solution of the involved problems is reduced to the solution of systems of linear algebraic equations with block-diagonal matrices.

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Hierarchical models are constructed when on the face surfaces of the laminated body either displacement vector components or mixed stress and displacement vector components are assumed to be known. The analysis of well-posedness of boundary value problems, in general, non-classical ones, is carried out (for more details see: http://www.viam.science.tsu.ge/others/gnctam/GeoMech4/abstracts.pdf pages 8-9).

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Nowadays pipelines have become the most popular means for natural gas transportation. Main reasons of pipeline constipation (emergency shutdown) are the following: generation of hydrates, freezing of water slugs, contamination and so on. Unfortunately practice shows that while natural gas is transported by pipelines changeability of gas pressure and temperature are causing formation of the gas liquid phase. There are several methods for avoiding gas hydrate problems: injection of thermodynamic inhibitors, use of kinetic hydrate inhibitors to sufficiently delay hydrate nucleation/intensification, and maintain pipeline operating conditions outside the hydrate stability zone by insulation, heating and controlling pressure. However, the above techniques needs knowledge of hydrates formation localization otherwise they may not be economical and practical. To take timely steps against generating of hydrates, it is necessary to study humidity and distribution of pressure and temperature. From existing methods the mathematical modeling with hydrodynamic method is more acceptable as it is very cheap and reliable and has high sensitive and operative features. In the present paper the problem of prediction of possible points of hydrates origin in the main pipelines taking into consideration gas non-stationary flow and heat exchange with medium is studied. For solving the problem of possible generation point of condensate in the pipeline under the conditions of non-stationary flow in main gas pipe-line the system of partial differential equations describing the hydrodynamics and thermal physics of a flow in the gas main in the presence of deposits of gas hydrate is investigated. Some results of numerical calculations for emergency detection are presented. Numerical calculations have shown efficiency of the suggested method. Acknowlegment. This work was supported by Shota Rustaveli National Scientific Foundation Grant #GNSF/ST09-614/5-21.

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We consider the beam equation with the conditions u(0)  u(l)=0, u ''(0)=u ''(l)=0 are some known constants. Equation beam is the stationary problem associated with the equation, which was proposed by Woinowsky- Krieger (J. Appl. Mech., 17 (1950), 35-35) as a model of deflection of an extensible dynamic beam. By means of Green’s function, Problem reduce to nonlinear integral equation...

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In the present work, the generation of large-scale zonal flows and magnetic field by short-scale collision-less electron skin depth order drift-Alfven turbulence in the ionosphere is investigated. The self-consistent system of two model nonlinear equations, describing the dynamics of wave structures with characteristic scales till to the skin value, is obtained. Evolution equations for the shear flows and the magnetic field is obtained by means of the averaging of model equations for the fast-high-frequency and small-scale fluctuations. It is shown that the large-scale disturbances of plasma motion and magnetic field are spontaneously generated by small-scale drift-Alfven wave turbulence through the nonlinear action of the stresses of Reynolds and Maxwell. Positive feedback in the system is achieved via modulation of the skin size drift-Alfven waves by the large-scale zonal flow and/or by the excited large-scale magnetic field. As a result, the propagation of small-scale wave packets in the ionospheric medium is accompanied by low-frequency, long-wave disturbances generated by parametric instability. Two regimes of this instability, resonance kinetic and hydrodynamic ones, are studied. The increments of the corresponding instabilities are also found.

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In this paper we consider geometrically nonlinear non-shallow spherical shells. By means of I. Vekua 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 = 1 in the hierarchy by I. Vekua. The concrete problem is solved.

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The purpose of this paper is to be explicitly solved the Neumann type boundary value problem (BVP) of the linear equilibrium theory of thermoelasticity with microtemperatures for the sphere and for the whole space with a spherical cavity. The obtained solutions are represented as absolutely and uniformly convergent series.

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The article focuses on the discussion of basic approaches to hypotheses testing, which are Fisher, Jeffreys, Neyman, Berger approaches and a new one proposed by the author of this paper and called the constrained Bayesian method (CBM). Wald and Berger sequential tests and the test based on CBM are presented also. The positive and negative aspects of these approaches are considered on the basis of computed examples. Namely, it is shown that CBM has all positive characteristics of the above-listed methods. It is a data-dependent measure like Fisher’s test for making a decision, uses posteriori probabilities like the Jeffreys test and computes error probabilities Type I and Type II like the Neyman-Pearson’s approach does. Combination of these properties assigns new properties to the decision regions of the offered method. In CBM the observation space contains regions for making the decision and regions for no-making the decision. The regions for no-making the decision are separated into the regions of impossibility of making a decision and the regions of impossibility of making a unique decision. These properties bring the statistical hypotheses testing rule in CBM much closer to the everyday decision-making rule when, at shortage of necessary information, the acceptance of one of made suppositions is not compulsory. Computed practical examples clearly demonstrate high quality and reliability of CBM. In critical situations, when other tests give opposite decisions, it gives the most logical decision. Moreover, for any information on the basis of which the decision is made, the set of error probabilities is defined for which the decision with given reliability is possible.

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The article focuses on the discussion of basic approaches to hypotheses testing, which are Fisher, Jeffreys, Neyman, Berger approaches and a new one proposed by the author of this paper and called the constrained Bayesian method (CBM). Wald and Berger sequential tests and the test based on CBM are presented also. The positive and negative aspects of these approaches are considered on the basis of computed examples. Namely, it is shown that CBM has all positive characteristics of the above-listed methods. It is a data-dependent measure like Fisher’s test for making a decision, uses posteriori probabilities like the Jeffreys test and computes error probabilities Type I and Type II like the Neyman-Pearson’s approach does. Combination of these properties assigns new properties to the decision regions of the offered method. In CBM the observation space contains regions for making the decision and regions for no-making the decision. The regions for no-making the decision are separated into the regions of impossibility of making a decision and the regions of impossibility of making a unique decision. These properties bring the statistical hypotheses testing rule in CBM much closer to the everyday decision-making rule when, at shortage of necessary information, the acceptance of one of made suppositions is not compulsory. Computed practical examples clearly demonstrate high quality and reliability of CBM. In critical situations, when other tests give opposite decisions, it gives the most logical decision. Moreover, for any information on the basis of which the decision is made, the set of error probabilities is defined for which the decision with given reliability is possible.

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The talk is dedicated to the Vekua type hierarchical models then on the face surfaces either displacements or stresses or mixed conditions are given.

Activity of anthropogenic factors resulted inthe considerable change of the area of underlying surface and water supplies inGeorgia. Namely there are observed decreasing of the following units:mowing, arable, unused lands, waterresources, shrubs and forests, owing to increasing of the production and building.Transformation of one type structural unit into another one, naturally, resultsin local climate change. Problem of desertification takes one of the importantplaces in the cycle of climate…

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The term of sustainable development means such development which gives the greatest pos­sible economic effect at high ecological and social guarantees [Susta­inable Management in Action (2005)]. For achievement of these pur­­po­ses very often there are used approaches based on risk assessment [Ribatet et al. (2008)]. General principle of this ap­pro­ach con­sists in planning such action to which correspond the maximal gua­ran­tees of obtaining desirable results actually. As a rule real processes are influenced by sets with the nature random factors, these processes are ran­dom and for their correct description and obtaining authentic solutions, to use the me­thods of probability theory and mathematical statistics is necessary. In the work we use the approach based on the method of testing statistical hypotheses, in particular Bayesian approach of testing ma­ny hypotheses [Kachiashvili et al. (2012), Kachiashvili and Mueed (2011)].

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The Comparison analysis of the Wald’s and Bayes-type sequential methods for testing hypotheses is offered. The merits of the new sequential test are: universality which consists in optimality (with given criteria) and uniformity of decision-making regions for any number of hypotheses; simplicity, convenience and uniformity of the algorithms of their realization; reliability of the obtained results and an opportunity of providing the errors probabilities of desirable values. There are given the Computation results of concrete examples which confirm the above-stated characteristics of the new method and characterize the considered methods in regard to each other.

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Pollution of the environment by solid, liquid and gas contaminants of artificial origin have been in the center of heightened attention of scientists for several decades and represents one of the most important problems of the contemporary science. Putting this problem forward is conditioned by the fact that most contaminants have a harmful effect on the health of living organisms, on the hygiene of the biosphere and on the regional and global climate alteration which represents one of the main problems of present days. The oil spillage caused by oil transportation by pipeline and railway results in serious deterioration of environment. Namely in the result of accidental spilling the oil can be spreading over the dry surface, it can cover the holes and depressions, it may be found in river through the peculiarities of relief and afterwards it can be transported into the seawater. As rule, the oil spilling, spreading and flowing over the dry surface is following by the oil penetration in subsoil. If this phenomenon proceeds intensively, then it may be happened to revealed oil in groundwater and it may be transported by groundwater flows. All these processes have very seriously impact on environment, and therefore, on human health. So it is very important to carry out preventive model studies of possible emergency situations. In this article distribution of petroleum and mineral oil into the soils in case of their emergency spilling on the flat surface containing pits is studied and analyzed. The behaviors of the infiltration process and diffusion parameters are studied. Also spreading of the spilled oil in the Georgian Black Sea coastal zone on the basis of a 2-D numerical model of oil distribution in the seawaters is simulated. Some results of numerical experiments are presented. Numerical experiments were carried out for different hypothetical sources of pollution in case of different sea circulation regimes dominated for the four seasons in the Georgian Black Sea coastal zone. Results of calculations have shown that risk of surface and subsurface waters pollution owing to oil emergency spilling is high

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Let us consider a solid crystal growth which is accompanied with a chemical reaction at the crystallizer. Crystallization is a chemical solid-liquid separation technique in which mass transfer of a solute from the liquid solution to a pure solid crystalline phase occurs (at the definite temperature). So it's formation depends on solubility conditions of the solute in the solvent, simultaneously with the chemical reaction . The crystallization process consists of three events :nucleation, crystal growth and supersaturation. When the supersaturation exhausted the solid-liquid system reaches equilibrium and crystallization is complete. Let at the definite temperature the single prismati begin to grow. This process could be described by the nonlinear reaction-diffusion equation with a moving initial-boundary conditions The unknown function is the supersaturation We suppose that supersaturation depends on time exponentially and reduce the initial problem to the Helmholtz equation with the homogeneous boundary conditions. By means of the conformal mapping method in which one parameter is small we reduced the initial problem to the linear elliptic differential equation and by the separation of variables we obtain the exact solutions.

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The present paper gives an up-dated exploratory survey of investigations concerning elastic cusped rods (i.e., the areas of their cross-sections equal zero at least at one rods’ ends), cusped plates, prismatic and standard shells (i.e., their thickness vanishes at least on some parts of their boundaries. The study of elastic cusped rods, plates, prismatic and standard shells within the framework of classical and so called refined theories mathematically leads to analysis of correct setting boundary value problems for even order ordinary and elliptic equations and systems with the equations order degeneration in the static case and of initial boundary value problems for even order equations and systems of hyperbolic type with the equations order degeneration in the dynamical case (in other words, the order of equations becomes less on some subsets of the boundary). Therefore, boundary conditions are not possible to set in the usual form in all the cases of sharpening (tapering) geometry.

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In this paper an algorithmic process to essh the velidity of the formulae with some property of the unranced logic is constructed.

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The main issue of the contemporary ecological problems is atmosphere purity and it’s prediction in space and time. Development of a method for air pollution prediction is based on studying of the spreading peculiarity of detrimental contaminants from the source of origination. The quality of the air pollution by detrimental contaminants depends not only on technological and constructive parameters, but on the factors like wind speed, atmospheric stratification, orography, etc. As a result, study of meteorological aspects of environmental pollution has been of significance. This article discusses some features of contamination with radioactive substances. Among the latter there is a clear example of radioactive contamination caused by nuclear explosions at nuclear power plants.

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In the paper 3D mixed boundary value problem of elasticity theory for the orthotropic beam with a rectangular cross-section is studied. By means of the Vekua theory the problem is reduced to two dimensional problem. The numerical solution is obtained by means of the finite difference schemes. The initial problem is reduced to the system of algebraic equations. The convergence of the iteration process is proved, the error is estimated.

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For the solution of a problem of elasticity, is formulated a numerical method which is called the method of a boundary elements, namely a method of fictitious loads. Considered body presents to the domain limited by coordinate axes of parabolic systems of coordinates. Curvilinear boundary is divided on the small size curves. In this case the considered domain is described more precisely, than in the case when the boundary domain is divided on small segments, and, as a result, the decision turns out more exact. For receiving this method of a boundary element there are used transformations differential and integrated expressions in curvilinear coordinates.

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A certain version of the Erdos problem is studied. More precisely, it is proved that there does not exist a finite constant c such that each plane set with the outer Lebesgue measure greater than c contains the vertices of a triangle of area 1. It is shown that a sentence ”each plane set E with Lebesgue outer measure +∞ contains the vertices of a triangle of area 1” is independent from the theory (ZF)&(DC). The Erdos problem is studied for the shy-measure in an infinitedimensional separable Banach space and it is established that any number from the interval [0,1[ is Erdos constant for such a measure. It is constructed an example of a thick (in the sense of shyness) subset of 2 l which does not contain vertices of a triangle of area 1.

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Viscoelastic materials play an important role in many branches of engineering, technology and, in recent years, biomechanics [1]. One of the simplest mathematical models constructed to describe the viscoelastic effects is the classical Kelvin-Voigt model [2]. The basic boundary value problems (BVPs) of the classical theories of viscoelasticity and thermoviscoelasticity for Kelvin-Voigt materials are investigated in [3]. The modern theories of viscoelasticity and thermoviscoelasticity for materials with microstructure have been a subject of intensive study in the last decade [4, 5]. The theory of thermoviscoelasticity for Kelvin-Voigt materials with voids is presented in [6]. Recently, the theory of thermoviscoelasticity for Kelvin–Voigt microstretch composite materials is introduced in [7]. The uniqueness and existence theorems of the BVPs of steady vibrations in the theory of viscoelasticity for Kelvin-Voigt materials with voids are proved in [8]. In the present paper the linear theory of thermoviscoelasticity for Kelvin-Voigt materials with voids [6] is considered and some basic results of the classical theory of thermoelasticity are generalized. Indeed, the fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions. The Green's formulas in the considered theory are obtained. The uniqueness theorems of the internal and external basic BVPs are proved. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The formulas of integral representations of Somigliana type of regular vector and regular (classical) solution are obtained. The basic properties of thermoelastopotentials and singular integral operators are given. Finally, the existence theorems for classical solutions of the basic BVPs of steady vibrations are proved by using of the potential method (boundary integral equations method) and the theory of singular integral equations.

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For integral functionals of the Gasser–Muller regression function and its derivatives, we consider the plug-in estimator. The consistency and asymptotic normality of the estimator are shown.

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We show that in Solovay’s model an arbitrary non-trivial closed ball in an infinite - dimensional non-separable Banach space of all real valued sequences ℓ ∞ is an infinitedimensionally Haar null set. This answers positively to the Problem 8 stated in [Shi H., Measure-Theoretic Notions of Prevalence, Ph.D.Dissertation (under Brian S.Thomson), Simon Fraser University, October 1997, ix+165] for non-separable Banach space ℓ ∞.

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We constructed estimates of maximal likelihood in infinite-dimensional Hilbert space. Some properties of consistency and asymptotic normality is investigated.

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The talk concerns 45 years long history of Ilia Vekua Institute of Applied Mathematics of Ivane Javaxishvili Tbilisi State University [1]. The Institute was founded by Georgian mathematician and mechanist Ilia Vekua on October 29, 1968. The aim of the Institute was to carry out research on important problems of applied mathematics, to involve University professors, teachers and students in research activities on topical problems of applied mathematics in order to integrate mathematics into the study processes and research, and to implement mathematical methodologies and calculating technology in the non-mathematical fields of the University. In 1978, the Institute was named after its founder and first director Ilia Vekua. At present, the Institute successfully continues and develops activities launched by his founder in four scientific directions: - Mathematical problems of mechanics of continua and related problems of analysis; - Mathematical modelling and numerical mathematics; - Discrete mathematics and theory of algorithms; - Probability Theory and mathematical Statistics. The institute sees its mission as threefold: - Carrying out fundamental and practical scientific research in applied mathematics, mathematical and technical mechanics, industrial mathematics and informatics, undertaking state and private sector contracts to provide expert services; - Offering the university a high-level computer technology base for University professors and teachers, research employees and students undertaking their scientific research activities; - Supporting PhD and post-graduate students to attain scientific grants, as well as through employment within the Institute and participation in scientific conferences. References 1. Ilia Vekua Insitute of Applied Mathematics of TSU. TSU Science. Vol. 2 (2012), 89-91.

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Prismatic and cusped prismatic shells are exposed. Relation of the prismatic shells to the standard shells and plates are analyzed. Cusped beams are defined. The Lipschitz boundaries are defined. Typical cross-sections of cusped prismatic shells and longitudinal sections of the cusped beams are illustrated. Moments of functions and their derivatives are introduced and their relations clarified.

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The oil spillage caused by oil transportation through pipeline and railway results in serious deterioration of soil, and as a consequence subsurface waters pollution. All these processes have very seriously impact on environment, and therefore, on human health. So it is very important to carry out preventive model studies of possible emergency situations. In this paper on the basis of nonlinear filtration equation of a liquid, petroleum and mineral oil distribution of into the soils in case of their emergency spilling on the flat surface containing pits is studied and analyzed

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In the present paper, the issues of the approximate solution of singular integral equation and pair systems of integral equations containing fixed-singularity are studied. The studied integral equations are obtained from the anti-plane problems of the elasticity theory for a composite (piece-wise homogeneous) orthotropic (in particular, isotropic) plane slackened by crack when it reaches or intersects the dividing boundary at the right angle. Algorithms of an approximate solution are designed by the collocation method, namely the method of discrete singularities. In both cases, (when the crack reaches or crosses the dividing border) behaviour of the solutions is studied and the stress intensity coefficients at the ends of the crack are calculated. Results of numerical computations are demonstrated. According to the obtained results, hypothetical predictions of the propagation of crack are made.

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The present talk is devoted to the construction and investigation of two-dimensional hierarchical models for biofilms occupying thin prismatic domains with variable thickness. We investigate the existence and uniqueness of solutions of the reduced twodimensional problems in suitable weighted Sobolev spaces. For the sake of simplicity we state the zero approximation; in the N-th approximation similar but more complicated assertions can be carried out.

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The paper is devoted to analytical solution construction of some application problems of thermoelasticity for a multilayer rectangular parallelepiped. There are solutions of some boundary value problems for a rectangular parallelepiped known in literature. In contrast with those works, our paper deals with a class of boundary value and boundary value contact problems, the great majority of which have been solved for the first time. It should be noted that some of the layers may consist of a non-contractible elastic material. Analytical solutions have been obtained and on their basis corresponding illustration are given. A description of a computer program is also presented by means of which computations and graph construction are performed.

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Investigation focuses on the consideration of basic approaches to hypotheses testing, which are Fisher, Jeffreys, Neyman, Berger approaches and a new one proposed by the author of this work and called the constrained Bayesian method (CBM). Wald and Berger sequential tests and the test based on CBM are presented also. The positive and negative aspects of these approaches are considered on the basis of computed examples. Namely, it is shown that CBM has all positive characteristics of the above-listed methods. It is a data-dependent measure like Fisher’s test for making a decision, uses a posteriori probabilities like the Jeffreys test and computes error probabilities Type I and Type II like the Neyman-Pearson’s approach does. Combination of these properties assigns new properties to the decision regions of the offered method. In CBM the observation space contains regions for making the decision and regions for no-making the decision. The regions for no-making the decision are separated into the regions of impossibility of making a decision and the regions of impossibility of making a unique decision. These properties bring the statistical hypotheses testing rule in CBM much closer to the everyday decision-making rule when, at shortage of necessary information, the acceptance of one of made suppositions is not compulsory. Computed practical examples clearly demonstrate high quality and reliability of CBM. In critical situations, when other tests give opposite decisions, it gives the most logical decision. Moreover, for any information on the basis of which the decision is made, the set of error probabilities is defined for which the decision with given reliability is possible.

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We consider for an elastic bodies of the dynamic boundary-value problems the theory of consolidation with double porosity.The problems are reduced to the corresponding problems for systems of equations for pseudooscillation by Laplace transformation relative to time. The solutions are represented in terms of metaharmonic functions. It is proved that the problem of pseudooscilation has a unique solution. Conditions are given for existece of inverse transformations that provide solutions for the initial problem.

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For the delay functional differential equation with the continuous initial condition variation formulas of solution are proved. Necessary optimality conditions are obtained for the initial data optimization problems.

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Cusped plates and beams, on the one hand, are very important details from the practical point of view, such plates and beams are often encountered in spatial structures with partly fixed edges, e.g., stadium ceilings, aircraft wings, submarine wings etc., in machine-tool design, as in cutting-machines, planning-machines, in astronautics, turbines, and in many other areas of engineering (e.g., dams); on the other hand, their theoretical analysis and calculation are mathematically connected with the study of very difficult problems for degenerate partial differential equations which are not covered by the general theory for degenerate partial differential equations. Some satisfactory results are achieved in this direction in the case of Lipschitz domains but in the case of non-Lipschitz domains there are a lot of open problems. To investigate such open problems is a main part of the the present talk. To this end there are used function-analytic, approximate and special methods (suitable to problems peculiarities). As results, boundary value problems in the zero approximation for I.Vekuas hierarchical models of cusped plates and beams in case of 3-D non-Lipschitz domains will be investigated.

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New methods of sequential analysis of Bayesian type of testing hypotheses are offered. The methods are developed on the basis of the specific statement of the Bayesian approach of testing hypotheses as constrained optimization problem instead of usual unconstrained problem [1, 2]. Unconstrained Bayesian statement of hypotheses testing minimizes the risk function which contains two types of errors: for incorrect rejection of hypotheses when they are correct and for incorrect acceptance of hypotheses when they are erroneous. We consider constrained statement of Bayesian approach which consists in the upper restriction of the probability of the error of one kind and the minimization of the probability of the error of the second kind. Such statement of the problem leads to the specificity of hypotheses acceptance regions. Particularly, in this case among of hypotheses acceptance sub-regions in the observation space we have the regions of the suspicion on the validity of several (more than one) tested hypotheses and the region of impossibility of acceptance of the tested hypotheses [3, 4]. Using these properties there are developed new sequential methods of testing hypotheses which give consistent, reliable and optimum results in the sense of the chosen criterion. The results of research of the properties of these methods are given. The examples of testing of hypotheses for the case of the sequential independent sample from the multidimensional normal law of probability distribution with correlated components are cited. On the basis of theses examples there are compared the Wald and the Berger [5, 6] and offered new sequential methods among them. The positive and negative sides of these approaches are considered on the basis of computed examples.

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In this paper two mathematical models for definition of gas accidental escape localization in the gas pipelines are suggested. The first model was created for leak localization in the horizontal branched pipeline and second one for leak detection in inclined section of the main gas pipeline. The algorithm of leak localization in the branched pipeline did 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 have been constructed. Some results of calculations for compound pipelines having two, four and five sections are presented. Also a method and formula for the leak localization in the simple inclined section of the main gas pipeline are suggested. Some results of numerical calculations defining localization of gas escape for the inclined pipeline are presented

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In the talk the Stokes flow over the ellipsoidal type bodies in a pipe is considered. The velocity of the flow is described by the axisymmetric Stokes system for the low Reynolds number with the appropriate boundary conditions. Effective solutions for different cases are obtained. The shear stresses and velocity are calculated. The graphics of the velocity profile and shear stresses are constructed.

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Pattern calculi extend the lambda calculus with patterns. Ÿ λ abstracts not only variables but also terms. Ÿ Pattern calculi integrate pattern matching capabilities into the λ-calculus. Ÿ Pattern calculi are expressive, but in general the confluence property is lost. Ÿ To recover confluence, some restrictions on patterns and their applications are imposed

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Cusped edges cannot be clamped (K-L, Vek) or be simply supported (K-L) in all the cases of sharpening. The corresponding (i.e., when it can be clamped or simply supported) criteria are established. In the case of geometrically non-linear (von-Kármán-Föppl type) model in contrast to linear model shearing forces and bending moments cannot be always correctly applied. The corresponding criteria are established.

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It is shown that At cusped edges mathematical moments of the displacement vector and stress tensor components become zero under assumption of boundedness of the displacement vector and stress tensor, respectively. If at cusped edges concentrated at a cusped edge (i.e., at a line) forces are applied then stresses become infinite (since the area of loading equals zero) but integrated stress vectors are finite at the cusped edge.

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Theorems on the continuous dependence of the solution on perturbations of the initial data and the right-hand side of equation are obtained.

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We consider hierarchical models for elastic prismatic shells, in particular, plates of variable thickness, when on the face surfaces displacements are known. 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.

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The structure of meso-scale air flow in the troposphere over the isolated model obstacle and real relief of Caucasus is investigated in conditions of nonstationarity of background undisturbed flow on the base of the 3-D hydrostatic numerical model. Calculations have shown that the non-stationary character of the undisturbed air flow can considerably change the structure of air flow above the mountain relief

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In the Earth atmosphere there are often observed non-periodical, non-ordinary air phenomenal events which are accompanying with material and even human damage. Such kind atmosphere phenomenal events may be: powerful wind vortex, strong local micro-orographic winds, different arising air currents in the atmosphere lower boundary layer and constantly dominated some regional geophysical “phenomenal” events. Over the territory of Georgia such kind of “phenomenal” events are observed over David Gareji depression and Surrami mountain plateau. In the present article on the bases of the hydrothermodynamic laws above mentioned phenomena is investigated. Namely it was proved that pressure in the wind vortex is arising proportionally with relief altitude and enlarged with augmentation of the angle between wind vortex axes and vertical direction. Also it was obtained that vertical component of the wind vortex was arising with altitude and it has exponential character. Obtained results are new and have as theoretical as well practical values

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It is put and solved numerically 2- dimentional (in a vertical plane x-z) a non-stationary problem about a mesometeorological boundary layer of atmosphere (MBLA). In it ecologically such actual processes, as a full cycle of development of a cloud and a fog and aerosol distribution are considered against of MBLA thermohydrodynamics. A number of abnormal meteoprocesses is simulated: Simultaneous existence of a stratus cloud and radiation fog; An incorporated vertical complex of a stratus cloud and radiation fog; Daily continuous overcast; Ensemble of humidity processes, particularly, three clouds and a fog which then were transformed to four clouds are simultaneously simulated. To the new the role of horizontal and vertical turbulence in formation of a tropical cyclone and a tornado and in mutual transformation of humidity processes are considered. Influence of some meteoparameters on aerosol distribution is investigated. Besides, such problems are resulted in a stage of computer realisation, as "secondary" pollution MBLA (capture already sedimented aerosols and its repeated carrying over); the account of cooling process, available on cloud and fog border; influence of cloudy shades on MBLA processes; the account of difficult temperature heterogeneity of an underlying surface

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The main goal of the project is to develop a nonlinear theory of wave structures in the atmospheres of the Earth and other planets such as Venus and Jupiter, incorporating all features mentioned above. The present proposal assumes the use of previous experience in this field gained by the members of the team. We plan to present the current state of research related to nonlinear Rossby waves, with a particular focus on the generation of large-scale eddies and zonal winds. Moreover, a special attention will be paid to the study of nonlinear planetary waves in the ionosphere. A part of the project will be dedicated to the interpretation of laboratory, numerical, and observational studies of Rossby waves in planetary atmospheres. A particular attention will be paid to the analysis of the ionospheric responses to man-made activities and extra-ordinary natural phenomena (earthquake, storms, hurricanes etc.).

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The work is a natural continuation of the article [1], in the sense that in order to create a general theory of programming should use the same method that is used for creation of notation theory [2]. In particular, for operators programming need to find a logical form and include them in a number of linguistic symbols of logical theory. As an example, the article summarizes the assignment operator, where the variable is assigned not only a numerical value, but also any term [3].

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The talk is devoted to an updated exploratory survey of results concerning elastic cusped shells, plates, and beams and cusped prismatic shell-fluid interaction problems. Mathematically, the corresponding problems lead to non- classical, in general, boundary value and initial-boundary value problems for governing degenerate elliptic and hyperbolic systems in static and dynamical cases, respectively, with the corresponding mechanical (physical) interpretations. Two principally different approaches of investigation are used: (1) to get results for 2D (two-dimensional) and 1D (one-dimensional) problems from results of the corresponding 3D (three-dimensional) problems and (2) to investigate directly governing degenerate and singular systems of 2D and 1D problems. In both the cases, it is important to study the relationship of 2D and 1D problems with 3D problems. On the one hand, it turned out that the second approach allows to investigate such 2D and 1D problems whose corresponding 3D problems are not possible to study within the framework of the 3D model of the theory of elasticity. On the other hand, the second approach is historically approved, since first the 1D and 2D models were created and only then the 3D model was constructed. Hence, the second approach gives a good chance for the further development (generalization) of the 3D model.

Spreading of oil pollution appearing as a result of an accidental release of the oil in the Georgian coastal area of the Black Sea is simulated. The real flow and turbulent fields are predetermined from the regional forecasting system, a core of which is a baroclinic regional model of the Black Sea dynamics. The splitting method is used for solution of the advection-diffusion equation of the diffusion model. Numerical experiments, which are carried out for different hypothetical pollution sources in case of different sea circulation modes, showed that character of regional circulation predetermines main features of spatial distribution of the oil pollution in the Georgian water area

Variable directions difference scheme for one nonlinear model of partial differential equations is con-sidered. The stability and convergence are studied. In two-dimensional case this system describes vein formation of young leaves. Numerical experiments for this two-dimensional case are done. These experiments agree with theoretical researches.

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In this talk one boundary value problem of Q-holomorphic vectors are solved.

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In the present talk the linear 2D theory of thermoelasticity with microtemperatures is considered. The representation of regular solution of the system of equations of steady vibrations in the considered theory is obtained. The fundamental and singular solutions for a governing system of equations of this theory are constructed. Finally, the single-layer, double-layer and volume potentials are presented.

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In the present talk by means of the I. Vekua method the system of differential equations for the nonlinear theory of non-shallow spherical shells is obtained. Using the method of the small parameter approximate solutions of I. Vekua’s equations for approximations N=0 is constructed. The small parameter ε = h/R, where 2h is the thickness of the shell, R is the radius of the sphere. Concrete problem is solved.

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There exist several methods of reduction of the three-dimensional problems to the two-dimensional one.

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Study of boundary value problems for the composite bodies weakened by cracks has a great practical significance. Mathematical model investigated boundary value problems for the composite bodies weakened by cracks in the first approximation can be based on the equations of anti-plane approach of elasticity theory for composite (piece-wise homogeneous) bodies. When cracks intersect an interface or penetrate it at all sorts of angle on the base of the integral equations method is studied in the works [1]–[3]. In the present article finite-difference solution of anti-plane problems of elasticity theory for composite (piece-wise homogeneous) bodies weakened by cracks is presented. The Mathematical Modeling differential equation with corresponding initial boundary conditions is approximated by finite-differential analogies in the rectangular quadratic area. Such kind set of the problem gives opportunity to find directly numeral values of shift functions in the grid points. The suggested calculation algorithms have been tested for the concrete practical tasks. The results of numerical calculations are in a good degree of approach with the results of theoretical investigations. References [1] A. Papukashvili, Antiplane problems of theory of elasticity piecewice-homogeneous orthotropic plane slackened with cracks. Bull. Georgian Acad. Sci. 169 (2004), No. 2, 267–270. [2] A. Papukashvili, About of one boundary problem solution of antiplane elasticity theory by integral equation methods. Rep. Enlarged Session Semin. I. Vekua Inst.Appl. Math. 24 (2010), 99–102. [3] A. Papukashvili, M. Sharikadze, and G. Kurdghelashvili, An approximate solution of one system of the singular integral equations. Rep. Enlarged Session Semin. I. Vekua Inst. Appl. Math. 25 (2011), 95–98.

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In the present article the solution of the composite (piece-wise homogeneous) bodies weakened by cracks using finite-difference method is studied. The plane is changed by a large square and the differential equations with boundary conditions are approximated by differential analogies. Such kind of statement of the problems gives opportunity to find numerical values of the stress functions in the grid points. The corresponding algorithms are composed and realized for the concrete practical tasks. The results of theoretical and numerical investigations are in a good conformation and are presented.

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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, as example, an unidirectional lamina with fibers parallel to -axis under shear strain is considered. Tension-compression is treated as well.

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The article presents a numerical model of the mesometeorological boundary layer of the atmosphere (MBLA), taking into account the propagation of aerosol in it. By varying the boundary conditions at the lower boundary of theMBLA , it is possible to simulate the secondary pollution of the environment.

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New estimate for the Bernoulli regression function is constructed. It is investigated some asymptotic properties of the estimate.

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Plasma vortices are often detected by spacecraft in the magnetospheric environment, for instance in the magnetosheath and in the magnetotail. Large scale vortices may correspond to the injection scale of turbulence, so that understanding their origin is important for understanding the energy transfer processes in the geospace environment. In a recent work, the THEMIS mission has detected vortices in the magnetotail in association with the strong velocity shear of a substorm plasma flow (Keiling et al., J. Geophys. Res., 114, A00C22 (2009), doi:10.1029/2009JA014114), which have conjugate vortices in the ionosphere. By analyzing the THEMIS data for that event, we find that several vortices in the magnetotail can be detected together with the main one, and that the vortices indeed constitute a vortex chain. The analysis is carried out by analyzing both the velocity and the magnetic field measurements for spacecraft C and D, and by obtaining the corresponding hodograms. It is found that both monopolar and bipolar vortices may be present in the magnetotail. The comparison of observations with numerical simulations of vortex formation in sheared flows is also discussed.

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Nonlinear interaction of the planetary electromagnetic Rossby-Khantadze waves with the local ionospheric shear zonal winds is investigated for the Earth's ionospheric E-layer.It is found that such interaction leads to the formation of solitary vortical structures moving along the latitude circles westward as well as eastward with the velocity different from the phase velocity of corresponding linear waves. The vorticies are weakly damped and long-lived. They cause geomagnetic pulsation stronger than the corresponding linear waves.

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The limiting distribution of the integral square deviation of a kernel type nonparametric estimator of the Bernoulli regression function is established.

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At present pipelines have become the most popular means of transporting natural gas. There are many scientific articles denoted to the problem of prediction of possible points of hydrates origin in the main pipelines for gas stationary flow. In this paper a mathematical model of hydrates origination in the gas main pipeline taking into consideration gas non-stationary flow is studied. For solving the problem of hydrates origination in the gas main pipeline taking into consideration gas non-stationary flow the system of partial differential equations is investigated. For learning the affectivity of the method one general test was created. Numerical calculations have shown efficiency of the suggested method. The results of numerical calculations are given

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Constructing the mathematical model for any kind of epidemic disease is an important task. One of the simple epidemic models is known as “SIR epidemic model”. According to this model, the host population is divided into three parts by their epidemiological status: “S” – susceptible - neither immune nor infected, “I”- infectives and “R”-the ones, who are recovered from the disease. The total size of host population is then N=S+I+R. As soon as disease occurs, individuals start to move to the susceptible class to the infective class, to the recovered class. This movement can be represented with respect to time by the differential equations. Various modifications of this model and the mathematical models for control disease by vaccinations are studied.

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see 50f8727d0e835Gogi-Pantsulaia-ENG.pdf

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It is put and solved numerically 2-dimentional (in a vertical plane x-z) a non-stationary problem about of a mesometeorological boundary layer of atmosphere. In it ecologically such actual processes, as a full cycle of development of a cloud and a fog and aerosol distribution are considered. A number of abnormal meteoprocesses is simulated: simultaneous existence of a stratus cloud and a radiation fog; an incorporated vertical complex of a stratus cloud and a radiation fog; daily continuous overcast ; ensemble of humidity processes; mutual transformation humidity processes. New the role of horizontal and vertical turbulence in formation of some meteoprocesses, in particular, a tropical cyclone and a tornado is considered. Influence of some meteoparameters on aerosol distribution is investigated. Becides such problems are simulated in a stage of computer realization, as the account of cooling on cloud and fog border; influence of cloudy shades on boundary layer processes; the account of difficult underlying surface temperature heterogeneity

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At present pipelines become the main practical means for liquid and gas substances transportation. Indeed oil, water and gas transportation by pipelines is the safest and cheapest method in comparison with railway, marine and motor transportation systems. That is way a large number of pipeline networks were constructed worldwide during the last 70 years for natural gas transportation. At the same time it should be noted that the gas delivery infrastructure is rapidly ageing. The main fault of the outdated pipelines is leak and as a consequence, explosion, fire and deterioration of environment. For instance leakage of methane which is the most principal greenhouse gases contributes to climate change. Owing to earthquakes, floods, corrosion and terrorist attacks escape of gas may occur at any time and location in pipeline networks, therefore, timely detection of leaks is important for the safe operation of pipelines, for minimization of environment contamination and economical loss. So elaboration of leak detection and location methods for gas pipeline system is an urgent and sensitive issue of nowadays. Unfortunately there have not been yet invented a fully perfect method for leak detection and localization so finding out the new methods and techniques for the leak fast detection and location in the pipelines is an urgent issue. There are many different methods that can detect natural gas pipe line leaks and location, among them we can note a very simple manual inspection using training dogs and also advanced satellite based detective systems. But mainly the various methods can be classified into non-optical and optical methods. In the paper are reviewed some methods for pipelines leak detection and location. Also in this paper we have created a new mathematical model defining the leak detection in oil and gas complex (having several branches) transmission pipelines for the gas stationary flow. The 9 mathematical model (an algorithm) does not required knowledge of corresponding initial hydraulic parameters at entrance and ending points of each sections of the pipeline (receiving of this information is rather difficult without using telemetric informational system). Numerical experiments gave positive results. We have created a new mathematical model defining of leak detection in the claimed gas transmission pipeline for the gas stationary flow. Previous numerical experiments are in good approaches with the observed values of the pipelines accidental leaks

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In this paper the expansion of regular solution for the equations of the theory of thermoelasticity with microtemperatures is obtained that we use for explicitly solving the Neumann boundary value problem (BVP) for the equations of the linear equilibrium theory of thermoelasticity with microtemperatures for the spherical ring. The obtained solutions are represented as absolutely and uniformly convergent series.

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By means of I. Vekua method 3-dimensional problems are reduced to 2-dimensional one’s, after that the method of a small parameter are used.

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We study incompressible fluid flow over ellipsoidal bodies moving in the infinite cylinder at the constant speed. We consider the case when the pressure fall is a constant and viscosity is rather small. In this case Navier-Stokes system is linearized and Stokes linear system is obtained. Exact solutions of this system are obtained in a moving coordinate system .

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The presentation concerns the scientific heritage of Professor Viktor Kupradze in the linear theory of three-dimensional elasticity. We will consider two main directions: Development of potential method for spatial problems of elasticity and Method of fundamental solutions. We describe main achievements of the worldwide known school of V. Kupradze in the theoretical study of boundary value problems of elastostatics, elastodymanics and elastic vibrations based on the boundary integral equations methods. We give also an overview of results related to the universal and easily realizable numerical method: Method of fundamental solutions. In the final part, we describe some new developments of the potential theory and treat some open problems.

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In the present article the problems for composite (piece-wise homogeneous) bodies weakened by cracks when cracks intersect an interface or penetrate it at rectangular angle is studied. Antiplane problems of the elasticity theory for piece-wise homogeneous orthotropic plane is reduced to the system (pair) of singular integral equations containing an immovable singularity with respect to the tangent stress jumps (problem A). First the behavior of solutions in the neighborhood of the crack endpoints is studied (see [1]). In a partial case when one half-plane has a rectilinear cut of finite length, which is perpendicular to the boundary, and one end of which is located on the boundary. We have one singular integral equation containing an immovable singularity (problem B). The question of the approached decision of one system (pair) of the singular integral equations is investigated. A general scheme of approximate solutions is composed by the collocation and asymptotic methods are presented...

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A boundary contact problem of elastic equilibrium is solved for a multilayer (tree layer) rectangular parallelepiped under external pint load. The corresponding boundary value contact problem of elasticity Is analytically solved where displacements are represented as infinite series, with each series term representing a product of trigonometric and exponential functions. On the basis of the obtained analytical solution a comprehensive iser-friendly wide-service program is created.

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The present talk is devoted to construction of differential hierarchical models for elastic prismatic shells with microtemperatures; it is organized as follows. In Section I. a brief survey of results concerning the linear theory for elastic materials with inner structureswhose particles, in addition to the classical displacement and temperature fields, possess microtemperatures is given. In Section 2 prismatic and cusped prismatic shells are exposed. Relation of the prismatic shells to the standard shells and plates are analyzed. In a lot of figures 3D illustrations of the cusped prismatic shells are given. Typical cross-sections of cusped prismatic shells are also illustrated. Moments of functions and their derivatives are introduced and their relations clarified. Section 3 contains hierarchical models for elastic prismatic shells with microtemperatures. To this end, a dimension reduction method based on Fourier-Legendre expansions is applied to basic equations of linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures. The governing equations and systems of hierarchical models are constructed with respect to so called mathematical moments of temperature and of stress and strain tensors, displacement and microtemperature vector components. Section 4 is devoted to deriving the governing relations and systems of the N = 0 approximation (hierarchical model) for elastic prismatic shells.

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The problem of contamination of the Georgian territory by radioactive products in case of possible accident at the Armenian Nuclear Power Station is studied. Mathematical model for computation of transporting and diffusion of radioactive substances with account of orography is developed. Some results of numerical calculations are presented.

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Analyses of a reliability of the main gas pipeline has shown high probability of the pipeline’s some sections damage and gas leakage. The leaks caused by damage of pipelines are usually very dangerous. Intensive leaks can stimulate explosions, fires and environment pollution, which can lead to the ecological catastrophe. In this case there can be an enormous economical loss. That is why the determination of damage place in pipelines in time is the significant problem. Generally gas leakage (as a result the gas pressure and expenditure alteration) is accompanied by non-stationary flow of in the pipelines. After some time of gas leakage (under some conditions) gas movement in the pipelines has stationary character. That is way it is necessary to study both the non-stationary and stationary stages of the gas movement in the pipelines having gas escape in the some sections of the main gas pipeline. In the present paper disclosing the location of large scale accidental gas escape from the complicated gas (oil) pipe-line for the both gas stationary and non-stationary flow is studied. For solving the problem it has been discussed early-made method, reason is that the exact analytical method has not been existed. We have created quite general test, the manner of the solution has been known in advance. We consider this question as a reverse task of hydraulic calculation problem. The algorithm does not required knowledge of corresponding initial hydraulic parameters at entrance and ending points of each sections of pipeline. The algorithm is based on mathematical model describing gas stationary movement in the simple gas pipeline and upon some results followed from that analytical solution and computing calculations.Comparison results of calculation with real data has shown the affectivity of the suggested methods

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In this paper the a mathematical model of hydrates origination in the gas main pipeline taking into consideration gas non-stationary flow is studied. For solving the problem of hydrates origination in the gas main pipeline taking into consideration gas non-stationary flow the system of partial differential equations is investigated. For learning the affectivity of the method one general test was created. Numerical calculations have shown efficiency of the suggested method.

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Solving the problem of identifying the place of emergency gas release from the main gas pipeline for a complex gas pipeline is not an easy task. It is important to know (detect) the place and intensity of an emergency gas release during unsteady flow, both in order to reduce gas losses and in environmental terms. Mathematical models are presented here for searching for leakage in branch main pipelines with non-stationary and stationary gas flow and for searching for leaks in inclined gas pipelines with stationary gas flow.

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Languages ​​where function and predicate symbols do not have a fixed rank, in recent years have become the object of intensive study due to the rather wide scope of their application [1]. . These operators are defined within the rational rules for introducing derived operators Sh. Pkhakadze [2], on the basis of which in a rank-free formal mathematical theory analogues of some of the obtained results were proved in the formal mathematical theory of N. Bourbaki [3]. [1] Kutsia T., Theorem Proving with Sequence Variables and Flexible Arity Symbols. In: M. Baaz and A. Voronkov, editors, Logic in Programming, Artificial intelligence and Reasoning. Prroceedings of the 9th International Conference LPAR 2002. Volume 2514 of Lecture Notesin Artificial Inteligence. Springer, 2002, 278–291. [2] Пхакадзе Ш. С. Некоторые вопросы теории обозначений. ТГУ, 1977. [3] Бурбаки Н. Теория множеств; M.: Наука, 1965.

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Differential hierarchical models for elastic prismatic shells with microtemperatures is constructed. Some problems for N = 0 approximation (hierarchical model) are studied.

In this study, spreading of the oil pollution in the Georgian Black Sea coastal zone on the basis of a 2-D numerical model of distribution of oil pollution is simulated. The model is based on a transfer-diffusion equation with taken into account reduction of oil concentrations because of physical - chemical processes. The splitting method is used for solution of the transfer-diffusion equation. Numerical experiments are carried out for different hypothetical sources of pollution in case of different sea circulation regimes dominated for the four seasons the Georgian Black Sea coastal zone. Some results of numerical experiments are presented

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In this contribution to the conference the distributions of petroleum and mineral oil into the soils, based on the integration of the nonlinear filtration equation of a liquid is studied. For this study some analytical solutions of nonlinear filtration equation of oil are presented. Analytical solution of filtration equation are obtained in case of presence as non-linear diffusion processes as well presences filtration processes and additionally occurrence of second source of oil. Some results are presented.

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A function f acting from the real line R into itself is called absolutely nonmeasurable if there exists no nonzero sigma-finite continuous (i.e., vanishing on all singletons) measure on R for which f is measurable. The existence of absolutely nonmeasurable functions cannot be proved within ZFC set theory, but follows from additional set-theoretical hypotheses, e.g., from Martin's Axiom. Some properties of absolutely nonmeasurable functions are considered in [1]. Theorem 1. The composition of any two absolutely nonmeasurable functions is absolutely nonmeasurable. Theorem 2. Under Martin's Axiom, every function acting from R into R is representable as a sum of two absolutely nonmeasurable injective functions. Theorem 3. Under Martin's Axiom, every additive function acting from R into R is representable as a sum of two absolutely nonmeasurable injective additive functions

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A functional differential equation \begin{equation} u^{(n)}(t)+F(u)(t)=0,\tag {$1$} \end{equation} is considered with continuous $F:C(R_+;R)\to L_{loc}(R_+;R)$. Oscillatory properties of proper solutions of (1) are studied. In particular sufficient conditions are given for equation (1) to have the property {\bf A} or {\bf B} ($\wa$ or $\wb$) which are optimal in a certain sense. Sufficient conditions for every solution of (1) to be oscillatory are obtained as well as existence conditions for an oscillatory solution. Chapter 6 is dedicated to boundary value problem (16.1)-(16.2). Sufficient conditions are established for the existence of a unique solution, a unique oscillatory solution and a unique bounded oscillatory solution of this problem.

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In the infinite area the elliptic equation with the cubic nonlinearity is studied. By means of the trigonometric representations the equation is equivalently reduced to another elliptic equation which is solved analytically. The solutions vanishing at infinity are obtained.

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The two-dimensional numerical model of an aerosol distribution from an instant point source against of thermohydrodynamics of a mesoscale boundary layer of atmosphere is developed. With its help it has been investigated the contribution and influence on considered ecological process of such basic factors, as an aerosol source height, an aerosol sedimentation velocity, a background wind, atmosphere stratification. In this work the special place is given to a problem of an surface pollution. It depends essentially on a source pollution height; it decreases with its increase, i.e. the ecological situation under the source improves. As the increase of an aerosol source height is connected with different technical and economic problems same effect can often by achieving by preliminary heating of an aerosol. This scenario is simulated By means of the present model. The first numerical experiments about of revelation of equivalence between increasing of pollution source height and degree of preliminary heating of an aerosol are received

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Let D be a point-set (finite or infinite) in the n-dimensional Euclidean space R^n. We say that this D is a quasi-Diophantine set if the distance between any two points from D is a rational number. Investigation of the combinatorial structure of various quasi-Diophantine sets in Euclidean spaces is a rather attractive and important topic. Properties of various quasi Diophantine point systems are considered in many works. Let X be a finite point-set in the Euclidean plane R^2. We shall say that a line segment l is an edge of X if there exist two points from X which are the end-points of l. This terminology is compatible with graph theory. Take any four points x, y, z and t from X such that the intersection of the line segments [x, y] and [z, t] is a singleton. The family of all singletons obtained in this manner will be denoted by I(X). Note that if X contains at least three points, then X ⊂ I(X). We shall say that a line segment l is admissible for X if its end-points belong to I(X) and there exists an edge of X containing l. Theorem 1. Let X be a finite quasi-Diophantine set in the space R^n. Then the length of each admissible line segment for X is a rational number. In particular, if the points of a quasi-Diophantine set {a, b, c, d} are the vertices of quadrangular in R^n and [a, b]∩[c, d] = {x}, then the set {a, b, c, d, x} is a quasi Diophantine set, too. Theorem 2. In the Euclidean plane R^2 there exists a pentagon X which is a quasi Diophantine set but for which I(X) is not quasi-Diophantine.

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Analogues of the results obtained in Burbak's quantization theory were confirmed in the Unranked quantization theory.

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Consider a boundary value problems (BVP) of statics of the theory of thermoelasticity with microtemperatures for the plane with a circular hole. The representation of solutions of the equation system is constructed with the help of the harmonic and metaharmonic functions. Using this representation the solution of the problems is given in the explicit form by using the Fourier series. There are the obtained the numerical solutions.

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Analyses of a reliability of the main gas pipeline’s exploitation has shown high probability of the main gas pipeline’s some sections damage and gas leakage and as a result the gas pressure and expenditure alteration when non-stationary processes are in progress [1-3]. After some time gas leakage (under some conditions), it is possible establishment a new stationary state of gas movement in the pipelines has stationary character. That is way it is necessary to study as a non-stationary stage as well the stationary stage of gas movement in the pipelines having gas escape in the some sections of the main gas pipeline [1-2]. It is known analytical method of determination a large-scale gas escape location on the simple section of main gas pipeline [1], using data of the gas pressures and expenditure at the entrance and ending of the gas pipeline. But this method cannot be used for main gas pipelines with several sections and branches if previously would not be discovered the location of the section with gas escape. The method offered by us is devoid from this default. So the problem can be formulated as follows: In the complex main gas pipeline with several branches and sections first of all the placement of the section having accidental gas escape is determined using minimal information (data of the gas pressures and expenditure at the main gas pipeline’s entrance and ending points before and after gas escape) and then defined location of the accidental gas escape in the determined section of main pipeline. In this article the above mentioned problem is studied. References [1] A. Bushkovsky, Characteristic System of Distribution of Parameters, Moscow, Nauka, 1979. [2] T. Davitashvili, G. Gubelidze and I. Samkharadze, Leak Detection in Oil and Gas Transmission Pipelines, in Book “Informational and Communication Technologie

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Study of boundary value problems for the composite bodies weakened by cracks has a great practical significance. Mathematical model investigated boundary value problems for the composite bodies weakened by cracks in the first approximation can be based on the equations of anti-plane approach of elasticity theory for composite (piece-wise homogeneous) bodies. When cracks intersect an interface or penetrate it at all sorts of angle on the base of the integral equations method is studied in the works [1]–[3]. In the present article finite-difference solution of anti-plane problems of elasticity theory for composite (piece-wise homogeneous) bodies weakened by cracks is presented. The Mathematical Modeling differential equation with corresponding initial boundary conditions is approximated by finite-differential analogies in the rectangular quadratic area. Such kind set of the problem gives opportunity to find directly numeral values of shift functions in the grid points. The suggested calculation algorithms have been tested for the concrete practical tasks. The results of numerical calculations are in a good degree of approach with the results of theoretical investigations. References [1] A. Papukashvili, Antiplane problems of theory of elasticity piecewice-homogeneous orthotropic plane slackened with cracks. Bull. Georgian Acad. Sci. 169 (2004), No. 2, 267–270. [2] A. Papukashvili, About of one boundary problem solution of antiplane elasticity theory by integral equation methods. Rep. Enlarged Session Semin. I. Vekua Inst.Appl. Math. 24 (2010), 99–102. [3] A. Papukashvili, M. Sharikadze, and G. Kurdghelashvili, An approximate solution of one system of the singular integral equations. Rep. Enlarged Session Semin. I. Vekua Inst. Appl. Math. 25 (2011), 95–98.

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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. Local circulation against which process fog- and cloudformation develops is caused by a daily march of temperature of an underlying surface. In the work process develops in time: in the end of the period of heating and in the beginning of cooling (evening) of an underlying surface simultaneous presence of a fog and of a cloud takes place. At the subsequent heating (at daybreak) of an underlying surface there is a gradual emerging and horizontal expansion of fog edges; they constantly lose touch with an initial fog and continue to exist irrespective of it as a layered cloud. So, we have three layered clouds and one fog. At the further heating of an underlying surface there is a gradual emersion of a fog. Therefore we receive ensemble from four layered clouds. Thus, during one day we have simultaneously a cloud and a fog, then three clouds and a fog, and at last four clouds. In modelling of the above-stated unordinary processes a determinative role are played relative humidity and turbulent regime of MBLA. The results of calculations received at selection of special meteorological conditions are physically quite logical and possible, but occurrence of such humidity processes ensembles in the nature are not known to us from materials of meteorological observation. Therefore it is possible to consider our work as numerical experiment. Further we will try to find meteorological process corresponding to our results

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There is being study stress-strain state of the piecewise homogeneous infinity elastic body, with an elliptic hole, when on internally surface body are four cracks. The top of each crack is strengthened in a certain area of the top with use a harder materials. The body is in plane-deformation state and therefore corresponding two-dimensional boundary-contact problem is considered. There is investigated dependences of deformation in a body on materials of the body (nearby tops of a cracks in a circle with radius r is other material), on size of radius r, on quantity of cracks and length. For some value of radius r and length of cracks by means of a method of boundary elements are received the numerical decisions, constructed corresponding schedules and physical and mechanical interpretations for the received results are made.

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see http://solmech2012.ippt.gov.pl/Solmech2012-Book%20of%20Abstracts.pdf pages 236-237

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The existence theorems of an optimal initial element are proved for the quasi-linear neutral functional differential equations.

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In the case of harmonic vibration we study the well-posedness of boundary value problems for elastic cusped prismatic shells in the zero approximation of I.Vekua’s hierarchical models.

The direct and inverse problems connected with the interaction between different vector fields of different dimension have been recently given much attention and intensively investigated in the mathematical and engineering scientific literature. In [Chinchaladze N., Gilbert R.P. Cylindrical Vibration of an Elastic Cusped Plate under Action of an Incompressible Fluid in Case of Approximation of I.Vekua's Hierarchical Models. Complex Variables, Vol.50, No. 7-11 (2005), 479-496 ] solid-fluid interaction problems are consider when the elastic plate obeys the approximation of I.Vekua's hierarchical model [Vekua I. Shell Theory: General Methods of Construction. Boston-London-Melbourne: Pitman Advanced Publishing Program, 1985]. The transmission conditions of the interaction problem between an elastic cusped plate and a fluid, when the fluid generates bending of the plate where established. The cylindrical bending of such plates under the action of either incompressible ideal or viscous fluid have been studied. In this paper, we investigate an elastic beam-fluid interaction problem. In the solid part the (0,0) approximation of hierarchical models and in the fluid part the Stokes equations are used. An inverse problem is considered as well.

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In the present article finite-difference solution of anti-plane problems of elasticity theory for composite (piece-wise homogeneous) bodies weakened by cracks is presented. The differential equation with corresponding initial boundary conditions is approximated by finite-differential analogies in the rectangular quadratic area. Such kind set of the problem gives opportunity to find directly numeral values of shift functions in the grid points. The suggested calculation algorithms have been tested for the concrete practical tasks. The results of numerical calculations are in a good degree of approach with the results of theoretical investigations.

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In the present paper, in the Cartesian system of coordinates, thermoelastic equilibrium of a rectangular parallelepiped consisting of three homogeneous isotropic layers is considered. Either symmetry or antisymmetry conditions are defined on the lateral faces of the parallelepiped, while either rigid contact or sliding contact conditions are defined on the contact surfaces. The problem is to define disturbances on the upper and lower sides of the parallelepiped so that stresses and normal displacement or displacements and normal stress on some planes parallel to the upper and lower sides inside the body would take the prescribed values.

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The nonlinear integral equation connected with non-linear non-stationary Schrödinger and diffusion equations with the appropriate initial-boundary conditions is considered. The approximate solution of this equation is obtained.

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In this talk some classes of generalized analytic vectors are considered.

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The problem of linear coupling for the irregular Carlmann-Vekua equations is investigated, when the coefficients of the equations belong to sufficiently wide function spaces representing the expansions of the classical spaces introduced by Vekua.

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I. Vekua suggested a simple method ensuring the boundary conditions of the face surfaces for shallow shells. In this talk the result is generalized for non-shallow shells.

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In the present talk the linear theory of thermoelasticity with microtemperatures is considered. The representation of regular solution for the equations of steady vibration of the 3D theory of thermoelasticity with microtemperatures is obtained. We use it for explicitly solving Dirichlet boundary value problem (BVP) for an elastic space with a spherical cavity. The obtained solutions are represented as absolutely and uniformly convergent series.

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A variant of variation-discrete method given in [Vashakmadze T.S. Some Remarks to Numerical Realisation of Ritz’s Method. (Russian) Semin. Instit. Appl. Math. Tbilisi, 1970.] is applied to solve some BVPs with Dirichlet conditions. First the Poisson equation and then the tension-compression problem of a 2D isotropic plate in a square [−1, 1]^2 is considered. Boundary condition for simplicity are assumed to be homogeneous. It is realized that the method applied has a higher level of accuracy, convergence, stability and a wider class of applicability when compared to the classical finite difference method. Other than those, the scheme obtained consists of four subsystems which can be solved independently and hence enhancing the use of parallel computations.

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The problem of the mesometeorological boundary layer of the atmosphere is posed and solved, taking into account the propagation of heavy aerosol in it. The effect on the process under consideration of such parameters as atmospheric stratification, aerosol settling rate, and aerosol dispersion sites has been studied.

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This work is devoted to investigation of nonlinear dynamics of planetary electromagnetic (EM) ultra-low-frequency wave (ULFW) structures in the rotating dissipative ionosphere in the presence of inhomogeneous zonal wind (shear flow). Planetary EM ULFW appears as a result of interaction of the ionospheric medium with the spatially inhomogeneous geomagnetic field. The shear flow driven wave perturbations effectively extract energy of the shear flow increasing own amplitude and energy. These perturbations undergo self organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation's front. Depending on the features of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices, or dipole vortex, or vortex streets and vortex chains.

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We constructed estimates of maximal likelihood in infinite-dimensional Hilbert space. Some properties of consistency and asymptotic normality is investigated.

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In this paper the solution in the boundary value problems of the theory of thermoelasticity with microtemperatures for the circular ring are consideret. The obtained solutions are represendet as absolutely and uniformly convergent series. For the numerical solutions there are programs.

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For a plate of variable thickness, included cusped ones, Reissner-Mindlin Type Model is considered. Setting of the BVPs are depend on the geometry of sharpening.

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Viscoelastic materials play an important role in many branches of engineering, technology and biomechanics. The modern theories of viscoelasticity and thermoviscoelasticity for materials with microstructure have been a subject of intensive study in recent years. Recently, the theory of thermoviscoelastic materials with voids is constructed by Iesan (2011). In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with voids is considered and the basic integral representations of regular vectors are obtained. the single-layer, double-layer and volume potentials are constructed and their basic properties are established. the uniqueness and existence of regular solutions of the boundary value problems are proved by means of the potential method.

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There are discussed several measurability properties of real-valued functions.

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An economic crisis forecasting method is proposed and an appropriate mathematical model is developed, which can be used to predict the crisis situation with more or less probability, which will enable interested countries and companies to implement anti-crisis measures in a timely and safe manner.

In work for an incompressible elastic homogeneous isotropic rectangular parallelepiped are set and the following nonclassical problems of thermoelasticity are analytically solved. On lateral sides of a parallelepiped, and also on its bottom side, conditions of symmetry or anti-symmetry are given. The problem is to define disturbances on the upper side of the parallelepiped so that stresses and normal displacement or displacements and normal stress on some planes parallel to the upper and lower sides inside the body would take the prescribed values.Considered tasks don't coincide with other nonclassical tasks of the theory of elasticity known in literature and have important applied value.

Brain is supplied by blood continually, as it is very important that nerve cells be supplied by oxygen and nutrients continually. When brain activity is increased influx of blood flow also increases. Accordingly it is decreased in other organs. Fluctuation of blood volume in the brain is closely depend upon alteration of oxygen and carbon dioxide contained in inhaling air. Violation in oxygen delivery to brain instantly changes its functioning. Suppression of blood flux to brain for 1.5 min. causes lose of consciousness, for 3min causes structural alterations of nerve cells, for 5min – death of nerve cells The different stresses at the human body leads mutations in genetic information. The cell division is regulated process and a loss of regulation can result of uncontrolled cell growth. There are over 100 types of cancers. More of 50% of them possess a mutant allele of the p53 gatekeeper gene. This is protein which regulates cell division. The mutation of this gene leads abnormal growth of previously normal cells and a result is a cancer. We derive the mathematical formula which shows connection of the oxygen consumption with the tumor volume. This formula enable estimate the death rate of astrocytes at the human brain caused by the cancer.

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Power series have good potential for modeling non-linear systems due to the large class of non-linear systems they cover and their connection to non-linear realization theory. In this article, we modeling nonlinear systems using some power series in both discrete and continuous time.

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Main reasons of pipeline constipation (emergency shutdown) are the following: generation of hydrates, freezing of water slugs, contamination and so on. To take timely steps against generating of hydrates, it is necessary to study humidity and distribution of pressure and temperature...

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In the present article two methods of solution of elasticity theory antiplane problems for piece-wise homogeneous orthotropic plane weakened by cracks by integral equations and finite-difference methods are studied. In the first case for orthotropic (particularly in isotropic case) plane antiplane theory elasticity problem is reduced to the system (pair) of singular integral equations containing immovable singularity with respect to the tangent stress jumps(see [1],[2]). The questions of behaviors of the decision in a vicinity of the ends of cracks and on dividing border are investigated. The general schemes of solving and carrying out numerical computation using spectral, collocation and asymptotic methods are given. In considering piece-wise homogeneous plane slackened with cracks antiplane elasticity theory by finite-differential methods the plane is replace by big-value square and differential equation and its responsible boundary conditions are approximated by different analogy. This statement of the problem gives possibility to find directly numerical values of displacement in grid knots. In cases of both methods the offered settlements have been approved for concrete practical problems and numerical results are in a good approximations with the results, obtained by theoretical studies

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In the paper numerical modelling humidity processes which take place in mesoscale boundary layer of atmosphere (MBLA) are considered. Under humidity processes fogs, layered clouds, cloudy processes and related questions are meant. 16 We investigate them not only from the point of view of weather forecast, sea, avia- and agricultural meteorology, but also ecology since polluting substances are accumulated in them. Also it is obligatory to consider the fact, that at formation of a fog and a layered cloud and, especially, at their simultaneous existence, allocation of the latent warmth of condensation of water steam takes place due to which the curve of temperature stratification varies and it takes the form of "broken line

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Boundary value contact problems on the thermoelastic equilibrium of multilayer cylindrical bodies are used as an example in considering problems of delocalization (localization) of stresses in such bodies. An analytical (exact) solution of the formulated problem is constructed in the form of infinite series. A sequence of solutions is as follows: first a general solution is constructed by means of harmonic functions. Each of these functions is represented by the corresponding series using the method of separation of variables. Conclusions on delocalization (localization) of stresses are made according to the conventional delocalization (localization) criterion. The considered problems (the two-dimensional case is meant) illustrate the advantage of the use of the polar coordinate system instead of the Cartesian one in the case where problems are solved by the BEM

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The present paper is devoted to the three-dimensional version of statics of the theory of thermoelasticity with microtemperatures. Some problems of the linear theory of thermoelasticity with microtemperatures will be considered in the upper half-space. Let on the boundary of the half-space one of the following boundary conditions be given: a) displacement vector, microtemperature vector and the temperature, b) displacement vector, microtemperature vector and a linear combination of normal component of microtemperature vector and a normal derivative of temperature, c) displacement vector, tangent components of microtemperature vector, a normal component of microstress vector and temperature, d) displacement vector, normal components of microtemperature vector, tangent components of microstress vector and the normal derivative of temperature. Using Fourier transform these problems are solved explicitly (in quadratures).

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The purpose of this paper is to consider the geometrically nonlinear shallow spherical shells. By means of I. Vekua method two-dimensional system of equations is obtained. Using the method of the small parameter, approximate solutions of these equations are constructed. The small parameter h/ R, where 2h is the thickness of the shell, R is the radius of the middle surface of the sphere. A concrete problem is solved.

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We consider 3D mixed boundary value problem of the elasticity theory for the orthotropic beam with the constant width, affected by static forces. We study this problem by means of two different methods: the variational-difference method and the finite-difference method and compare the accuracy. The results are applied to one problem of micromechanics.

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There are several methods for preventing the formation and intensification of gas hydrates. Among them: the injection of thermodynamic and kinematic inhibitors; maintain the normal functioning of pipelines from the outside by insulating, heating, regulating pressure, etc. However, all of the above methods require knowledge of the hydrate formation zone on the pipeline. Currently, there are existing mathematical models for determining the zone of hydrate formation in a stationary gas flow, but the gas flow in pipelines is non-stationary. Thus, we present a mathematical model for determining the zone of hydrate formation during non-stationary gas flow in main pipelines.

With the help of computer modeling, mathematical modeling and the use of numerical analysis, it is possible to predict water quality parameters, control and manage pollution processes. This kind of observation and prediction is economically beneficial and saves the costs that would be necessary to organize and conduct experiments; sometimes this approach is the only way to study the relevant phenomena. Thus, mathematical modeling of diffusion processes in the environment and the study of pollution problems is one of the most relevant and interesting problems in applied and computational mathematics. Therefore, mathematical modeling and the models themselves are constantly being improved, refined and, in some cases, even simplified. In fact, there is a wide variety of nonlinear mathematical models that describe pollution processes, but in this paper we will focus on only one nonlinear mathematical model that describes the transport and diffusion of pollution in water bodies and present some results of numerical calculations.

The study of some extreme phenomena during the transportation of gas and oil through main pipelines is also interesting from a theoretical point of view. The presented model takes into account the main physical, chemical and geographical parameters that characterize extraordinary processes, phenomena and the environment as a whole. Namely, the infiltration of oil and gas into the soil, taking into account the heterogeneity of the soil; oil seepage into the soil, taking into account high pressure and heterogeneity of soil porosity; the corresponding mathematical models are non-linear differential equations and systems in partial derivatives. New analytical solutions of the non-stationary diffusion equation are obtained, which describe the infiltration of oil into the soil.

Boundary value problems with cracks of slackened piecewise homogeneous bodies occupy a given place in the theory of elasticity. It’s interesting a case, when the cracks across the boundary or reach to the boundary. Anti-plane problems of theory of elasticity by using the theory of analytical functions are presented in this paper. The problems lead to a system of singular integral equations with immovable singularity with the respected to leap of the tangent stress. The problems behavior of solutions at the boundary are studied.

The brain consumes up to twenty percent of the energy used by the human body, more than any other organ. Let us consider the oxygen diffusion process at the human brain capillary across the radial direction. We consider a diffusion process accompanied with a chemical reaction at the cylindrical area with the radius and length . We suppose that this process is axi-symmetrical and the substance moves along the inner boundary of the cylinder at a constant speed. In this case the process is described by the axi-symmetric reaction-diffusion with the appropriate initial-boundary conditions. The diffusion coefficient of oxygen at the brain tissue was calculated experimentally in the last century. We linearized the equation and consider it in the rectangular area. By the separation of variables we obtain the exact solution and hence estimated the oxygen consumption rate by the brain at the normal condition of the human body. Nitrate (NO3) toxicosis in humans occurs through enterohepatic metabolism of nitrate to ammonia.Nitric oxidize the iron atoms in hemoglobin from ferrous iron (2+) to ferric iron (3+) rendering it unable to carry oxygen. This process lead to leak of oxygen in organ tissue and dangerous condition called methemoglobininemia, which leads neurodegenerations. In case of NO consumption the oxygen consumption is disrupted and without oxygen cells can not consume nutrients which lead their death.

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Thus, we present a mathematical model for determining the zone of oil infiltration in soils. We performed numerical experiments for these four soil types, which are more common on the Georgian section of the TRACECA route. Numerical calculations have shown that the process of oil infiltration in soils proceeds qualitatively in the same way, i.e. in all considered soils, one can distinguish the stage of oil absorption by the soil and the stage of oil spreading to the depth and width of the soil.

For solution of the initial value problem (1)- (3) asymmetric difference schemes have been constructed and used. The analysis of the algorithms and numerical experiments has been performed.

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Necessary optimality conditions of the initial data are obtained for the nonlinear delay functional-differential equations with the mixed initial condition.

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It is known, that results obtained from (1) work have huge theoretical and practical value. It allows to effectively use extensions, obtained by adding the abbreviational symbols, for the theory to be effectively studied. It deeply reveals connections between different mathematical theories. Also, it is practically necessary for the mathematical researches to be automated and for the creation of the kind of systems, which will be able to process mathematical tests. References [1] Пхакадзе Ш. С.. Некоторые вопросы теории обозначений. Тбилиси: Изд. ТГУ, 1977. C. 195. [2] Глушков В. М.. Некоторые проблемы теории автоматов и искусственного интелекта. Кибернетика. 1970. № 2. C. 3–13. [3] Rukhaia Kh. М., Tibua L. М. One Method of constructing a formal system. Applied Mathematics,Informatics and Mechanics(AMIM). 2006. Т. 11. № 2. C. 3–15.

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Air emissions from industrial facilities and motor vehicles and monitoring of the atmosphere quality in the most industrialized cities of Georgia: Tbilisi Rustavi, Qutaisi, Zestafoni and Batumi are presented. Fuel consumption and emissions from transport sector in Tbilisi is investigated. Using mathematical simulation, distribution of concentration of harmful substances NOx at Rustavely Avenue, the crossroad of King David Agmashenebeli and King Tamar Avenue, where traffic is congested, is studied. Some results of numerical calculation are given.

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In this paper a mathematical model (an algorithm) defining a placement of a section having gas accidental escape in complex main gas pipeline with several sections and branches is suggested. The algorithm does not required knowledge of corresponding initial hydraulic parameters at entrance and ending points of each sections of pipeline (receiving of this information is rather difficult without using telemetric informational system). The algorithm is based on mathematical model describing gas stationary movement in the simple gas pipeline and upon some results

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Study of propagation in the space and time of air flow, generated in the small time by the action of high-power phenomenon, has huge theoretical and especially practical value. As usual, these phenomena propagate during the small time on the relatively small territory, but their results are long and important. Especially interesting is the advective propagation on mountainous territory. Even low height hills slow down the velocity of flow motion and often changes its direction and sometimes even to the opposite direction. Exactly such regional peculiarity is characteristic for some regions of Georgia, among them Tskhinvali and Sachkhere territory, where military actions took place. Then in the region, the conditions are developed, theoretical justification of which, as we think, is given in this article.

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In the present work the Cauchy problem for an abstract evolution equation with Lipschitz‐continuous operator is considered. We took the case when the main operator represents a sum of positive definite self‐adjoint operators. The fourth order accuracy decomposition scheme is constructed for approximate solution of this problem. The constructed scheme contains two starting vectors on the zero and first layers, respectively. The starting vector on the zero layer is given, and to calculate the starting vector on the first layer with the necessary accuracy, we use recursive algorithm, where the interval is halved on each next step of the recursion. Using the scheme, numerical calculations for different model problems are carried out.

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Cauchy problem for a quasi‐linear abstract hyperbolic equation is considered, where the main operator is self‐adjoint positive definite and represents a finite sum of its similar operators. For this problem the fourth order accuracy three‐layer decomposition scheme is constructed on the basis of rational splitting of the cosine operator‐function. To calculate the starting vectors necessary for its realization with the corresponding accuracy, we suggest a recursive algorithm, whose number of steps has the same order as the logarithm of division number of the time interval.

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It is traditional any big town has the satellite-town. In this case we actually have two thermal “islands”. In this work the two-dimensional, non-stationary problem about local circulation over these “islands” and sreading in it of an aerosol from different sources (point, surface, instant, constantly acting etc.) is statement. Now the problem only describing thermohydrodynamics and humidity processes is programmed and numerically realised on the computer. Qualitatively real space-time fields of speed of air, temperatures, pressure, water-vapor and liquid-water mixing ratios are received. On the basis of this model optimum control of two virtual objects-towns (distances variation between them, forecasting of meteorological fields, creation of recreational zones, influence of prevailing wind, playing of different ekometeorological interactions scenario etc.) is possible

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Rather actual it is considered research temperature inversion processes in connection with the environmental problems which are taking place in a mesoboundary layer of atmosphere (MBLA). there is an accumulation of polluting substances in these inversion layers. Such anomalies arise at formation of clouds and fogs when allocation of the latent warmth of condensation of water steam takes place There was local circulation over thermal ”island” is numerically simulated at its periodic heating with the help of the developed by us numerical model and under certain physical conditions is received against the MBLA thermohydrodynamics simultaneously four humidity processes: it is a fog and a cloud direct over ”island” and two clouds on each side. Because of them aforementioned inversion layers arise. This interesting numerical experiment is quite logical physically. On the future we plan to get corresponding materials of meteorological supervision for the purpose of comparison of theoretical results with experimental data

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A number of alternative options were assessed during preliminary work relating to the export of oil and gas from the Shah Deniz field. There was considering the best transportation method for oil and gas (followed by a detailed assessment of the best pipeline corridor, the best possible markets) culminating in the final route selection. Gas and oil transportation by pipeline and railway routs from the Shah Deniz field via Azerbaijan and Georgia was defined as the most acceptable commercial and environmental solution. The probability of crashes for pipeline transport rises with the age of the oil pipelines in service, and with the extent of their network. But there are able to take place non ordinary situations too. As foreign practice of pipeline exploitation shows, that the main reason of crashes and spillages (and fires as a consequence) is destruction of pipes as a result of corrosion, defects of welding, natural phenomena and so on (including terrorist attacks and sabotage). In West Europe it has been found, that 10 - 15 leakages occur every year in a pipeline network of around 16,000 km length resulting in a loss of 0.001 per sent of transferred products. The proposed transport corridor via Georgia is characterized by very diverse ecological conditions and by abundant biodiversity. The route crosses a multitude of minor watercourses with broad seasonal variations of surface water flow. Six major river crossing occur along the route on the territory of Georgia. Ground water along the route is also abundant and generally of high quality. So that it is necessary: to design a new high-quality soil pollution models by oil, to develop new algorithms and means of the control and detection of emergency places of underground water pollution. We have created a new numerical model and scheme describing oil infiltration into soil. The constructed scheme is investigated and error of the approximated solution is estimated. Using this scheme, we have carried out numerical experiments for four types of soils dominated within the transport corridor of Georgia. The results of calculations are presented

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At present pipelines have become the most popular means of transporting natural gas. As our practice shows while transportation of natural gas by pipelines over the territory of Georgia, pressure and temperature changes cause formation of a liquid phase owing to partial condensation of the gaseous medium. There are many scientific articles denoted to the problem of prediction of possible points of hydrates origin in the main pipelines. There are several methods for avoiding gas hydrate problems, but generally modern methods for prevention of hydrate formation are based on the following techniques: injection of thermodynamic inhibitors, use of kinetic hydrate inhibitors to sufficiently delay hydrate nucleation/intensification, and maintain pipeline operating conditions outside the hydrate stability zone by insulation, heating and controlling pressure. However, the above techniques may not be economical and practical From existing methods the mathematical modelling with hydrodynamic method is more acceptable as it is very cheap and reliable and has high sensitive and operative features. In the present paper the problem of prediction of possible points of hydrates origin in the main pipelines taking into consideration gas non-stationary flow and heat exchange with medium is studied. We have created a new method prediction of possible points of hydrates origin in the main pipelines taking into consideration gas non-stationary flow. For solving the problem of possible generation point of condensate in the pipeline under the conditions of non-stationary flow in main gas pipe-line the system of partial differential equations is investigated. For learning the affectivity of the method quite general test was created. Numerical calculations have shown efficiency of the suggested method

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As usual, elements of the theory of plane curves are included in various lecture courses of Higher Mathematics (Analytic Geometry, Differential Geometry, Calculus). The beginning courses of Analytic Geometry are primarily devoted to algebraic curves of first and second degree (Le., conical sections) and their properties. In particular, it is demonstrated that any such a curve admits a rational parameterization. In more advanced courses, various parameterizations ration of much more complicated curves are discussed in Calculus lectures and courses of Differential Geometry. These topics are interesting and important from the purely theoretical point of view and from the view-point of applications (in rigid and arch type constructions). The extensive study of these topics is justified from the didactic and methodological stand-point. In our opinion, it is desirable in various mathematics lectures to pay more attention to questions related to general algebraic curves and their properties. This is very reasonable, because the properties of algebraic curves are tightly connected with Diophantine equations in classical Number Theory and the algebra of polynomials. Among topics which seem to be of interest to students, we may propose the following ones: (1) Newton's Theorem on barycenters of sections of an algebraic curve by parallel straight lines (this is one of the first theorems concerning a general property of algebraic curves); (2) the problem of a rational parameterization of an algebraic curve (which is almost trivial for conical sections but is decided negatively for cubic curves); (3) the fact that the envelope of an algebraic family of algebraic curves is also algebraic; (4) the fact that the evolute of an algebraic curve is also algebraic. As a summary, we may state that the discussion of algebraic curves in Higher Mathematics courses should be more thorough and extensive, which will provide under-graduate students with additional valuable information from algebra and geometry.

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The problem connected with the quantum properties of the hexagonal type crystalline structures is considered. The movement of particles at such structures is described by means of the Schrodinger equation. In the stationary case this equation is reduced to the Helmholtz equation with the suitable boundary conditions. We estimated the spectrum of the Helmholtz equation in the hexagonal type stripe by means of the conformal mapping method.

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In the last decades the study of regularities of space-temporal distribution of anthropogenic admixtures in the Black Sea becomes extremely important and urgent because of sharp deterioration of an ecological situation of this unique sea basin. Among different pollutants an oil and oil products present the most widespread and dangerous kind of pollution for the separate regions of the World ocean including the Black Sea. They are able to cause significant negative changes in hydrobiosphere and to infringe natural exchange processes of energies and substances between the sea and atmosphere. Potentially the most dangerous regions from the point of view of oil pollution are coastal zones of the sea, which are exposed to the significant anthropogenic loadings. Modeling of distribution of oil spill enables to estimate pollution zones and probable scales of influence of the pollution on the water environment with the purpose to reduce to a minimum negative consequences of oil pollution in case of emergencies. In this study, spreading of the oil pollution in the Georgian Black Sea coastal zone on the basis of a 2-D numerical model of distribution of oil pollution is simulated. The model is based on a transfer-diffusion equation with taken into account reduction of oil concentrations because of physical - chemical processes. The splitting method is used for solution of the transfer-diffusion equation. The surface current field is determined from the regional baroclinic model of the Black Sea dynamics developed at M Nodia Institute if Geophysics of Iv. Javakhishvili State University. The regional domain, which was limited by Caucasus and Turkish shorelines and the western liquid boundary coinciding with a meridian 39.360E was covered with a grid having 193 x 347 points and grid step equal to 1 km spacing. Numerical experiments are carried out for different hypothetical sources of pollution in case of different sea circulation regimes. Numerical experiments showed that character of regional sea circulation predetermines main features of spatial distribution of the oil pollution in the Georgian Black Sea coastal zone

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see http://rmi.tsu.ge/~gmu/II_Annual_Conference/comming/Abstracts_Batumi_2011_Final.pdf page 133

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The elastic equilibrium of the rectangular parallelepiped is considered, when the symmetry or asymmetry conditions are given on the lateral faces of the parallelepiped and stresses on the upper and lower faces are equal or zero. The problem consists in choosing a temperature distribution on the upper and lower faces so that normal or tangential displacements on those faces take the prescribed values. The solution of the problem is found in analytic form by means of the method of separation of variables.

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In 1905, G. Vitali prove the existence of a non-measurable point set (in the Lebesgue sense). With this classical result absolutely new type of researches started in Mathematics, which also reflected in receiving various Equidecomposability Paradoxes in Euclidean space. Namely, the analysis of the proof of Vitali's Theorem has revealed that the existence of a non-measurable set is closely related to uncountable forms of the Axiom of Choice and certain group-theoretical features of the Lebesgue measure. As it has become clear later on, this factor has been recognized in a different form in remarkable results obtained by F. Hausdorff, S. Banach, A. Tarski, J. Mycielski and others. They results may be considered as a continuation of the research initiated by Vitali whose theorem was generalized in various directions along with obtaining many further equidecomposability paradoxes. A special attention deserves a discovery by J.von Neumann that the existence of a free subgroup of the motion group of three-dimensional Euclidean space, with two independent generators, plays a very important role for the aforementioned theorems and similar statements. The purpose of this report is to review the above-mentioned researches and demonstrate an advisability of their inclusion in an appropriate (more or less adapted) form in the modern courses of Higher Mathematics. This is justified by a lot of interesting connections of the topic with other mathematical disciplines

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In this paper, I defined existence and generality quantifiers with the tau operator and showed some of its properties.

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The mathematical model of growth of the 1D nanocrystals (nanoneedles) in the crystallizer is considered. The process is described by the reaction-diffusion equation with the suitable initial-boundary conditions. We admit, that growth rate depends on time exponentially. In 1D the equation is simplified and by the separation of variables the exact solutions are obtained. Hence the growth rate of the 1D crystal is estimated.

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The functional classes of generalized analytic vectors for generalized Beltrami systems are introduced and investigated. Some properties of these classes which turned to be useful in order to solve the discontinuous boundary value problems are established.

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Different altitudes of the Earth's ionosphere are investigated to reveal some nonlinear solitary structures as the response to the interaction of seismo-electromagnetic emissions of different frequencies with charged particles of the ionospheric gas. It is shown that 1) high-frequency electron cyclotron seismo-electromagnetic emission can generate nonlinear Rossby solitary vortical anticyclone structures in the ionospheric D-layer; 2) nonlinear propagation of low-frequency seismic-origin internal-gravity perturbations in the stably stratified ionospheric E-layer under the excitation of nonlinear solitary vortices can enhance atomic oxygen radiation at wavelength 557.7 nm. Detection of the new oscillation branches and energetically reinforcing nonlinear structures may serve as precursors to earthquake.

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We consider a differential equation u(n)(t)+p(t)∣∣u(σ(t))∣∣μ(t)signu(σ(t))=0, Consider the following general type equation u (n) (t) + F(u)(t) = 0, where n ≥ 2, F : C(R+; R) → Lloc(R+; R) is a continuous mapping. Investigation of asymptotic behavior of solutions of the general equation is interesting on its own. In particular, we used to succeed in obtaining such results for this equation which as a rule were new for linear ordinary differential equations as well. It is an interesting fact that investigation of such general equations enabled me to single out new classes (“almost linear” differential equation) of ordinary differential equations which had not been considered earlier. The obtained results make it clear that the considered equations are interesting and constitute a transitory stage between linear and nonlinear equations. There are considered “almost linear” (essentially nonlinear) differential equation and the sufficient (necessary and sufficient) conditions are established for oscillation of solutions. Some of the already got results are published in the following papers [1–4].

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Nikoloz Muskhelishvili’s famous monograph in elasticity theory [1] had and has a great worldwide influence on investigations in the corresponding fields of mechanics and mathematics. His student and distinguished successor I. Vekua constructed hierarchical models of shells which are the natural continuation of the ideological program of the Georgian Mathematical school founded by N. Muskhelishvili. If we ignore members containing unknown so called moments and their first order derivatives in the governing system of prismatic shells, then the system obtained can be divided in two groups. The first one will contain systems of the 2D elasticity, while the second one will contain Poisson’s equations. This fact makes possible to apply Muskhelishvili’s methods based on the theory of analytic functions of one complex variable. In 1977 I. Vekua [2] wrote: “If we consider shells with cusped edges, then the thickness vanishes on the boundary or on it’s part. In this case we have an elastic system of that equations with degenerations on the boundary. At present, the investigation of the class of such equations is carried out rather intensively (cf. A. Bitsadze, M. Keldysh, S. Tersenov, G. Fichera). However, in the study of the equations generally, and the above system, in particular, only the first steps are made (cf. G. Jaiani)”. The present updated survey is mainly devoted to contribution of Georgian Scientists (in alphabetic order) G. and M. Avalishvili, N. Chinchaladze, D. Gordeziani, G. Jaiani, S. Kharibegashvili, N. Khomasuridze, B. Maistrenko, D. Natroshvili, N. Shavlakadze, and G. Tsiskarishvili in this field. Some of these results are obtained with R. Gilbert, A. Kufner, B. Miara, P. Podio-Guidugli, B.-W. Schulze, and W.L. Wendland within the framework of international projects. As a background along with I. Vekua’s hierarchical models, Timoshenko’s geometrically nonlinear and Kirchoff-Love’s models of plates of variable thickness are used [3]. An exploratory survey with a wide bibliography of results concerning elastic cusped shells, plates, and rods, and cusped prismatic shell-fluid interaction problems one can find in [4]. In the present talk we mainly confine ourselves to cusped prismatic shells and rods. References [1] N. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen, ¨ 1953. [2] I. Vekua, Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston-London-Melbourne, 1985. [3] S. Timoshenko, Theory of Plates and Shells. McGRAW-HILL BOOK COMPANY, INC, New York-Toronto-London, 1959. [4] G. Jaiani, Cusped Shell-like Structures. SpringerBriefs in Applied Sciences and Technology. DOI: 10.1007/978-3-642-22101-9.

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At present, pipelines have become the most popular means of natural gas transportation, but while natural gas is transported by pipelines, pressure and temperature changes cause partial condensation of the gas and formation of a liquid phase-hydrates. There are several methods for avoiding of gas hydrates origination and intensification. Among them are: injection of thermodynamic inhibitors, use kinetic hydrate inhibitors, maintain pipeline normal operating conditions from outside by insulation, heating and controlling of pressure. However, to use the above techniques it is necessary to know the hydrate origination zone. From existing methods of definition hydrates origin zone the mathematical modelling with hydrodynamic method is more acceptable as it is very cheap and reliable. Nowadays there are existing mathematical models which can define possible section of hydrates formation in the main pipeline but for stationary flow of gas. But as we know in reality gas flow in pipelines is non-stationary. In the present paper the problem of prediction of possible points of hydrates origin in the main pipelines taking into consideration gas non-stationary flow and heat exchange with medium is studied. For solving the problem of possible generation point of condensate in the pipeline under the conditions of non-stationary flow in main gas pipe-line the system of partial differential equations is investigated. For learning the affectivity of the method a quite general test is created. Numerical calculations have shown efficiency of the suggested method. Acknowledgment. The research has been funded by Shota Rustaveli National Science Foundation Grant # GNSF/ST09-614/5-210.

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Let us consider the system of singular integral equations containing an immovable singularity (see [1]) The system singular integral equations of the singular integral equations is solved by a collocation method, in particular, a method of discrete singular (see [2]) in both cases uniform, and non-uniformly located knots. The designated project has been fulfilled by financial support of the Georgian National Science Foundation (Grant # GNSF 09 − 614 − 5 − 210). References [1] A. Papukashvili, Antiplane problems of theory of elasticity for piecewice-homogeneous ortotropic plane slackened with cracks. Bull. Georgian Acad. Sci. 169 (2004), No. 2, 267–270. [2] S. M. Belotserkovski and I. K. Lifanov, Numerical methods in the singular integral equations and their application in aerodynamics, the elasticity theory, electrodynamics. (Russian) Nauka, Moskow, 1985.

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This work deals with 3D mixed boundary value problem of the elasticity theory for the isotropic case body. It is supposed that static forces effect the body.We obtained the approximate solutions by means absolutely stable explicit finite-difference schemes.

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