The analytic relation between solutions of the original Cauchy problem and a corresponding perturbed problem is established. In the representation formula of solution, the effects of the discontinuous initial condition and perturbation of the initial data are revealed.

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The article considers the case of plane deformation for the Cowin–Nunziato linear model, which describes the static equilibrium of elastic bodies with voids. The general solution of the system of two-dimensional equations corresponding to this model is represented by any two harmonic functions and the solution of the Helmholtz equation. Based on the general solution and using the method of fundamental solutions, an algorithm is presented that allows one to approximately solve the corresponding boundary value problems. Approximate solutions of various boundary value problems for square domains with circular holes are constructed using this algorithm.

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The present Lecture Notes is devoted to singular partial differential equations, i.e., to partial differential equations with the order degeneracy. It is foreseen as a Lecture Course for the Advanced Courses of TICMI and the Elective Course within the framework of master programs in Mathematics and in Applied Mathematics. The results stated in the course are applied in investigations of cusped prismatic shells and bars and of motion of fluids in angular ducts.

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This handbook of formal analysis in cryptography is very important for secure communication and processing of information. It introduces readers to several formal verification methods and software used to analyse cryptographic protocols. The chapters give readers general knowledge and formal methods focusing on cryptographic protocols. Handbook of Formal Analysis and Verification in Cryptography includes major formalisms and tools used for formal verification of cryptography, with a spotlight on new-generation cryptosystems such as post-quantum, and presents a connection between formal analysis and cryptographic schemes. The text offers formal methods to show whether security assumptions are valid and compares the most prominent formalism and tools as they outline common challenges and future research directions. Graduate students, researchers, and engineers worldwide will find this an exciting read.

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This book constitutes the refereed proceedings of the 19th International Conference on Unity of Logic and Computation, CiE 2023, held in Batumi, Georgia, during July 24–28, 2023. The 23 full papers and 13 invited papers included in this book were carefully reviewed and selected from 51 submissions. They were organized in topical sections as follows: ​Degree theory; Proof Theory; Computability; Algorithmic Randomness; Computational Complexity; Interactive proofs; and Combinatorial approaches.

Action detection in densely annotated, untrimmed videos is a challenging and important task, with important implications in practical applications. Not only the right actions must be discovered, but also their start and end times. Recent advances in deep neural networks have pushed forward the action detection capabilities, in particular the I3D network. This paper describes a network with attention, which is based on the I3D features and includes state-of-the-art blocks, namely: MLP-Mixer and Vision Permutator. A light version of the original network is proposed, called PDAN light, which has 22.5% fewer parameters than the original PDAN, while improving the accuracy a 1.98% on average; and the MLP-Mixer-based architecture which has 34.5% fewer parameters than the original PDAN, while improving the accuracy a 0.95% on average

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We analyze the Schwarz–Christoffel mapping with emphasis on its geometric properties and apply it to give an algorithm for numerical computation of the modulus of a polygonal quadrilateral. We represent the generalized modulus of the quadrilaterals by formal power series and give an algorithm for the computation of its coefficients by a hypergeometric function in a special case. To illustrate the effectiveness of the algorithm some computational results are given.

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We consider equilibrium configurations of three mutually repelling point charges with the Coulomb interaction confined to a simple arc of a constant length with fixed positions of ends. For given values of charges, length of the arc, and distance between its ends, we compute all possible equilibrium configurations. We also study the behavior of equilibrium configurations for variable values of charges and show that the only possible bifurcation is a pitchfork bifurcation. Similar results are presented for elastic loop obeying Hook’s law and for charges interacting via a Riesz potential.

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This book is dedicated to the branch of statistical science which pertains to the theory of hypothesis testing. This involves deciding on the plausibility of two or more hypothetical models based on some data. This work will be both interesting and useful for professional and beginner researchers and practitioners of many fields, who are interested in the theoretical and practical issues of the direction of mathematical statistics, namely, in statistical hypothesis testing. It will also be very useful for specialists of different fields for solving suitable problems at the appropriate level, as the book discusses in detail many important practical problems and provides detailed algorithms for their solutions.

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In the paper the free boundary problem for the stationary and non-stationary 2D viscous fluid flow of large viscosity and low Reynolds number is considered. For the definition of velocity components the Stokes system of equations with the appropriate initial-boundary conditions is studied. The case of the solenoidal body force and a harmonic pressure is considered. By means of the methods of conformal mapping and integral equations the unique solution of the system is obtained. The profiles of free surfaces for the different pressure are plotted by means of Maplesoft.

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The paper is devoted to the homogeneous Dirichlet problem for the vi- bration problem of cusped porous elastic prismatic shell-like bodies in case of the zero approximation of the hierarchical models (see Jaiani, G.: Hierarchical models for viscoelastic Kelvin-Voigt prismatic shells with voids. Bulletin of TICMI, 21(1), 33-44 (2017)). 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.

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The goal of the present paper is to study the propagation of action potential in cardiac tissue using the cable equation. The paper discusses one-dimensional models of continuously coupled myocytes. Electrical behavior in cardiac tissue is averaged over many cells. Therefore, the transmembrane potential behavior for a single cell is studied. Using the monodomain model, in the absence of current at the beginning and end of the cable (cell), the initial boundary problem is posed and solved analytically. The paper also discusses a one-dimensional mathematical model of conduction in discretely coupled myocytes. The electrical behavior in the tissue is studied in individual myocytes, each of which is modeled as a continuum connected through conditions at the cell boundaries, which represent gap junctions. A stationary passive problem with Dirichlet boundary conditions is stated and solved analytically using the bidomain model. The problems are solved by the method of separation of variables. Numerical modeling of transmembrane potential propagation is performed using MATLAB software. Transmembrane isopotential contours, and 2D and 3D graphs corresponding to the obtained numerical results are presented.

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Artificial Intelligence is developing using statistical and logical approaches. 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. Such formalisms allow representation and reasoning about uncertain knowledge. Uncertainty happens in many areas, like medicine, manufacturing, weather forecasts, prediction (of e.g. voting intentions, natural disaster), etc. Ontologies are machine-processable formalisms for knowledge representation. Their purpose is to describe objects according to domain of interests. This knowledge is used by (automated) reasoning systems for query answering. Probabilistic ontologies are obtained by adding a probabilistic interpretation to the constraints forming the ontology, and adapting corresponding reasoning methods to handle these probabilities. The RFEPO is an interdisciplinary project and 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 address the problem of enumerating all maximal clique-partitions of an undirected graph and present an algorithm based on the observation that every maximal clique-partition can be produced from the maximal clique-cover of the graph by assigning the vertices shared among maximal cliques, to belong to only one clique. This simple algorithm has the following drawbacks: (1) the search space is very large; (2) it finds some clique-partitions which are not maximal; and (3) some clique-partitions are found more than once. We propose two criteria to avoid these drawbacks. The outcome is an algorithm that explores a much smaller search space and guarantees that every maximal clique-partition is computed only once. The algorithm can be used in problems such as anti-unification with proximity relations or in resource allocation tasks when one looks for several alternative ways to allocate resources.

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In the paper, we consider a three-dimensional mathematical problem of fluid-solid dynamical interaction, when an anisotropic elastic body occupying a bounded region Ω+ is immersed in an inviscid fluid occupying an unbounded domain Ω−=R^3∖Ω+. In the solid region, we consider the generalized Green–Lindsay's model of the thermo-electro-magneto-elasticity theory. In this case, in the domain Ω+ we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function), while we have a scalar acoustic pressure field in the unbounded domain Ω−. The physical kinematic and dynamical relations are described mathematically by the appropriate initial and boundary-transmission conditions. Using the Laplace transform, the dynamical interaction problem is reduced to the corresponding boundary-transmission problem for elliptic pseudo-oscillation equations containing a complex parameter τ. We derive the appropriate norm estimates with respect to the complex parameter τ and construct the solution of the original dynamical problem by the inverse Laplace transform. As a result, we prove the uniqueness, existence, and regularity theorems for the dynamical interaction problem. Actually, the present investigation is a continuation of the paper [Chkadua G, Natroshvili D. Mathematical aspects of fluid-multiferroic solid interaction problems. Math Meth Appl Sci. 2021;44(12):9727–9745], where the fluid-solid interaction problems for elliptic pseudo-oscillation equations associated with the above mentioned generalized thermo-electro-magneto-elasticity theory are studied by the potential method and the theory of pseudodifferential equations.

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In this paper 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 three-dimensional 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.

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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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In recent decades, the study of the spatial and temporal distribution of anthropogenic impurities in the Black Sea has become extremely important and relevant, due to the sharp deterioration of the environmental situation in this unique marine basin. Among various pollutants, oil and oil products are the most common and dangerous type of pollution for certain areas of the World Ocean, including the Black Sea. They can cause significant negative changes in the hydrobiosphere and disrupt the natural processes of energy and matter exchange between the sea and the atmosphere. The most potentially dangerous zones in terms of oil pollution are the coastal zones of the sea, which are subject to significant anthropogenic pressure. Modeling the spread of oil spills makes it possible to estimate the extent of pollution zones and the degree of possible impact on the aquatic environment in order to minimize the negative consequences of oil pollution in case of emergencies.In this paper, the migration of spilled oil in the coastal zone of the Black Sea of Georgia is modeled on the basis of a two-dimensional numerical model. The model is based on the numerical integration of the transport-diffusion equation, taking into account the change in the concentration of spilled oil caused by physical and chemical processes. For the numerical integration of the complete system of hydrothermodynamic equations, the splitting method is used. Numerical experiments were carried out in the coastal zone of the Black Sea of Georgia for different regimes of marine circulation existing in all four seasons under the conditions of various hypothetical sources of pollution. Some results of numerical experiments are presented

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The problem is discussed based on the Kelvin-Voigt model of a viscous elastic triangular plate with a circular hole. We mean that the outer part of the boundary of the plate is acted upon by rigid stamps with a rectilinear base with normal forces with a given principal vector acting on them (or known constant normal displacements of the boundary points), while the inner boundary is loaded with normal forces of a given constant intensity. Based on the methods of the theory of boundary value problems of conformal reflection and analytic functions, the searchable complex potentials are built efficiently (in an analytical form).

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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The main purpose of this work presents some explicit solutions of boundary value problems in the theory of thermoelasticity for solids with double porosity. Special representations are constructed for the general solution of basic equations. They are expressed through of elementary functions, whose properties are well known. Applying these concepts, in the proposed work, the boundary value problems of statics of the theory of thermoelasticity for an elastic circle with double porosity are solved explicitly, in the form of absolutely and uniformly converging series.

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In this paper, the necessary conditions of optimality of delay parameters, of the initial vector, of the initial and control functions are proved for the nonlinear optimization problem with constant delays in the phase coordinates and controls. The necessary conditions are concretized for the optimization problem with the integral functional and fixed right end.

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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 article studies the stress-strain state of a spongy bone of an implanted jaw. A spongy bone can be considered as a multi porous area with its fissures and pores as most evident components of a double porous system. The work studies the stress-strain state of a spongy jaw-bone 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 jaw-bone. The problem is solved by using the boundary element methods, which are based on the solutions of Flamant (BEMF) and Boussinesq’s (BEMB) problems. The cases of various lengths of an implant diameter are considered. Stresses contours (isolines) in the jaw-bone are drafted and the results obtained by BEMF and BEMB for the different diameters implants are compared. Keywords: spongy bone, implanted jaw, contact problem, boundary element method, Flamant problem, Boussinesq’s problem.

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The issue of approximate solution for ordinary second-order nonlinear differential equation with Sturm-Liouville boundary conditions by the multipoint difference method is studied, when the sufficient conditions for the existence and uniqueness of the solution are met. To find both the solution and its derivative, the method developed in [1] and its modification for the class of less smooth functions is used. The case when the right side of the differential equation is an oscillating function will be studied separately. In this article, the solution of the difference analog is performed at each iteration stage by summing the finite part of the trigonometric rows with variable coefficients, which is related to the issue of selecting the optimal method in the numerical implementation of the "fast Fourier transform".

The paper discussed the process of reduction of the boundary value problem corresponding to elastic thin-walled structures, when the initial problem is replaced by a finite or countable system of two-dimensional differential problems. From an ideological point of view, despite the so-called Among the abundance and variety of "specified theories", we highlight the methods of building models developed by I.Vekua and A.Tvalchrelidze in the direction of ideological innovation. According to our approach, not only in the case of thin walls with finite height, a priori nonlinear boundary conditions on face surfaces are satisfied, which were not considered in countable number of models (even in linear cases) due to their "natural" boundary conditions, which, as K. Rectoris showed, leads to an unstable process.

In the present note, some results about Mazurkiewicz type sets are discussed in the context of their measurability and the Baire property.

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In the paper, the theorems implying the existence of a Rademacher series, convergent to each real-valued function, piecewise continuous over (0, 1) on certain dense subsets of (0, 1), are announced. The set of all Rademacher series with the above-mentioned property is fully described. Among the elements of this set are both almost everywhere convergent and almost everywhere divergent Rademacher series.

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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.

ეს სტატია აღწერს ექსპერიმენტებს, რომლებიც შესრულებულია ციანობაქტერიების გარკვეული ექსტრემალური შტამების გადარჩენის, ზრდის, სპეციფიკური ადაპტაციისა და ბიორემედიაციის პოტენციალის შესასწავლად ატმოსფერული შემადგენლობის, ტემპერატურისა და წნევის სიმულაციის ფარგლებში, რომელიც მოსალოდნელია მარსის მომავალ სათბურში. პირველადი სახეობები მოპოვებულია საქართველოში მარსის ანალოგიური უბნებიდან. შედეგები ნათლად აჩვენებს, რომ სპეციფიკური ბიოქიმიური ადაპტაცია საშუალებას აძლევს ამ ავტოტროფებს მეტაბოლიზდეს AMG-ში (Artificial Martian Ground) და დაგროვდეს ბიოგენური ნახშირბადი და აზოტი. ამრიგად, ამ აღმოჩენებმა შეიძლება ხელი შეუწყოს მომავალი მარსის სოფლის მეურნეობის განვითარებას, ისევე როგორც სიცოცხლის მხარდაჭერის სისტემების სხვა ასპექტებს მარსის საცხოვრებელ სადგურებზე. კვლევამ აჩვენა, რომ კარბონატული ნალექი და აზოტის ფიქსაცია, რომელსაც ახორციელებს ციანობაქტერიული საზოგადოებები, რომლებიც აყვავდებიან იმიტირებული მარსის სათბურის პირობებში, ჯვარედინი ბიოლოგიური პროცესებია. ამავდროულად, პერქლორატების არსებობა (დაბალი კონცენტრაციით) მარსის ნიადაგში შეიძლება გახდეს ჟანგბადის და, ირიბად, წყალბადის საწყისი წყარო ფოტო-ფენტონის რეაქციების საშუალებით. ამ ექსპერიმენტების შედეგად მიიღეს სხვადასხვა კარბონატები, ამონიუმის და ნიტრატების მარილები. ეს გავლენას ახდენს AMG-ისა და მისი კომპონენტების pH-ზე, მარილიანობასა და ხსნადობაზე, ამიტომ გაუმჯობესდა AMG-ის მწირი ბიოგენური თვისებები, რაც აუცილებელია სასოფლო-სამეურნეო კულტურების მდგრადი ზრდისთვის. ამრიგად, შესაძლებელია მიკროორგანიზმების გამოყენება ხელოვნური მარსის ნიადაგის ბიოლოგიური რემედიაციისა და უწყვეტი ადგილზე განაყოფიერებისთვის.

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Under GCH, there are described the cardinalities of all Hausdorff topological groups G such that there is a nonzero Borel measure on G having the card ⁢ (G) {{\rm card}(G)} -Suslin property and quasi-invariant with respect to an everywhere dense subgroup of G. Some connections are pointed out with the method of Kodaira and Kakutani (1950) for constructing a nonseparable translation invariant extension of the standard Lebesgue (Haar) measure on the circle group {{\bf S}_{1}}.

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The notion of a generalized random variable is introduced in terms of extensions of a given probability measure. Some properties of generalized random variables are considered.

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We consider some extensions of invariant (quasi-invariant) measures on a ground set E, which have a π-base of cardinality not exceeding card(E).

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For invariant (quasi-invariant) σ-finite measures on an uncountable group, the behavior of absolutely negligible sets with respect to the algebraic sums is studied.

The limiting distribution of the statistic, which describes the mutual deviation of the projection type estimates from each other of distribution density in p ≥ 2 independent samples is established. The goodness-of-fit test is constricted. Various examples are given.

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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.

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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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In the frame of shallow water model the influence of free surface action on nonlinear propagation of planetary ULF magnetized Rossby waves in the weakly ionized ionospheric D-, E-, and F-layers is revealed. Relevant nonlinear dynamic equations satisfying several conservation laws are obtained and investigated. The role of Hall and Pedersen conductivities is investigated explicitly. It is shown that while potential vorticity is conserved in the ionospheric D-,and E-layer it is broken by Pedersen conductivity in the ionospheric F-layer. Similar to KdV nonlinearity two new scalar nonlinearities due to Pedersen conductivity are revealed in the F-layer. Obtained results extend and complement known theoretical investigations and are especially relevant for nonlinear vortical propagation of magnetized Rossby waves in the weakly ionized ionospheric plasma.

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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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A mathematical model is formulated for an initial-boundary value problem associated with the J. Ball integro-differential equation, which serves as a mathematical description of the dynamic state exhibited by a beam. The solution to this problem is approximated through a combination of the Galerkin method, a stable symmetrical difference scheme, and the Jacobi iteration method. This paper desires to present an approximate solution to a practical problem, specifically focusing on the numerical results obtained from the initial-boundary value problem pertaining to a specific iron beam. Notably, the effective viscosity of the material is considered to be dependent on its velocity

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We study the matching problem of regular tree languages, that is, "∃σ:σ(L)⊆R?" where L,R are regular tree languages over the union of finite ranked alphabets Σ and X where X is an alphabet of variables and σ is a substitution such that σ(x) is a set of trees in T(Σ∪H)∖H for all x∈X. Here, H denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if L and R are regular, then the problem "∃σ∀x∈X:σ(x)≠∅∧σ(L)⊆R?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when L,R are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if L⊆T(Σ∪X) and R⊆T(Σ) are regular tree languages over finite or infinite trees, then the problem "∃σ∀x∈X:σ(x)≠∅∧σio(L)⊆R?" is decidable. Here, the subscript "io" in σio(L) refers to "inside-out". Moreover, there are only finitely many maximal solutions σ, the maximal solutions are regular substitutions and effectively computable. The corresponding question for the outside-in extension σoi remains open, even in the restricted setting of finite trees.

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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 ontologies, where information is vague and incomplete. Such kind of ontologies have applications in many different fields, such as medicine, biology, e-commerce and the like. They require concepts from both, fuzzy and unranked theories. This is a follow-up paper of a one by the authors, where an unranked fuzzy logic was introduced. Here we develop a tableau method for reasoning over such logic. The unranked fuzzy logic is an extension of many-valued logics with sequence variables and flexible-arity function and predicate symbols. The unranked fuzzy tableau calculus corresponds to Hájek’s witnessed fuzzy logics and is therefore complete.

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In this paper we define an unranked nominal language, an extension of the nominal language with tuple variables and term tuples. We define the unification problem for unranked nominal terms and present an algorithm solving the unranked nominal unification problem.

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There is a Mazurkiewicz set in the Cohen–Halpern–Levy model.

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In the paper the unsteady incompressible fluid flow over the infinite and finite prismatic bodies is studied. Mathematically this problem is modeled as 3D Navier-Stokes equations (NSE) for the fluid velocity components with the appropriate initial-boundary conditions. The study of the fluid flow over the bodies with the sharp edges is the important problem of Aerodynamics and Hydrodynamics. We admit that near the sharp edges the velocity components are non-smooth. By the methods of mathematical physics the bounded novel exact solutions are obtained for the specific pressure. The profile of the velocity is plotted for the different parameters by means of “Maple”.

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We study the matching problem of regular tree languages, that is, "∃σ:σ(L)⊆R?" where L,R are regular tree languages over the union of finite ranked alphabets Σ and X where X is an alphabet of variables and σ is a substitution such that σ(x) is a set of trees in T(Σ∪H)∖H for all x∈X. Here, H denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if L and R are regular, then the problem "∃σ∀x∈X:σ(x)≠∅∧σ(L)⊆R?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when L,R are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if L⊆T(Σ∪X) and R⊆T(Σ) are regular tree languages over finite or infinite trees, then the problem "∃σ∀x∈X:σ(x)≠∅∧σio(L)⊆R?" […]

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In this paper, in a general formulation, we solve the initial-boundary value problems of dynamics for an isotropic elastic body with a double voids. Using the Laplace transform, these problems are reduced to pseudo-oscillation boundary value problems the solutions of which are represented explicitly as absolutely and uniformly converging series. It is proved that inverse transforms yield solutions of the initial dynamic problems. The uniqueness of regular solutions of the considered problems is investigated.

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Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is a sum of operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. Each addend is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval classic difference problems are solved in parallel (independently from each other) respectively with the operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. The weighted average of the received solutions is announced as an approximate solution at the right end of the local interval. Convergence of the proposed scheme is proved and the approximate solution error is estimated, as well as the error of the difference analogue for the first-order derivative for the case when the initial problem data satisfy the natural sufficient conditions for solution existence.

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In the Banach space, the analytical semi-group is approximated by a sequence of natural degrees of the fractional-linear operator functions. It is proved that the order of the approximation error in the domain of the generating operator is O(n^{-2}ln(n)). We also consider the approximation of the semi-group exp(-tA) (t≥0) by weighted means when A is a self-adjoint positive definite operator, which can be represented as a finite sum of self-adjoint positive definite operators. It is proved that the order of the approximation error in the operator norm is O(n ^ {- 1/2} ln(n)).

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We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved.

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In the present paper we give a necessary and sufficient condition on monotone weights, which guaranties the Nörlund means of Walsh-Fourier series converge in L_1 norm and C_W norm. We also discuss the almost everywhere summability by Nörlund means and we prove that our condition is sufficient for almost everywhere convergence of Nörlund means for all integrable functions.

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This article is concerned with the coupled linear quasi-static theory of thermoelasticity for porous materials under local thermal equilibrium. The system of equations is based on the constitutive equations, Darcy's law of the flow of a fluid through a porous medium, Fourier's law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of governing equations is expressed in terms of displacement vector field, the change of volume fraction of pores, the change of fluid pressure in pore network and the variation of temperature of porous material. 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 boundary value problems (the Dirichlet type boundary value problem for a sphere and the Neumann type boundary value problem for a space with a spherical cavity) are solved explicitly. The obtained solutions are given by means of the harmonic, bi-harmonic and metaharmonic 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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We discuss several generalizations of Riemann-Hilbert boundary value problem and Riemann-Hilbert monodromy problem formulated in terms of representations of compact Lie groups and motivated by recent applications in mechanics and modern mathematical physics. A relevant setting for generalization of Riemann-Hilbert boundary problem is provided by linear representations of compact Lie groups and loop groups. Generalizations of Riemann-Hilbert monodromy problem are developed in the framework of principal bundles of compact Lie groups and meromorphic connections on Riemann surfaces. The systematic use of Lie groups and homological concepts reveals new connections between these classical problems and clarifies the role of piecewise constant coefficients.

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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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We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

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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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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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A short consideration of the existing approaches of statistical hypothesestesting is given below. Among classical methods, comparatively new ConstrainedBayesian Method and its peculiarities are introduced. A brief description of theessences of these methods is given. Recommendation for choosing a concretemethod for statistical hypotheses testing is given finally

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

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The paper deals with the approximate solving of an inverse problem for the nonlinear delay differential equation, which consists of finding the initial moment and delay parameter based on some observed data. The inverse problem is considered as a nonlinear optimal control problem for which the necessary conditions of optimality are formulated and proved. The obtained optimal control problem is solved by a method based on an improved parallel evolutionary algorithm. The efficiency of the proposed approach is demonstrated through various numerical experiments.

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The theory of low-frequency internal gravity waves (IGWs) is readdressed in the stable stratified weakly ionized Earth’s ionosphere. The formation of dipolar vortex structures and their dynamical evolution, as well as, the emergence of chaos in the wave-wave interactions are studied both in presence and absence of the Pedersen conductivity. The latter is shown to inhibit the formation of solitary vortices and the onset of chaos.

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In the paper, the limiting distribution is established for an integral square deviation of estimates of Bernoulli regression functions based on two group samples. Based on these results, the new test is constructed for the hypothesis testing on the equality of two Bernoulli regression functions. The question of consistency of the constructed test is studied, and the asymptotic of the test power is investigated for some close alternatives.

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The nonlinear propagation of internal gravity waves in the weakly ionized, incompressible Earth’s ionosphere is studied using the fluid theory approach. Previous theory in the literature is advanced by the effects of the terrestrial inhomogeneous magnetic field embedded in weakly ionized ionospheric layers ranging in altitude from about 50 to 500 km. It is shown that the ionospheric conducting fluids can support the formation of solitary dipolar vortices (or modons). Both analytical and numerical solutions of the latter are obtained and analyzed. It is found that in absence of the Pedersen conductivity, different vortex structures with different space localization can be formed which can move with the supersonic velocity without any energy loss. However, its presence can cause the amplitude of the solitary vortices to decay with time and the vortex structure can completely disappear owing to the energy loss. Such energy loss can be delayed, i.e., the vortex structure can prevail for relatively a longer time if the nonlinear effects associated with either the stream function or the density variation become significantly higher than the dissipation due to the Pedersen conductivity. The main characteristic dynamic parameters are also defined which are in good correlation with the existing experimental data.

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The nonlinear propagation of internal gravity waves in the weakly ionized, incompressible Earth’s ionosphere is studied using the fluid theory approach. Previous theory in the literature is advanced by the effects of the terrestrial inhomogeneous magnetic field embedded in weakly ionized ionospheric layers ranging in altitude from about 50 to 500 km. It is shown that the ionospheric conducting fluids can support the formation of solitary dipolar vortices (or modons). Both analytical and numerical solutions of the latter are obtained and analyzed. It is found that in absence of the Pedersen conductivity, different vortex structures with different space localization can be formed which can move with the supersonic velocity without any energy loss. However, its presence can cause the amplitude of the solitary vortices to decay with time and the vortex structure can completely disappear owing to the energy loss. Such energy loss can be delayed, i.e., the vortex structure can prevail for relatively a longer time if the nonlinear effects associated with either the stream function or the density variation become significantly higher than the dissipation due to the Pedersen conductivity. The main characteristic dynamic parameters are also defined which are in good correlation with the existing experimental data.

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In this work, we focused on the study of the influence of the rotation of the Earth’s atmosphere on the properties of evanescent acoustic-gravity waves, which we studied earlier in the absence of rotation. It is shown that evanescent acoustic-gravity waves (AGW) with a continuous spectrum can exist in an atmosphere rotating with an angular frequency X below the frequency 2 X (the Coriolis parameter). It is also shown that the rotation of the atmosphere also leads to a modification of the previously discovered continuous spectrum of evanescent AGWs with frequencies higher than the Coriolis parameter, which fills the entire ‘‘forbidden” region in the diagnostic diagram between freely propagating acoustic and internal gravity waves. It is concluded that the AGW spectrum in the diagnostic diagram consists of regions of acoustic and gravity waves, as well as two regions of evanescent waves, and is continuous. The found new spectrum expands the full spectrum of evanescent waves and indicates the need to search for evanescent waves at ultra-low frequencies. The result is obtained for high-latitude regions from a system of linear hydrodynamic equations for perturbations that take into account the rotation of the Earth’s atmosphere, by imposing an additional spatial relation on the components of the perturbed velocity vector of the elementary volume of the medium, which proposed by us for the first time. This made it possible to obtain an infinite number of solutions describing evanescent acoustic-gravity waves propagating in an isothermal atmosphere. The specified connection between the components of the perturbed velocity is characterized by the a parameter, which can only take real values. It has been established that the detected spectrum of evanescent acoustic-gravity waves can exist only at 0 < a < 1, while the previously found spectrum of these waves, modified by taking into account the Earth’s rotation, is realized at arbitrary values of a. Analytical and numerical analysis of the obtained solutions is carried out. It is shown that these solutions, at certain values of the parameter a, pass into the previously studied evanescent modes.

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The terrestrial magnetosheath is characterized by large-amplitude magnetic field fluctuations. In some regions, and depending on the bow-shock geometry, these can be observed on several scales, and show the typical signatures of magnetohydrodynamic turbulence. Using Cluster data, magnetic field spectra and flatness are observed in two intervals separated by a sharp transition from quasi-parallel to quasi-perpendicular magnetic field with respect to the bow-shock normal. The multifractal generalized dimensions Dq and the corresponding multifractal spectrum f(α) were estimated using a coarse-graining method. A p-model fit was used to obtain a single parameter to describe quantitatively the strength of multifractality and intermittency. Results show a clear transition and sharp differences in the intermittency properties for the two regions, with the quasi-parallel turbulence being more intermittent

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In this paper, a finite difference scheme for one system of nonlinear partial differential equations is constructed and investigated. Investigated model is based on the well-known system of Maxwell's equations and represents some of its generalizations. The one-dimensional case with threecomponent magnetic field is considered. The convergence of the scheme under consideration is studied and estimate of the error of the approximate solution is obtained

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We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved.

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In the present paper we study a uniform unsteady incompressible creeping fluid flow in the finite pipes with the polygonal cross-section. Our purpose is to define the velocity of the flow for the certain pressure. The velocity components satisfy the Stokes linear system of equations (STS) with the appropriate initial-boundary conditions. The STS represents a linearized Navier-Stokes equations (NSE) for the small Reynolds number. In our work STS is studied for the specific pressure. We used the conformal mapping method and the Poisson formula and reduced the Stokes system to the system of Fredholm integral equations. By means of the stepwise approximation method the unique solution of this system is obtained. Several cases of the fluid flow in the pipes having regular polygons as a cross-section are considered.

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The paper 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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In the present work the boundary value problems 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 internal boundary value problems of elastic equilibrium of the homogeneous isotropic body bounded by coordinate lines of the parabolic coordinate system, when on parabolic border normal or tangential stresses are given. Exact solutions are obtained using the method of separation of variables. Numerical results and corresponding graphs of above-mentioned problems are presented.

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In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain $\Omega\subset R^3$ , when the boundary surface $S=\partial\Omega$ is divided into two disjoint parts $S_D$ and $S_N$ where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. 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 the 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. 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 $W^1_2(\Omega)$. Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the $L_p$ -based Besov spaces, which under appropriate boundary data implies $C^\alpha$-Hölder continuity of the solution to the mixed BVP in the closed domain $\bar\Omega$ with $\alpha=\frac{1}{2}−\varepsilon$, where $varepsilon>0$ is an arbitrarily small number.

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This paper outlines the famous mathematical contributions due to Bogdan Bojarski (1931–2018), an active member of the ISAAC society. His results constitute a part of the modern complex analysis. We outline the Gokhberg-Krein-Bojarski criteria of the stability of partial indices. This result makes clear the algebraic and topological structure of boundary value problems. The special attention is payed to one of the favorite topics discussed by Bojarski during his life, quasiconformal mappings. His results on boundary value problems including the Riemann-Hilbert problem for a multiply connected domain, and their applications to composites are briefly presented. The paper contains unique, perhaps first published, pictures.

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In this paper we prove Liouville theorem for the irregular nonhomogeneous Cauchy-Riemann equation depended on parameter's and we show that qualitative properties of generalized analytic vectors strongly depend on the asymptotic parameters.

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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. 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.

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We construct hierarchical models for the heat conduction in standard and prismatic shell-like and rod-like 3D domains with non-Lipschitz boundary, in general.

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Imitation models for computing the environmental water pollution level depending on the intensity of pollution sources created by the author over the years are presented. For this purpose, an additive model of a non-stationary random process is considered. For the modeling of its components, models that consider only dilution and self-purification processes are proposed for waste water and three-dimensional turbulent diffusion equations for river waters, and multidimensional Gaussian Markov series are proposed for modeling the random component. The purpose, the capabilities and the peculiarities of such imitation models are discussed taking into account the peculiarities of the water objects. The modular principle of creating imitation models is proposed to facilitate their development and use

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Based on a three-dimensional hydrostatic mesoscale model, the air flow over the complex relief of the South Caucasus (Georgia) is studied. Numerical experiments have shown a strong influence of the Likhi Range on the movement of monsoon air currents between the Main and Lesser Caucasus Ranges. Also, in order to use the effect of an increase in the speed of the air flow after the flow around the Likhy Range, the strong wind regime and its statistical characteristics in the region of the Rioni River for the period 1960-2021 were studied. It was determined that in terms of energy, the main range of wind speeds for the Kutaisi region is 16-20 m/s, which provide automatic operation of wind farms and are an important basis for the development of wind farms in the western regions of Georgia.

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Morphological synthesis of Georgian words requires to compose the word-forms by indication unchanged parts and morphological categories. Also, it is necessary by using a stem of the given word to get by the computer all grammatically right word-forms. In case of morphological analysis of Georgian words, it is essential to decompose the given word into morphemes and get the definition each of them. For solving these tasks we have developed some specific approaches and created software. Its tools are efficient for a language, which has free order of words and morphological structure is like Georgian. For example, a Georgian verb (in Georgian: “ ” - ts’era, in English: Writing) has several thousand verb-forms. It is very difficult to express morphological analysis’ rules by finite automaton and it will be inefficient as well. Splitting of some Georgian verb-forms into morphemes requires non-deterministic search algorithm, which needs many backtracks. To minimize backtracking, it is necessary to put constraints, which exist among morphemes and verify them as soon as possible to avoid false directions of search. Sometimes the constraints can be as a description type of specific cases of verbs. Thus, proposed software tools have many means to construct efficient parser, test and correct it. We realized morphological and syntactic analysis of Georgian texts by these tools. Besides this, for solving such problems of artificial intelligence, which requires composing of natural language’s word-form by using the information defining this word-form, it is convenient to use the software developed by us.

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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 equations with the suitable initial-boundary conditions. We modify 2D NSE and find new class of solutions. It is supposed that near sharp edges the velocity components are non-smooth and by the methods of mathematical physics we obtain novel exact solutions of NSE for the specific pressure.

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The paper studies the elastic equilibrium of a homogeneous isotropic incompressible elliptic cylinder with a hole, when normal or tangential stresses are applied on its internal and external surfaces. The cylinder is in a state of plane deformation. Thus, the boundary value problems are set and solved for an incompressible confocal elliptic ring in an elliptic coordinate system. The boundary value problems for a confocal elliptic ring are given with the superposition of the internal and external problems of an ellipse. For incompressible bodies, equilibrium equations and Hooke’s law are written in the elliptic coordinate system, boundary value problems are set and solutions are presented with two harmonic functions, which are obtained by a method of separation of variables. Two test problems for a confocal elliptic semiring are solved and the graphs relevant to the numerical values are drafted. One problem concerns the change in the deformed state of the incompressible confocal elliptic semiring in relation with the change in the axes of the elliptical hole, while in the second problem the deformation process of the rubber shaft with the elliptical hole is investigated.

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In the report the three-dimensional system of equations of equilibrium for solids with voids is considered. From this system of equations, using a reduction method of I. Vekua, we receive the equilibrium equations for the shallow shells. Further we consider the case of plates with constant thickness in more detail. Namely, the systems of equations corresponding to approximations N = 0 is 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.

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In the present paper, we explicitly solve, in the form of absolutely and uniformly convergent series, a two-dimensional boundary value problem of statics in linear theory elasticity for an isotropic elastic disk consisting of empty pores. The uniqueness theorem for the solution is proved. For a particular problem numerical results are given.

In the article, using the method of fundamental solutions, approximate solutions are constructed for some boundary-value problems of tension-compression and bending of homogeneous isotropic plates of constant thickness with holes. In this case, the elastic equilibrium of the plates is described by the refined system of equations of I. Vekua in the case of the $N=1$ approximation.

The problem of finding an equistrong contour in a rectangular viscoelastic plate is considered by using the Kelvin–Voigt model. It is assumed that normal contractive forces with prescribed principal vectors (or with constant normal displacements) are applied to the rectangle sides by means of a linear absolutely rigid punch, while an unknown part of the boundary (an unknown equistrong contour) is free from external forces. The equistrength of an unknown contour lies in the fact that tangential normal stress at each point of the contour admits the same values. To solve the problem, we use the methods of conformal mappings and of boundary value problems of analytic functions. The equation of an unknown contour, as a function of a point and time, is constructed effectively (analytically).

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In this paper 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 the parabolic border normal stress is given. The exact solution is obtained by the method of separation of variables. Using the MATLAB software, the numerical results are obtained at some characteristic points of the body and relevant 2D and 3D graphs are presented

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In the paper, several classes of measures and their cardinality number are discussed

A new estimator for a Bernoulli regression function based on Bernstein polynomials is constructed. Its consistency and asymptotic normality are studied. A criterion for testing the hypothesis on the form of a Bernoulli regression function and a criterion for testing the hypothesis on the equality of two Bernoulli regression functions are constructed. The consistency of these two criteria is studied.

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The Nadaraya–Watson kernel-type nonparametric estimate of Poisson regression function is studied. The uniform consistency conditions are established and the limit theorems are proved for continuous functionals on C[a, 1 − a], 0 < a < 1/2.

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In the present paper, theorems concerning the Walsh series with gaps are formulated. Rademacher series are particular cases of considered Walsh series. Formulas to calculate coefficients of such a Walsh series by means of values of the sum of this series at certain two points are presented. These two points vary depending on the index of the coefficient being calculated.

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A form of the system of differential equations is established, which satisfies the sensitivity coefficients of a controlled differential model of the immune response considering perturbations of the delay parameter, the initial and control functions.

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In the paper, the nonlinear controlled functional integral equation corresponding to the quasilinear neutral functional differential equation with two types controls is constructed. A structure and properties of the integral kernel are established. Equivalence of the functional integral equation and the neutral functional differential equation is established also.

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The nonlinear controlled integral equation corresponding to the quasi-linear controlled neutral differential equation is constructed. The structure and properties of kernel of the integral equation are established. For the neutral and integral equations theorems on the existence and uniqueness of solution are provided. The equivalence of the integral and neutral differential equations is established.

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Human Activity Recognition and particularly detection of abnormal activities such as falls have become a point of interest to many researchers worldwide since falls are considered to be one of the leading causes of injury and death, especially in the elderly population. The prompt intervention of caregivers in critical situations can significantly improve the autonomy and well-being of individuals living alone and those who require remote monitoring. This paper presents a study of accelerometer and gyroscope data retrieved from smartphone embedded sensors, using iOS-based devices. In the project framework there was developed a mobile application for data collection with the following fall type and fall-like activities: Falling Right, Falling Left, Falling Forward, Falling Backward, Sitting Fast, and Jumping. The collected dataset has passed the preprocessing phase and afterward was classified using different Machine Learning algorithms, namely, by Decision Trees, Random Forest, Logistic Regression, k-Nearest Neighbour, XGBoost, LightGBM, and Pytorch Neural Network. Unlike other similar studies, during the experimental setting, volunteers were asked to have smartphones freely in their pockets without tightening and fixing them on the body. This natural way of keeping a mobile device is quite challenging in terms of noisiness however it is more comfortable to wearers and causes fewer constraints. The obtained results are promising that encourages us to continue working with the aim to reach sufficient accuracy along with building a real-time application for potential users.

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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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An updated comprehensive exploratory survey of the literature on elastic cusped standard and prismatic shells and bars, in particular, cusped plates, and to the corresponding singular partial differential equations and systems is given. The governing systems of equations of statics and dynamics in the cases of compression–tension and bending are derived from I. Vekua’s hierarchical models of the generic N th-order approximation, in particular, for cusped elastic plates. In the static case, the well-posedness of the basic BVPs for cusped plates is investigated. The BCs at the cusped edge are non-classical, in general, and depend on the kind of thinning. The corresponding criteria are established. In the special cases, the BVPs are solved in the explicit form

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We consider the question whether a given real-valued non-negative upper semi-continuous function on a topological space E is the oscillation function of a Borel real-valued function defined on the same space E.

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The generalized nonmeasurability of certain classical point sets (such as Vitali sets, Bernstein sets, and Hamel bases) is considered in connection with CH and MA.

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It is shown that the cardinality continuum is not measurable in the Ulam sense if and only if for every nonzero σ-finite diffused measure µ on R2 there is a µ-nonmeasurable uniform subset of R2 . Several related results are also considered.

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It is proved that there exists a T2-negligible set in the plane R2 , which simultaneously is S2-absolutely nonmeasurable

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In this research the quasi-static boundary value problem of the coupled theory of elasticity for porous materials is examined. The problem of equilibrium of a spherical layer is reviewed and the explicit solution of the Dirichlet boundary value problem is given as a absolutely and uniformly convergent series.

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There are considered main methods of machine learning and their realization algorithms and examples of use. In particular, issues such as dependency recovery, decision theory, information theory, probabilistic distributions, probabilistic models, linear regression models, Bayesian models, linear classification models, machine learning algorithms, neural networks, and nuclear methods. In methodological guidelines there is suggested to discuss these theoretical issues and solving relevant practical tasks in seminar courses with doctoral students. Methodical guidelines are recommended for PhD students in Informatics

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In this research the quasi-static boundary value problem of the coupled theory of elasticity for porous materials is examined. The problem of equilibrium of a spherical layer is reviewed and the explicit solution of the Dirichlet boundary value problem is given as absolutely and uniformly convergent series. 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 displacements without friction. The problem consists of determining the corresponding complex potentials characterizing the equilibrium of the plate by the Kelvin-Voigt model.

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Statistical hypotheses testing is one of the basic direction of mathematical statistics the methods of which are widely used in theoretical research and practical applications. These methods are widely used in medical researches too. Scientists of different fields, among them of medical too, that are not experts in statistics, are often faced with the dilemma of which method to use for solving the problem they are interested. The article is devoted to helping the specialists in solving this problem and in finding the optimal resolution. For this purpose, here are very simple and clearly explained the essences of the existed approaches and are shown their positive and negative sides and are given the recommendations about their use depending on existed information and the aim that must be reached as a result of an investigation

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The construction of solutions in explicit form is especially important from the point of view of its application, since it makes it possible to effectively carry out a quantitative analysis of the problem under study.This paper investigates the processes of deformation of solids in the quasi-static case. Two-dimensional boundary value problems of Dirichlet and Neumann for an elastic body with double porosity are considered. In Using the Laplace transform, these problems are reduced to auxiliary boundary value problems. Special representations of solutions to auxiliary boundary value problems are constructed using elementary functions that allow reducing the original system of equations to equations of a simple structure and facilitate the solution of the original problems. Auxiliary boundary value problems are solved for a specific elastic body - a porous disk. Solutions to these problems are obtained in the form of series. Conditions are provided that ensure the absolute and uniform convergence of these series and the use of the inverse Laplace theorem. It is proved that the inverse transforms provide a solution to the initial problems.

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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 circular ring when the stress tensor and the equilibrated stress vector are given on the boundary.

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The present work studies the change in the strength of a quite long isotropic thick-wall pipe (circular cylinder) for the varying pipe diameter, wall thickness and material. The pipe is in the plane deformed state, i.e. plane deformation is considered. Based on the problems of statics of the Theory of Elasticity, a mathematical model to calculate the strength of the thick-wall pipe was developed and the problems of statics of the Theory of Elasticity were set and solved analytically in the polar coordinate system. The analytical solution was obtained by the method of separation of variables, which is presented by two harmonious functions. The dependence of the pipe strength on the thickness and material of the pipe wall, when (a) normal stress is applied to the internal boundary (internal pressure) and external boundary is free from stresses and (b) normal stress is applied to the external boundary (external pressure) and the internal boundary is free from stresses, is studied. In particular, the minimum thicknesses of the walls of homogenous isotropic circular cylinders of different materials and diameters with a plane deformed mode when the pressures in the cylinders do not exceed the admissible values were identified. Some numerical results are presented as tables, graphs and relevant consideration.

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The paper considers the concrete problems of the punch for a viscoelastic half-plane by the Kelvin–Voigt model. It is known that many buildings and composite materials exhibit viscoelastic properties which are reflected in Hooke’s law in which the stresses are proportional both to the deformations and to their derivatives in time. The purpose of the present paper is to study the some concrete problems of the punch for a viscoelastic half-plane by means of Kolosov–Muskhelishvili’s method for the Kelvin–Voigt model and get formulas for the distribution of the tangential and normal stresses under the punch.

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An 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 omega={-xi1Full article

For an optimal control problem involving neutral differential equation x˙(t) = A(t) ˙x(t − σ) + f(t, x(t), x(t − τ ), u(t)) +g(t, x(t), x(t − τ ), u(t − θ)), t ∈ [t0, t1] existence theorems of an optimal element are provided. Under element we imply the collection of delay parameters σ and τ, the initial moment and vector, control and finally moment.

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In the paper non-stationary 3D incompressible viscous fluid flow over the point, the infinite line, the plane, the rectangular prism and the octahedron is studied. The corresponding Navier-Stokes equations (NSE) with the appropriate initial-boundary conditions are considered. (NSE is a very important equation and has various applications in Plasma Physics, Astrophysics, magma physics, geophysical fluids, biophysics, nanofluids etc. NSE describes significant characteristics of different fluids. The exact solutions are obtained in a very few cases and especially in 2D. In the paper the novel exact non-smooth solutions blow-up in time are obtained for the specific pressure and initial conditions by means of the methods of Mathematical Physics. Besides, the solutions for the turbulent flows are given. Those solutions are new and are applied to the solving of the problem of some substance propagation in the space by the turbulent flow. The profiles of the velocity and substance distribution are constructed by means of “Maple” for the different parameters.) The results have applications to the description of atmospheric and ocean currents, nanosciences

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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. By the integral equation method the existence and uniqueness of the solution is proved. 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 creeping flow in the infinite pipes of the different configuration is considered in case when pressure is regulated and unknown function is depending on time exponentially. By means of integral equation method the existence and uniqueness of the solution is proved. The exact solutions are obtained in case of lemniscat and ellipse.

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The linear stability and Hopf bifurcation of a solution of the initial-boundary value problem for one system of nonlinear partial differential equations (NPDEs) is studied. A blow up result is given.

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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 singular integral equation containing a fixed-singularity with respect to characteristic function of disclosure of crack when the crack reaches the dividing border of interface with the right angle. The method of discrete singularity is applied to finding a solution of the obtained singular integral equation. The corresponding new algorithm is constructed and realized. In this work, behavior of solutions in the neighborhood of the crack endpoints is studied by a method of discrete singularity with uniform division of an interval by knots. The results of computations are represented.

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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 [1]. 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 [2]. In this article, a climate change parameter such as temperature has been investigated by SD and DD methods with an emphasis on SD. R E F E R E N C E S 1. Stocker, T.F. Climate Change 2013: The Physical Science Basis: Working Group I Contribution to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, 2014, 267-270. 2. Davitashvili, T., Kutaladze, N., Kvatadze, R., Mikuchadze, G. Effect of dust aerosols in forming the regional climate of Georgia. Scalable Computing: Practice and Experience, 19, 2 (2018), 50-57.

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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.

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In the paper, a purely implicit four-layer semi-discrete scheme for an abstract evolution equation is reduced by means of an perturbation algorithm to two-layers schemes. Based on the latter schemes, an approximate solution of the initial problem is constructed. The approximate solution error estimate is obtained in the Hilbert space by using associated polynomials

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PρLog [11] is a rule-based system that supports programming with individual, hedge, function and context variables. It extends Prolog with rule-based programming capabilities to manipulate sequences of terms, also known as hedges. The four kinds of variables help to traverse tree forms of expressions both in horizontal and vertical directions, in one or more steps. It facilitates to have expressive pattern matching that often helps to write short and intuitive code. Another important feature of PρLog is the use of strategies. They provide a mechanism to control complex rule-based computations in a highly declarative way. With the help of strategies, the user can combine simpler transformation rules into more complex ones. In this way, PρLog conveniently combines the whole Prolog power with rule-based strategic programming features. PρLog is based on ρLog calculus [17], where the inference system basically is the SLDNF-resolution with normal logic program semantics [14]. It has been successfully used in the extraction of frequent patterns from data mining workflows [20], XML transformation and web reasoning [7], modeling of rewriting strategies [9] and access control policies [18], etc. The ρLog calculus has been influenced by the ρ-calculus [4, 5], which, in itself, is a foundation for the rule-based programming system ELAN [2]. There are some other languages for programming by rules, such as, e.g., ASF-SDF [3], CHR [12], Maude [6], Stratego [21], Tom [1]. The ρLog calculus and, consequently, PρLog differs from them, first of all, by its pattern matching capabilities. Besides, it adopts logic programming semantics (clauses are first class concepts, rules/strategies are expressed as clauses) and makes a heavy use of strategies to control transformations. Earlier works about ρLog and its implementation in Mathematica include [16, 19, 15]. PρLog is available at https://www.risc.jku.at/people/tkutsia/software/prholog

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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 termination, soundness, completeness, and minimality. The minimal complete set of matchers for such a system can be infinite, but our algorithm computes its finite representation in the form of solved set. From the practical side, we identify two finitary cases and impose restrictions on the procedure to get an incomplete algorithm, which, based on our experiments, describes the input-output behavior and properties of Mathematica's flat and orderless pattern matching.

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We consider bending of thin plates with polygonal and curvilinear edges and indicate analogies and differences between the boundary conditions and boundary value problems arising in these two cases if the polygon is inscribed in the curvilinear contour and the number k of vertices of the polygon tends to infinity. We believe that the so-called Sapondzhyan paradox that arises when solving the boundary value problems for supported plates with a curvilinear contour and a k-gonal contour inscribed in it as k → ∞ can be called a paradox only by misunderstanding. Sapondzhyan’s paradox was studied in several papers briefly surveyed in the monograph [1]. Apparently, the interpretation of “paradoxes” and the results proposed in the present paper are published for the first time. Sapondzhyan’s paradox can be generalized to the case of bending of the so-called sliding-fixed plates (i.e., the generalized shear force and the rotation angle are zero on the plate contour) with a curvilinear contour and a k-gonal contour inscribed in it as k → ∞. In the case of three-dimensional elasticity problems, we present boundary conditions and boundary value problems similar to those listed above and consider the situations resulting in “paradoxes” similar to those arising in plate bending. We give the corresponding explanations and interpretations.

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The paper deals with the problem of creating the theory of distributions and is an attempt to show that Andrea Razmadze stood behind the creation of this theory. He was the first to have introduced the class of finite-jump functions, which are considered to be both native solutions (extremals) of some variational problems and foundations for creating the generalized functions.

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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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The seasonal changes of (Ey/E0), which was obtained from the WKB (Wentzel, Kramers, Brillouin) solution, for (-300N; 300S geographic latitudes and (390, 410, 450, 500, 550 and 600 Km) altitudes in the equatorial anomaly region were investigated. (Ey/E0) ratio takes minimum value in latitudes, where the electron density is at maximum levels and takes maximum value in places where the electron density is at minimum levels, respectively. It is possible to say that in places where the electron density is at maximum, the wave transfers energy to the medium, otherwise it receives energy from the medium.

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Taking into account the action of inhomogeneous zonal wind (shear flow), nonlinear dynamic equations describing the propagation of planetary ULF magnetized Rossby waves in the ionospheric D-, E-, and F-layers are obtained and investigated. The influence of existence of charged particles through Hall and Pedersen conductivities on such dynamic equations is studied in detail. It is shown that the existence of shear flow and Pedersen conductivity can be considered as the presence of an external energy source. The possibility of a barotropic instability of the magnetized Rossby waves is shown. Based on the Rayleigh's theorem, the appropriate stability conditions are defined in case of the ionospheric D- and E-layers. It is indicated that magnetized Rossby waves under the action of shear zonal flow correspond to states with negative energy. Some exponentially localized vortical solutions are found for the ionospheric D- and E-layers.

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In the paper, we consider a three-dimensionalmodel of fluid-solid interaction when a thermo-electro-magneto-elastic body occupying a bounded region $\Omega^+$ is embedded in an inviscid fluid occupying an unbounded domain $\Omega^-=R^3\backslash \Omega^+$. In this case, we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function) in the domain $\omega^+$, while we have a scalar acoustic pressure field in the unbounded domain $\omega^-$. The physical kinematic and dynamic relations are described mathematically by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, we prove the uniqueness and existence theorems for the corresponding boundary transmission problems in appropriate Sobolev-Slobodetskii and H\"older continuous functional spaces.

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One-dimensional nonlinear Maxwell-type system is considered. The initial-boundary value problem with mixed type boundary conditions is discussed. It is proved that in some cases of nonlinearity there exists critical value of the boundary data $\psi_{c}$, such that for a suffciently small positive values of $\psi$ the steady state solution is linearly stable. But as $\psi$ passes through a critical value $\psi_{c}$, the stability changes and a Hopf bifurcation may takes place. The finite difference scheme is constructed. Results of numerical experiments with graphical illustrations are given.

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The paper is devoted to the investigation of mixed boundary-transmission problems for composed elastic structures consisting of two contacting anisotropic bodies occupying two three dimensional adjacent regions with a common contacting interface, being a proper part of their boundaries. It is assumed that the contacting elastic bodies are subject to different mathematical models. In particular, we consider Green-Lindsay's model of generalized thermo-electro-magnetoelasticity in one elastic component, while in the other one, we considered Gree Lindsay's model of generalized thermo-elasticity. The interaction of the thermo-mechanical and electro-magnetic fields in the composed piecewise elastic structure is described by the fully coupled systems of partial differential equations of pseudo-oscillations, obtained from the corresponding dynamical models by the Laplace transform. These systems are equipped with the appropriate mixed boundary-transmission conditions which cover the conditions arising in the case of interfacial cracks. Using the potential method and the theory of pseudodifferential equations on manifolds with a boundary, the uniqueness and existence theorems in suitable function spaces are proved, the regularity of solutions is analyzed and singularities of the corresponding thermo-mechanical and electro-magnetic fields near the interfacial crack edges are characterized. The explicit expressions for the stress singularity exponents are derived and it is shown that they depend essentially on the material parameters. A special class of composed elastic structures is considered, where the so-called oscillating stress singularities do not occur.

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We investigate mixed type boundary–transmission problems of the generalized thermo-electro-magneto-elasticity theory for complex elastic anisotropic layered structures containing interfacial cracks. This type of problems is described mathematically by systems of partial differential equations with appropriate transmission and boundary conditions for six dimensional unknown physical field (three components of the displacement vector, electric potential function, magnetic potential function, and temperature distribution function). We apply the potential method and the theory of pseudodifferential equations and prove uniqueness and existence theorems of solutions to different type mixed boundary–transmission problems in appropriate Sobolev spaces. We analyze smoothness properties of solutions near the edges of interfacial cracks and near the curves where different type boundary conditions collide.

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Weisz proved-among others–that for f∈ LlogL the Fejér means σ (t, u) n, m of conjugate transform of two-parameter Walsh-Fourier series ae 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 purpose of this paper is to construct explicit solutions of the Dirichlet type and the Neumann type boundary value problems of the theory of elasticity for a sphere and for a space with spherical cavity with a double voids structure. The solutions of considered boundary value problems are presented as absolutely and uniformly convergent series.

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This paper studies the linear theory of thermoelastic materials with inner structure whose particles,in addition to the classical displacement and temperature fields, possess microtemperatures. The present work considers the 2D equilibrium theory of thermoelasticity for solids with microtemperatures. This paper is devoted to the explicit solution of the Neumann type boundary value problem for an elastic plane, with microtemperatures having a circular hole. Special representations of the regular solutions of the considered equations are constructed by means of the elementary (harmonic, biharmonic and meta-harmonic) functions. Using the Fourier method, we presented the solution of the Neumann type boundary value problem for the plane with circular hole in the explicit form.

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This paper is concerned to study quasi-static boundary value problems of coupled linear theory of elasticity for porous circle and for plane with a circular hole. The Dirichlet type boundary value problem for a circle and the Neumann boundary value problem for a plane with a circular hole are solved explicitly. All the formulas are presented in explicit ready-to-use form. The solutions are represented by means of absolutely and uniformly convergent series.

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We study oscillatory properties of solutions of the functional difference equation of the form ∆(n)u(k) + F(u)(k) = 0, where n ≥ 2, F : S(N; R) → S(N; R) (By S(N; R) denote the set of discrete functions whose set of values is R). Sufficient conditions for the above equation to have the co-cold Property B are established.

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We study oscillatory properties of solutions of the Emden–Fowler type difference equation $\Delta^{n} u(k)+p(k)|u(\sigma (k))|^{\lambda}signu(\sigma(k))=0$, where $n\geq 2, 0<\lambda<1, p:N\rightarrowR_{+}, \sigma :N \rightarrow N$ and $\sigma (k) \geq k+1$ for $k \in N.$ Sufficient conditions of new type for oscillation of solutions of the above equation are established. 0)(())(()()()(kusignkukpkun , where 10,2n , NNRNp :,:   and σ(k) ≥ k + 1, for  k ∈ N. Sufficient conditions of new type for oscillation of solutions of the above equation are established.

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We consider the equilibrium points of the electrostatic potential of three mutually repelling point charges with Coulomb interaction placed at the vertices of a given triangle T. It is proven that for each point P inside the triangle T, there exists a unique collection of positive point charges, called stationary charges for P in T, such that P is a critical point of the electrostatic potential of these point charges placed at vertices of T in a fixed order. Explicit formulas for stationary charges are given, which are used to investigate the existence and geometry of stable equilibria arising in this setting. In particular, symbolic computations and computer experiments reveal that for an isosceles triangle T, the set S(T) of points P that are stable equilibria of their stationary charges is a non-empty open set containing the incenter of a triangle T. For a regular triangle, using symbolic computations, it appears possible to verify that the formulas for stationary charges define a stable mapping in the sense of Whitney having a deltoid caustic with three ordinary cusps. An interpretation of our results in terms of electrostatic ion traps is also given, and several plausible conjectures are presented.

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In this paper, we investigate the problem of simulation of quantum-mechanical dynamical system with prescribed properties using of exactly solvable Hamiltonians. A method for solving time-dependent matrix Schrodinger equations in an explicit analytical form will be developed. It is based on exactly solvable time-independent problems, a certain choice of the initial conditions, and special time-dependent gauge transformations converting time-independent problems to time-dependent ones. An application of the exactly solvable time-dependent problems thus obtained to the construction of a universal set of gates for quantum computers is also presented.

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We present several generalizations of our recent results on the electrostatic interpretation of points in the plane with respect to a given non-degenerate triangle T. First, we extend the definition of the so-called stationary charges in such a way that their existence and uniqueness hold for all points in the complement of three straight lines defined by the sides of T. Next, we show that, for any point P outside of T, stationary charges cannot have the same sign, and describe possible combinations of signs. For a regular triangle T and point P outside of T, it is also shown that the stationary charges of P have exactly two saddle-points and this defines a differentiable involution in the complement of T. The main results are complemented by a few typical examples and several related conjectures.

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In this paper we show that in the expository texts on linear Algebra, the notion of a basis could be introduced by an argument much weaker than Gauss’ reduction method. Our aim is to give a short proof of a simply formulated lemma, which in fact is equivalent to the theorem on frame extension, using only a simple notion of the kernel of a linear mapping, without any reference to special results, and derive the notions of basis and dimension of a finite dimensional vector space in a quite intuitive and logically appropriate way, as well as obtain their basic properties, including a lucid proof of Steinitz’s theorem

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 The constrained Bayesian method (CBM) of testing statistical hypotheses and their 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. The obtained results clearly demonstrate an advantage of CBM in comparison with the listed approaches

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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 this paper, the Black Sea upper mixed layer (UML) structures in mid-February by using a 3-D numerical model of the Black Sea dynamics (BSM-IG, Tbilisi, Georgia) are investigated. In order to present the turbulent mixing peculiarities more clearly, a new version of the classical Pacanowski{Philander parameterization formulated by Bennis et al. (2010) for vertical turbulent viscosity and diffusion coefficients is integrated in the BSM-IG. The Black Sea UML homogeneity is estimated using criterion of temperature (△T = 0:2◦C) and salinity (△S = 0:15 psu). Besides, mixed layer structures have been investigated according to both values of the Richardson number: RiT and RiS, respectively. As result analysis shows: in February UML structures in the temperature fields correspond to the Richardson number specificity, basically, but mixed layer homogeneity reduced in the salinity fields, when Richardson number changed in the following range 0:07 < RiS ≤ 1, especially, in deep waters of the sea basin.

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In this study, numerical simulations, together with remote sensing products, are used for the first time to study the transport of Aeolian dust from deserts to Georgia. The results of calculations performed by the Weather Research and Forecasting Chemistry model (WRF-Chem) from December 2017 to November 2018 showed that nine cases of dust transfer to the territory of Georgia were recorded. Two of them, which took place on March 22–24 and July 25–26, 2018, are modeled and discussed in this article. Comparison of the calculation results with the data of observations of PM10 particulate matter and satellite products of the Cloud Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) and the Moderate Resolution Imaging Spectroradiometer (MODIS) showed that the chosen WRF-Chem model satisfactorily simulates the transport of desert dust to the territory of Georgia in the complex orography of the Caucasus. In addition, Aeolian dust aerosol transported from deserts turned out to be a significant pollutant and influenced the climate in Georgia. Indeed, the calculations of the WRF-Chem model showed that during the period under study, dust was transferred to the territory of Georgia equally from the deserts of Africa, the Middle East, and Central (West) Asia. It should be noted that among them, the transfer of dust from the Karakum and Kyzylkum deserts was recorded twice, the traces of which have not yet been recorded on the glaciers of the Caucasus (Elbrus and Kazbeg)

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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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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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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 necessary conditions of optimality of delays parameters, of the initial vector, of the initial and control functions are proved for the optimization problem with constant delays in the phase coordinates and controls. The necessary conditions are concretized for the optimization problem with the integral functional and with the fixed right end.

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The paper considers a three-dimensional system of differential equations describing the thermoelastic static equilibrium of homogeneous isotropic materials with voids. In the Cartesian coordinate system the general solution of the mentioned system of equations is constructed using harmonic and metaharmonic functions. On the basis of the constructed general solution using the method of separation of variables boundary value problems for a semi-infinite prism and a rectangular parallelepiped are analytically solved. The corresponding boundary-contact problems for a multilayer rectangular parallelepiped are also considered.

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In the present paper the coupled linear theory of double-porosity viscoelastic materials is considered and the basic boundary value problems (BVPs) of steady vibrations are investigated. Indeed, in the beginning, the systems of equations of motion and steady vibrations are presented. Then, Green's identities are established and the uniqueness theorems for classical solutions of the BVPs of steady vibrations are proved. The fundamental solution of the system of steady vibration equations is constructed and the basic properties of the potentials (surface and volume) are given. Finally, the existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (the boundary integral equations method) and the theory of singular integral equations.

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This paper concerns with the coupled linear theory of viscoelasticity for porous materials. In this theory the coupled phenomenon of the concepts of Darcy’s law and the volume fraction is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions and its basic properties are presented. Green’s identities are obtained and the uniqueness theorems for the regular (classical) solutions of the BVPs of steady vibrations are proved. The surface and volume 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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In this paper, the linear theory of thermoviscoelastic binary porous mixtures is considered and the basic boundary value problems (BVPs) of steady vibrations are investigated. Namely, the fundamental solution of the system of equations of steady vibrations is constructed explicitly and its basic properties are established. Green’s identities are obtained and the uniqueness theorems for classical solutions of the internal and external basic BVPs of steady vibrations are proved. The surface and volume potentials are constructed and their basic properties are given. The determinants of symbolic matrices of the singular integral operators are calculated explicitly and the BVPs are reduced to the always solvable singular integral equations for which Fredholm’s theorems are valid. Finally, the existence theorems for classical solutions of the internal and external BVPs of steady vibrations are proved by means of the potential method and the theory of singular integral equations.

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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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In the paper, for the perturbed controlled nonlinear differential equation with the constant delay in the phase coordinates and in controls a formula of the analytic representation of a solution is obtained in the left semi-neighborhood of the endpoint of the main interval. The novelty here is the effect in the formula related with perturbation of the initial moment.

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The pattern calculus described in this paper integrates the functional mechanism of the lambda-calculus and the capabilities of pattern matching with star types. Such types specify finite sequences of terms and introduce non-determinism, caused by finitary matching. We parametrize the calculus with an abstract matching function and prove that for each concrete instance of the function with a finitary matching, the calculus enjoys subject reduction property.

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The presented book is devoted to the certain combinatorial and set-theoretical aspects of the geometry of Euclidean space and consists of two parts. The material of this book is primarily devoted to various discrete geometric structures and, respectively, to certain constructions of algorithmic type which are associated with such structures. Typical questions of combinatorial, discrete and convex geometry are examined and discussed more or less thoroughly. There are indicated close relationships between the questions of geometry and other areas of discrete mathematics.

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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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It is shown that any function acting from the real line R into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function x→exp(x^2) cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

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For certain families of topologies, the existence of a common Sierpiński–Zygmund function (of a common Sierpiński–Zygmund function in the strong sense) is established. In this connection, the notion of a Sierpiński–Zygmund space (of a Sierpiński–Zygmund space in the strong sense) is introduced and examined. The behavior of such spaces under some standard topological operations is considered.

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It is proved that there exists a translation invariant extension µ of the two-dimensional Lebesgue measure λ_2 on the plane R^2 such that µ is metrically isomorphic to λ_2 and all linear sections of some µ-measure zero set are absolutely nonmeasurable

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It 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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Attribute-based access control (ABAC) is an access control paradigm whereby access rights to system resources are granted through the use of policies that are evaluated against the attributes of entities (user, subject, and object), operations, and the environment relevant to a request. Many ABAC models, with different variations, have been proposed and formalized. Since the access control policies that can be implemented in ABAC have inherent rule-based specifications, it is natural to adopt a rule-based framework to specify and analyse their properties. We describe the design and implementation of a software tool implemented in Mathematica. Our tool makes use of the rule-based capabilities of a rule-based package developed by us, can be used to specify configurations for the foundational model ABACα of ABAC, and to check safety properties.

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We consider some measurability properties of the uniform subsets of the Euclidean plane R2. Furthermore, it is shown that there exists an uniform subset of the plane which is simultaneously a Hamel basis of the plane

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This paper is concerned with the study of the 2D boundary value problems for transversely isotropic elastic half-plane with double porosity. Explicitly is solved the basic BVPs for half-plane. For finding explicit solutions of the basic BVPs the potential method and the theory of Fredholm integral equations are used. The Poisson type formulas are constructed.

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The paper is devoted to the construction and study of the decomposition type semidiscrete scheme for one nonlinear multi-dimensional integro-differential equation of parabolic type. Unique solvability of the first type initial-boundary value problem is given as well. The studied equation is some generalization of integro-differential model, which is based on the well-known Maxwell system arising in mathematical simulation of electromagnetic field penetration into a medium.

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For the nonlinear controlled functional differential equations with several constant delays, 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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For the nonlinear optimization problem with delays necessary optimality conditions are obtained: for delays in the form of equality; for control functions in the form of linearized integral maximum principle.

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The article studies the stress-strain mode of a spongy bone of an implanted jaw. A spongy bone can be considered as a multiporous area with its channels and pores as most evident components of a double porous system. The work studies the stress-strain mode of a spongy jaw bone 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 jaw-bone. The problem is solved by using the boundary element methods, which are based on the solutions of Flamant and Boussinesq’s problems. The cases of various lengths of an implant diameter are considered. Stress isolines in the bone are drafted and the results obtained by BEMF and BEMB for the different-diameter implants are compared.

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This report represents the part of works, dedicated to the creation of consistent 2D boundary value problems corresponding, to elastic thin-walled structures (TWS), an analysis for K´arm´an type system of DEs without variety of ad hoc assumptions, since in the classical form of this system, one of them represents the condition of compatibility. Then we find the general solution of nonlinear systems by development methodology of generalized analysis functions theory for some class of complex systems of DEs, containing the integrals both of Volterra and Fredholm type

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For the perturbed controlled nonlinear delay differential equation with the discontinuous initial condition, a formula of the analytic representation of solution is proved in the left neighborhood of the endpoint of the main interval. In the formula, the effects of perturbations of the delay parameter, the initial vector, the initial and control functions are detected

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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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For an optimal problem involving quasi-linear neutral differential equation with the general boundary conditions and the phase restrictions, existence theorems of optimal element are provided. Under element we imply the collection of initial and final moments, delay parameters, initial vector, initial functions and control.

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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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The free boundary problem for the Stokes flow is studied in the upper half-plane. It is assumed that the pressure can be regulated and depends on time exponentially. The Pressure at the free boundary is constant. The problem is reduced to the system of linear integral equations with the weakly singular kernel. The sufficient conditions of the existence and uniqueness of solutions is proved. In case of small parameters the approximate solution is obtained by means of the step-wise approximation method. The profile of a free boundary is constructed in case of the harmonic pressure.

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The paper considers the problem of pressure of a rigid punch onto a viscoelastic half plane in the presence of friction. The problems of the linear theory of viscoelasticity attracted the attention of many scientists first of all due to the fact that building and composite materials (concrete, plastic polymers, wood, human fabric, etc.) exhibit significant viscoelastic properties and, thus, calculations of constructions for strength, with regard for the viscoelastic properties, are now becoming increasingly important. Thanks to this fact, various methods of calculating the above mentioned problems were proposed, one of which is the Kelvin–Voigt differential model on which the present paper is based. Using the methods of a complex analysis elaborated in the plane theory of elasticity by N. I. Muskhelishvili and his followers, the unknown complex potentials, characterizing viscoelastic equilibrium of a half-plane, are constructed effectively and the tangential and normal stresses under the punch are defined.

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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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Currently, the problem of climate change is an urgent issue in the South Caucasus region, as well as in Georgia, where increased trends of average annual temperaturewith heavy precipitation, hail, floods and drought have becomemore frequent. To prevent the consequences of these events (accidents) in a timelymanner, it is necessary to take more effective steps in providing scientific information on extreme events a regional and local scale, the official environmental authorities, society and the scientific community. In this work, on the one hand, a comparative study of three cumulus parameterization and five microphysical schemes of the Weather Research and Forecast (WRF) v.3.6 model, is carried out for four exceptional precipitation phenomena that have occurred in Georgia (Tbilisi) in the summer of 2015 and 2016. On the other hand, the Real-time Environmental Applications and Display System (READY) is used to study these phenomena. Predicted events are evaluated by a thorough examination of weather radar data. Some results of numerical calculations based on the WRF and READY systems are presented and analyzed.

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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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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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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 paper we consider solving constraints over several similarity relations, 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, and discuss applications.

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The paper is devoted to the construction and study of the additive averaged semi-discrete scheme for one nonlinear multi-dimensional integro-differential equation of parabolic type. The studied equation is based on the well-known Maxwell’s system arising in mathematical simulation of electromagnetic field penetration into a substance.

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In the present work 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 $x_3$, i.e. equal to $const.\times x^κ_3$, $κ = const. ∈ [0, 1)$. It is assumed that all other functions depend on time $t$ and spatial variable $x_3$, with prescribed charge density ($f_e$) and volume force component ($Φ_3$). The well-posedness of initial-boundary value problem is studied. The displacement vector ($u_3$) as well as electric (χ) and magnetic (η) potentials that arise during the deformation are represented as absolutely and uniformly convergent series. The conditions on the volume force components $Φ_1$ and $Φ_2$, which guarantee the strain state under consideration, are established.

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This book commemorates the 75th birthday of Prof. George Jaiani – Georgia’s leading expert on shell theory. He is also well known outside Georgia for his individual approach to shell theory research and as an organizer of meetings, conferences and schools in the field. The collection of papers presented includes articles by scientists from various countries discussing the state of the art and new trends in the theory of shells, plates, and beams. Chapter 20 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

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The nonlinear propagation of coupled Alfvén and ion-acoustic waves is investigated in homogeneous, magnetized and collisionless electron-ion plasma. The ion parallel motion is included in the model so that ion-acoustic wave couples with Alfvén wave in plasmas. In linear limit, the dispersion relation of coupled Alfvén and ion-acoustic waves is derived and its dispersion effects are investigated analytically as well as numerically using laboratory plasma parameters, given in the literature. Using reductive perturbation method, the Korteweg–de Vries (KdV) equation is derived for nonlinear coupled Alfvén and ion-acoustic waves in plasmas by applying appropriate periodic boundary conditions. The cnoidal wave solution of coupled Alfvén and ion-acoustic waves in terms of Jacobian elliptic function (cn) is obtained and its soliton solution is also investigated. The numerical plots of cnoidal wave and soliton structures with variations of plasma β (defined as the ratio of kinetic pressure to magnetic pressure) and obliqueness of the wave propagation with respect to external magnetic field are presented for illustration. It is found that low amplitude compressive structures of nonlinear coupled Alfvén and ion-acoustic waves exist in a magnetized plasma, which moves with sub-Alfvénic wave speed.

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Uniqueness theorems for function series with respect to systems of finite functions, Lebesgue measurable and finite functions, and some orthonormal systems of functions are formulated

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We consider the time-harmonic acoustic wave scattering by a bounded anisotropic inhomogeneous obstacle embedded in an unbounded anisotropic homogeneous medium assuming that the boundary of the obstacle and the interface are Lipschitz surfaces. We assume that the obstacle contains a cavity and the material parameters may have discontinuities across the interface between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical model is formulated as a boundary-transmission problem for a second order elliptic partial differential equation of Helmholtz type with piecewise Lipschitz-continuous variable coefficients. The problem is studied by the so-called nonlocal approach which reduces the problem to a variational-functional equation containing sesquilinear forms over a bounded region occupied by the inhomogeneous obstacle and over the interfacial surface. This is done with the help of the theory of layer potentials on Lipschitz surfaces. The coercivity properties of the corresponding sesquilinear forms are analyzed and the unique solvability of the boundary transmission acoustic problem in appropriate Sobolev-Slobodetskii and Bessel potential spaces is established.

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To describe the nonlinear propagation of electrostatic drift and ion-acoustic waves (DIAWs), the generalized Hasegawa-Mima equation containing both vector (Jacobian) and scalar (Korteweg-de Vries type) nonlinearities is obtained for electron-positron-ion (EPI) plasmas. In addition, density and temperature non-homogeneities of electrons and positrons are taken into account. Appropriate set of 3D equations consisting of generalized Hasegawa-Mima equation for the electrostatic potential and equation of parallel to magnetic field motion of ions are obtained to describe the formation of coherent dipole and large-scale monopole vortices.

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The paper deals with the three-dimensional Robin type boundary-value problem (BVP) for a second-order strongly elliptic system of partial differential equations in the divergence form with variable coefficients. The problem is studied by the localized parametrix based potential method. By using Green’s representation formula and properties of the localized layer and volume potentials, the BVP under consideration is reduced to the a system of localized boundary-domain singular integral equations (LBDSIE). The equivalence between the original boundary value problem and the corresponding LBDSIE system is established. The matrix operator generated by the LBDSIE system belongs to the Boutet de Monvel algebra. With the help of the Vishik–Eskin theory based on the Wiener–Hopf factorization method, the Fredholm properties of the corresponding localized boundary-domain singular integral operator are investigated and its invertibility in appropriate function spaces is proved.

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We investigate the solvability of three-dimensional dynamical mixed boundary value problems of electro-magneto-elasticity theory for homogeneous anisotropic bodies with interior cracks. Using the Laplace transform technique, the potential method, and the theory of pseudodifferential equations, we prove the existence and uniqueness theorems and analyze asymptotic properties of solutions near the crack edges and near the lines where the different boundary conditions collide.

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In this paper we study the a. e. exponential strong (C, 1, 0) summability of of the 2-dimensional trigonometric Fourier series of the functions belonging to L (log+L)2.

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We prove that if the lacunary partial sums of the Fourier series of every square summable function concerning each one-dimensional orthonormal system Φ1,..., Φd converge almost everywhere, then the product system Φ1×···× Φd also has a similar property for a quite general type of partial sums.

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Three-layer semi-discrete schemes for second order evolution equations are studied in the Hilbert space using Chebyshev polynomials. A priori estimates are proved for approximate solutions, as well as for difference analogues of first and second order derivatives. Using these a priori estimates we obtain estimates of the approximate solution error and, taking into account the smoothness of the solution of the continuous problem, the rate of convergence of an approximate solution with respect to step is estimated. The paper also discusses three-layer semi-discrete schemes for a second order complete equation and for an equation with a variable operator.

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The object of the present paper is to consider the Dirichlet boundary value problem of the coupled linear quasi-static theory of elasticity for porous isotropic elastic infinite strip. The general representations of a regular solutions of a system of considered equations for a homogeneous isotropic medium are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. Using the Fourier method, the Dirichlet BVP is solved effectively (in quadratures) for the infinite strip.

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In the present paper, we consider an elastic circle and a plane with circular hole with voids and microtemperatures. Representations of a general solution of a system of equations for a homogeneous isotropic thermoelastic medium with voids and microtemperatures are constructed by means of the elementary (harmonic, bi‐harmonic and meta‐harmonic) functions. The Dirichlet type boundary value problems for a circle and for a plane with circular hole are solved explicitly. The obtained solutions are represented in the form of absolutely and uniformly convergent series.

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This book is devoted to asymptotic properties (exponential stability, instability, oscillation and nonoscillation, existence of solutions with specific asymptotic)for second and the higher order functional differential equations. This equations include delay differential equation, integro-differential equations and equations with distributed delay. Until now most known asymptotic and stability results on functional differential equations have been obtained for scalar and vector equations of the first order and only a few for equations of second and higher order. For higher order operator – differential equations oscillation of solutions are investigate.

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

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In the report a special approach of a large complex structure algorithms’ programming is presented, which is based on a description of a project by formal grammars. Restrictions are placed on a grammar rule. Such a description is traduced in a logical expression, the execution of which gives the resolution of the problem. The approach is demonstrated on the example of a morphological analysis of a natural language’s word.

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In the present chapter, the boundary value problems are considered in a parabolic coordinate system. In terms of parabolic coordinates, the equilibrium equation system and Hooke’s law are written, and analytical (exact) solutions of 2D problems of elasticity are constructed in the homogeneous isotropic body bounded by coordinate lines of the parabolic coordinate system. Analytical solutions are obtained using the method of separation of variables. The solution is constructed using its general representation by two harmonic functions. Using the MATLAB software, numerical results and constructed graphs of the some boundary value problems are obtained.

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In this paper we describe a feasible construction of universal set of quantum gates using monodromy matrices of Fuchsian system. Fuchsian systems are considered as Schrödinger type equations and it is shown that such quantum systems are exactly solvable. We also show that dynamics of trapped cold ions may be described by a Fuchsian system which also describes the critical points of logarithmic potential associated with equilibrium positions of trapped ions in line geometry. Two different approaches to the inverse problem are also discussed.

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We prove that, for any Fuchsian system of differential equations on the Riemann sphere, there exists a rational matrix function whose partial indices coincide with the splitting type of the canonical vector bundle induced from the Fuchsian system. From this, we obtain solution of the Riemann-Hilbert boundary value problem for piecewise constant matrix function in terms of holomorphic sections of vector bundle and calculate the partial indices of the problem.

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We consider equilibrium configurations of three mutually repelling point charges with the Coulomb interaction confined to a simple arc of a constant length with fixed positions of ends. For given values of the charges, the length of the arc, and the distance between its ends, we calculate all possible equilibrium configurations. We also study the behavior of equilibrium configurations for variable values of charges and show that a unique possible bifurcation is the pitchfork bifurcation. Similar results are presented for elastic loop obeying Hooke's law and for charges interacting via a Riesz potential.

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We discuss stable equilibrium points of electrostatic potential of three constrained unit point charges with Coulomb interaction. The main aim is to describe all triples of points on a unit circle in the Euclidean plane such that unit charges placed at those points form a trapping configuration, i.e. possess a stable equilibrium. We show that trapping configurations exist and give an explicit description of the set of all trapping configurations in the reduced configuration space (Theorem 1). We also prove that the set of all stable equilibria arising in this way is a circle of radius approximately equal to 0.4 (Theorem 2). We also consider the case where three equal charges are constrained to two concentric circles. Here one has to distinguish two cases: two charges on the inner circle, and two charges on the outer circle. In the first case we show that trapping configurations always exist and describe the set of all stable equilibria arising in this way (Theorem 3). In the second case, we give a criterion of existence of trapping configurations in terms of the ratio of the two radii (Theorem 4). In conclusion several possible generalizations are outlined and relations to other mathematical models of constrained point charges are briefly discussed.

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By statistical processing of Georgian Cancer Registry data of 2015-2016, clustering (grouping) of Georgian populations was realized, according to the intensity of the cancer disease prevalence, for the purpose of priority distribution of existed resources and means in the country and for the reduction of the number of patients and improvement of the quality of treatment. Cluster analysis methods of mathematical statistics were used for the study, which was directly implemented using universal statistical software package SPSS. The concept of disease index was introduced for achieving the intruded purpose. Its several variants were determined. The study results using indexes showed that it is possible to group objectively populated areas and regions of the country by intensity of dissemination of cancer disease

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The original computer technologies for controlling and managing the ecological condition of the environmental water objects, developed under the guidance and direct participation of the author, are described in the article. In particular, their purpose, capabilities and peculiarities are briefly described. There is also given a short description of problems solved by using them.

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It is shown that Euler integrals of the first and second kind are expressed by the Dirac delta function in the domain of their singularity. Analytical extension of Euler integrals are considered as distributions on main functional space and some calculations in spirit of generalized functions in complex domain are given

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Constrained Bayesian methods (CBMs) and the concept of false discovery rates (FDRs) for testing directional hypotheses are considered in this article. It 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 a multiple case when we consider m (m>1) sets of directional hypotheses. When guaranteeing restriction on the desired level, a Bayesian sequential method can be applied, the stopping rules of which are proper and the sequential scheme for making a decision strongly controls the mixed directional FDR. Computational results of concrete examples confirm the correctness of the theoretical outcomes

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The limiting distribution of the statistic of the homogeneity test of chi-square is established in case of a simultaneous increase of the number of observations and the number of interval partitions in case of “close” alternatives of Pitman type. Also, it is compared with another test based on the integral square deviation of a nonparametric kernel estimate of density. It is shown that the limiting power of the above-mentioned test is greater than the limiting power of Pearson's Chi-square test.

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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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Functionally Graded Couette flow when viscous coefficients vary from zero $\mu(x_2) \in C^1,$ $\mu(0) =0,$ $\mu(x_2) > 0$ for $x_2 > 0$, in particular, as a power function of a width of a duct, where the fluid is contained at rest at the initial moment, is considered the peculiarities of non-classical setting BCs at the wall of the duct, where viscosity coefficients vanish, is investigated.

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In the paper, 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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We establish the limit distribution of the square-integrable deviation of two nonparametric kernel-type estimations for the Bernoulli regression functions. The criterion of testing the hypothesis of two Bernoulli regression functions is constructed. The question as to its consistency is studied. The power asymptotics of the constructed criterion is also studied for certain types of close alternatives.

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The nonlinear dynamics of internal gravity waves (IGW) in stably stratified dissipative ionosphere with non-uniform zonal wind (shear flow) is studied. Due to the nonlinear mechanism nonlinear solitary, strongly localized IGW vortex structures can be formed. Therefore, a new degree of freedom of the system and accordingly, the path of evolution of disturbances appear in a medium with shear flow. Depending on the type of shear flow velocity profile the nonlinear IGW structures can be the pure monopole vortices, the transverse vortex chain or the longitudinal vortex street in the background of non-uniform zonal wind. Accumulation of these vortices in the ionosphere medium can create the strongly turbulent state.

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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 processes did not split and for this aim we cited von Karman type system without variety of ad hoc assumptions since, in the classical form of this system of DEs one of them represents the condition of compatibility but it is not an equilibrium equation. 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.

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Present study evaluates hindcast over the Caucasus Region of the multi-model system, comprising from 4 ERA-Interim-driven regional climate models (RCM) and the high resolution GCM-MRI-AGCM3 of Meteorological Research Institute (MRI). In total, five climate models simulations were assessed against the CRU observational database. Present work focuses on the mean surface air temperature. The study shows the performance of the members of ensembles in representing the basic spatiotemporal patterns of the climate over the territory of Georgia for the period of 1991–2003. Different metrics covering from monthly and seasonal to annual time scales are analyzed over the region of interest: spatial patterns of seasonal mean, annual cycle of temperature, as well monthly mean temperature bias and inter annual variation. The results confirm the distinct capabilities of climate models in capturing the local features of the climatic conditions of the Caucasus Region. This work is in favor to select models with reasonable performance over the study region, based on which a high-resolution bias-adjusted climatic database can be established for future risk assessment and impact studies

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The purpose of this note is to analyze degenerate one-dimensional case of such type equations. 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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In this article monthly maximums and minimums of 2-meter air temperature from three GCMs of CMIP5 [1] database has been statis-tically downscaled using RCMES [2] package, with four different methods for 27 selected meteorological stations on the territory of Georgia. Stations have been selected all over the territory of Georgia, based on the completeness of their air temperature series through-out the entire period of 1961−2010, their credibility (measured by the number of non-missing data) and to cover as much complex climate features of the territory as possible. The downscaling methods have been trained for the period of 1961-1985 and validated for the period of 1986-2010. 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 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 multimodel ensemble, with mean and spread. Future change tendencies have been assessed in comparison of the period of 1986–2010 but was also compared with previous 25-years period (1961-1985) to compare future changes with the magnitudes of past tendencies

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This book presents peer-reviewed papers from the 4th International Conference on Applications of Mathematics and Informatics in Natural Sciences and Engineering (AMINSE2019), held in Tbilisi, Georgia, in September 2019. Written by leading researchers from Austria, France, Germany, Georgia, Hungary, Romania, South Korea and the UK, the book discusses important aspects of mathematics, and informatics, and their applications in natural sciences and engineering. It particularly focuses on Lie algebras and applications, strategic graph rewriting, interactive modeling frameworks, rule-based frameworks, elastic composites, piezoelectrics, electromagnetic force models, limiting distribution, degenerate Ito-SDEs, induced operators, subgaussian random elements, transmission problems, pseudo-differential equations, and degenerate partial differential equations. Featuring theoretical, practical and numerical contributions, the book will appeal to scientists from various disciplines interested in applications of mathematics and informatics in natural sciences and engineering. G.Jaiani, D.Natroshvili (Eds.)

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The paper reviews the static equilibrium of a micropolar porous elastic material. We assume that the body under consideration is an elastic Cosserat media with voids, however, it can also be considered as an elastic microstretch solid, since the basic differential equations and mathematical formulations of boundary value problems in these two cases are actually identical. As regards the three-dimensional case, the existence and uniqueness of a weak solution of some boundary value problems are proved. The two-dimensional system of equations corresponding to a plane deformation case is written in a complex form and its general solution is presented with the use of two analytic functions of a complex variable and two solutions of the Helmholtz equations. On the basis of the constructed general representation, specific boundary value problems are solved for a circle and an infinite plane with a circular hole.

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The present paper deals with plane strain problem for linear elastic materials with voids. In the spirit of N.I. Muskhelishvili the governing system of equations of the plane strain is rewritten in the complex form and its general solution is represented by means of two analytic functions of the complex variable and a solution of the Helmholtz equation. The constructed general solution enables us to solve analytically a problem for a circle and a problem for the plane with a circular hole.

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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 and appear in various diffusion problems. Unique solvability, asymptotic behavior of the solution of the initial-boundary value problem and convergence of the finitedifference scheme are given.

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The initial boundary value problems of dynamics are considered for the isotropic elastic body with double porosity. By the Laplace transform these problems are reduced to boundary value problems of pseudo-oscillations. Special representations are constructed for the general solution of pseudo-oscillation equations by means of metaharmonic functions. Such an approach facilitates the solution of problems and their solutions are written explicitly in form of absolutely and uniformly converging series. It is proved that inverse transforms yield solutions of initial dynamic problems. The question concerning the uniqueness of regular solutions of the considered problems is investigated.

In this paper, special representations of a general solution of a system of differential equations of linear thermoelasticity are constructed for materials with voids by means of elementary functions, which make it possible to reduce the initial system of equations to equations of simple structure. These representations are used for solving static two-dimensional boundary value problems of thermoelasticity for a disk with void pores. The solutions are represented in explicit form as absolutely and uniformly convergent series. The uniqueness theorems are proven for regular solutions of the considered problems.

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We consider a boundary-value problem for the nonlinear integrodifferential equation, simulating the static state of the Kirchhoff beam. The problem is reduced to a nonlinear integral equation, which is solved by using the Picard iterative method. The convergence of the iterative process is established and the error is estimated.

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Investigation and numerical solution of one nonlinear integro-differential equation of parabolic type is considered. Integro-differential models of this type appear in various diffusion problems. Unique solvability of the initial-boundary value problem is fixed and convergence of the semi-discrete and finiti-difference schemes are studied

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For the nonlinear functional differential equation with several constant delays, the variation formulas for its solution are proved, in which the effects of perturbations of delays and the initial moment are detected.

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Existence theorems of optimal element are given for the nonlinear two-stage optimal problem with the constant delays in the phase coordinates, with the general boundary conditions and the phase restrictions, with the general functional and the continuous intermediate condition. Under element we imply the collection of initial moment and delay parameters, initial function and vector, moment change of system stage, finally moment and controls.

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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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In the present paper we consider the materials with voids. The two dimensional system of equations, corresponding to a plane deformation case, 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 the plane with a circular hole.

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In the present paper, the linear theory of viscoelasticity for binary porous mixtures is considered. The fundamental solution of the system of steady vibration equations is constructed, and its basic properties are established. Green’s identities of this theory are obtained. The uniqueness theorems for classical solutions of the internal and external basic boundary value problems (BVPs) of steady vibrations are proved. The surface and volume potentials are introduced, and their basic properties are established. The determinants of symbolic matrices of the singular integral operators are calculated explicitly, and the BVPs are reduced to the always solvable singular integral equations for which Fredholm’s theorems are valid. Finally, the existence theorems for classical solutions of the internal and external BVPs of steady vibrations are proved by means of the potential method and the theory of singular integral equations.

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The following question is considered: when an uncountable commutative group of homeomorphisms of a second category topological space contains a subgroup, no orbit of which possesses the Baire property?

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In this paper random measures and their nonlinear transformations in an infinite dimensional linear space are considered. The conditions of absolute continuity for this measures are obtained in case of cylindrical type transformation of a space. Explicit formula for Radon-Nikodym derivative is given

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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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In the present paper the 2D equations of thermoelasticity with diffusion, microtemperatures and micro-concentrations are considered. The fundamental and singular matrices of solutions are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions.

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A statistical hypothesis is a formalized record of properties of the investigated phenomenon and relevant assumptions. The statistical hypotheses are set when random factors affect the investigated phenomena, i.e. when the observation results of the investigated phenomena are random. The properties of the investigated phenomenon are completely defined by its probability distribution law. Therefore, the statistical hypothesis is an assumption concerning this or that property of the probability distribution law of a random variable. Mathematical statistics is the set of the methods for studying the events caused by random variability and estimates the measures (the probabilities) of possibility of occurrence of these events. For this reason, it uses distribution laws as a rule. Practically all methods of mathematical statistics one way or another, in different doses, use hypotheses testing techniques. Therefore, it is very difficult to overestimate the meaning of the methods of statistical hypotheses testing in the theory and practice of mathematical statistics.

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In the present paper the 3D equations of thermoelasticity for materials with voids is considered. The representation of general solution of the system of equations is constructed by means of elementary (harmonic, meta-harmonic and bi-harmonic) functions, which makes it possible to solve the BVPs for a sphere. The Dirichlet type BVPs for the sphere with voids and for the space with spherical cavity are solved explicitly. The obtained solutions are represented in the form of absolutely and uniformly convergent series.

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For the nonlinear perturbed controlled differential equation with constant delay in the phase coordinates a formula on the representation of solution is proved. In the formula the effects of perturbations of the delay parameter and initial and control functions are detected.

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For the controlled differential equation with delay is established a form of equation in variations. An example is discussed.

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For the nonlinear controlled functional differential equations with delays in the phase coordinates and controls considering the discontinuous (continuous) initial condition the local (global) variation formulas of solutions are obtained. For the optimization problems with general boundary conditions and functional the necessary optimality conditions are proved: for the initial and final moments in the form of inequalities and equalities; for delays containing in the phase coordinates and for the initial vector in the form of equalities; for the initial and control functions in the form of the integral maximum principle.

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We study the exponential uniform strong summability of two-dimensional Vilenkin–Fourier series. In particular, it is proved that the two-dimensional Vilenkin–Fourier series of a continuous function f is uniformly strongly summable to a function f exponentially in the power 1/2. Moreover, it is proved that this result is best possible.

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For the nonlinear optimization problem with delays necessary optimality conditions are proved: for delays in the phase coordinates and controls in the form of equality; for control functions in the form of linearized integral maximum principle

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In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p < 2) with subsequence of triangular partial means \(S_{2^A}^\Delta(f)\) of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence \(S_{a(n)}^\Delta (f) \rightarrow f\) holds, where a(n) is a lacunary sequence of positive integers.

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An analytical solution of two-dimensional problems of elasticity in the region bounded by hyperbola in the elliptic coordinates is constructed using the method of separation of variables. The stress-strain state of a homogenous isotropic hyperbolic body and that with a hyperbolic cut is studied when there are non-homogenous (non-zero) boundary conditions given on the hyperbolic boundary. The graphs for the numerical results of some test problems are presented.

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In this paper 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 paper sets non-classical problems, and formulates the problems of stress and displacement localization for a homogeneous isotropic elastic half-plane based on them. The problems are solved with a boundary element method. Test examples are given in the work showing the value of the normal stress to be applied to the part of the half-plane boundary to obtain the pre-given localized stress or displacement at the midpoint of the segment located inside the body. By using MATLAB software, the numerical results are obtained and corresponding graphs are constructed.

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The paper considers elastic stress distributions in infinite space with hyperbolic notch when normal or tangential stresses are given on the boundary of notch. The work considers plane deformation. So, exact (analytical) solution of two dimensional boundary value problems of elasticity in the domain with hyperbolic boundary in the elliptic coordinate system is constructed using the method of separation of variables. The stress–strain state of a homogeneous isotropic infinite body with a hyperbolic cut is studied when there are non-homogeneous (nonzero) boundary conditions given on the hyperbolic cut. Finally, the numerical simulation is performed to the stress and displacement distributions over a finite size volume surrounding the notch and relevant graphs for the numerical results of some test problems 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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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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In this paper a generalization of the classical Liouville theorem for the solutions of special type elliptic systems and some nonclassical interpretations of this theorem are obtained.

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An initial-boundary value problem is considered for the Timoshenko type nonlinear integro-differential equation. In particular, considered is an initial-boundary value problem for the J.Ball integro-differential equation, which describes the dynamic state of a beam. The solution is approximated using the Galerkin method, stabile symmetrical difference scheme and Jacobi iteration method. The algorithm has been approved on tests. The results of recounts are represented in tables.

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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. The stability and convergence of that scheme are studied. Numerical experiments are carried out. The appropriate graphical illustrations and tables are given.

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Nonlinear elliptic equations describe wide range of physical phenomena and those equations with the different kind of nonlinearity were considered by numerous authors. Nonlinear elliptic equation connected with the cubic nonlinear Schrodinger type equation (NLS) is considered in the in the infinite area. The non-smooth effective solutions of this equation exponentially vanishing at infinity are obtained. Several examples are given. The profiles of linear waves and symmetric solitary waves connected with those solutions are plotted by using Maple.

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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 prismatic nanostructure with the hexagonal cross-section. 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 quantum properties of materials essentially depend on their molecular structure. In the paper the mathematical model of quantum properties of 2D carbon nanostructures is studied from the nonrelativistic viewpoint. The energy levels of electrons in such structures are connected with the spectral problem for the stationary Schrödinger equation in the areas of hexagonal configuration. By means of the conformal mapping method the Schrödinger equation is reduced to the degenerated elliptic equation in the rectangle with the appropriate boundary conditions. This equation is solved analytically. The eigenvalues and eigenfunctions are obtained in new variables and consequently, the possible energy levels of electrons in 2D carbon nanostructures are derived numerically.

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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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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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One type of integro-differential systems arising in mathematical modeling of the process of penetration of the magnetic field into a substance is studied. The model is based on the system of Maxwell equations. Uniqueness and large time behavior of solution of the corresponding initial-boundary value problem for the aforementioned model are given. Convergence of the fully discrete scheme is proved. A wide class of nonlinearity is studied.

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Properties and peculiarities of linear 3D - propagation of electromagnetic internal gravity waves in an ideally conducting incompressible medium embedded in a uniform magnetic field are studied. Local and non-local (plane waves) approaches are applied. It is shown that ordinary internal gravity waves couple with Alfven waves. Associated partial differential equations and dispersion relations are obtained. New branches of oscillations are revealed. The results obtained may be applicable to the Earth's ionosphere and solar atmosphere.

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The magnetoacoustic nonlinear periodic (cnoidal) wave and solitons in multi-ion plasmas are studied, and its Korteweg–de Vries (KdV) equation is derived by applying the reductive perturbation (RP) method with appropriate boundary conditions. The dynamics of all species in a multi-ion plasma, which consists of warm (light) ions, cold (heavy) ions, and hot electrons, are taken in the electromagnetic field. Two types of multi-ion plasmas, i.e., H+-O+-e (positive ion plasma) and H+−O−2−e (negative ion plasma), are chosen, which can exist in different layers of the ionosphere. It is found that for the H+-O+-e plasma case, the nonlinear structure forms compressive magnetoacoustic wave pulses (which move with the super-Alfvénic speed of light ions in the lab frame) and depends on plasma parameters such as the external magnetic field intensity and light ion temperature and density. It is also found that the rarefactive magnetoacoustic wave structures (move with the sub-Alfvénic speed of light ions in the lab frame) are also formed in H+-O+-e plasma if heavy and light ions' density ratio is increased beyond a certain limit. In the case of negative ion (i.e., H+−O−2−e) plasma, again, compressive magnetoacoustic cnoidal waves and soliton structures are formed at a comparatively lower value of heavy to light ions' density ratio (χ) with the positive ion plasma case. Again, these compressive magnetoacoustic structures switch to the rarefactive ones in negative ion plasma when density ratio χ is increased beyond a certain limit. The parametric analysis and numerical plots are illustrated, and the obtained theoretical results are found to be consistent with the Freja experimental observations in the upper ionosphere.

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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 paper is devoted to the dynamical problem of 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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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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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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Methods of accounting for water phase transformations in some numerical models of the mesoscale atmospheric boundary layer are considered. With the help of our model, a number of humidity processes such as fog, cloud, fog and cloud ensemble were simulated. A new classification of foehns is carried out; the possibility of their modeling in the case of a plane problem is given

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In this paper, theorems are stated regarding geometrical realizations of finite families of sets, for a given countable family of sets the existence of families of point sets combinatorially strictly ω-isomorphic to this given family of sets, independent families of triangles in the Euclidean plane, also, the existence and extensions of ρ-at-sets, ρ-rt-sets, ρ-ot-sets and ot-sets. These theorems are a continuation of the corresponding results of [6, 7, 8, 5, 3, 4]

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In the present paper ε-uniqueness multiple function systems are considered. A theorem representing a possibility of calculation of the limit of a convergent in the Pringsheim sense multiple function series with respect to an ε-uniqueness multiple function system via application of iterated limits is formulated

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In the present paper, the Λ summable single and multiple function series are considered. The notion of a sequence of Cantor’s Λ functionals, which represents formulas of reconstruction of coefficients of a single function series and is also a generalization of Fourier formulas for calculation of coefficients of an orthonormal function series is introduced. A theorem representing a possibility of reconstruction of coefficients of a multiple function series via iterated application of Cantor’s Λ functionals is formulated

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It is well known that the Lebesgue and F. Riesz theorems show an interrelation between the convergence in measure and the convergence almost everywhere of a sequence of functions; the first one is a sufficient and the second one is a necessary condition of convergence in measure of a sequence of functions. In the present paper we formulate a theorem representing a necessary and sufficient condition of convergence in measure of a sequence of functions.

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We investigate multifield problems for complex elastic anisotropic structures when in different adjacent components of the composed body different refined 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 the other adjacent region. This type of mechanical problem is described mathematically by systems of partial differential equations with appropriate transmission and boundary conditions. In the GTEME model part we have six-dimensional unknown physical field (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 field (three components of the displacement vector and temperature distribution function). The diversity in dimensions of the interacting physical fields are taken into consideration in mathematical formulation and analysis of the corresponding boundary-transmission problems. We apply the potential method and the theory of pseudodifferential equations and prove the uniqueness and existence theorems of solutions to different type boundary-transmission problems in appropriate Sobolev spaces.

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Oscillation criteria generalizing a series of earlier results are established for n-th order linear differential equations

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In this paper we prove the following: let 1≤ p≤ 2, then the set of the functions from the space Lp (I²) with a subsequence of spherical partial sums of the double Walsh-Fourier series convergent in measure on I2 is of first Baire category in Lp(I²). We also prove that for each function f∊ L2(I²) ae convergence SR (n)(/)--> f holds, where R (n) is a lacunary sequence of positive integers.

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Morphological synthesis of Georgian words requires to compose of the word-forms by indication unchanged parts and morphological categories. Also, it is necessary by using a stem of the given word to get by the computer all grammatically right word-forms. In case of morphological analysis of Georgian words, it is essential to decompose the given word into morphemes and get the definition each of them. For solving these tasks we have developed some specific approaches and created software. Its tools are efficient for a language, which has free order of words and morphological structure is like Georgian. For example, a Georgian verb (in Georgian: "წერა"-ts'era, in English: Writing) has several thousand verb-forms. It is very difficult to express morphological analysis' rules by finite automaton and it will be inefficient as well. Resolution of any problems of complete morphological analysis of Georgian words is impossible by finite automata. Splitting of some Georgian verb-forms into morphemes requires non-deterministic search algorithm, which needs many backtracks. To minimize backtracking, it is necessary to put constraints, which exist among morphemes and verify them as soon as possible to avoid false directions of search. Sometimes the constraints can be as a description type of specific cases of verbs. Thus, proposed software tools have many means to construct efficient parser, test and correct it. We realized morphological and syntactic analysis of Georgian texts by these tools. Besides this, for solving such problems of artificial intelligence, which requires composing of natural language's word-form by using the information defining this word-form, it is convenient to use the software developed by us. In the presented article, we describe the software tools and its application for Georgian language.

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აღწერილია სპეციალური მიდგომები და მათი კომპიუტერული რეალიზაცია (პროგრამები) ქართული ენის გამოყენებით შემდეგი პრობლემების გადასაწყვეტად: 1. ქართული ტექსტის გახმოვანება; 2. ქართული სიტყვების სრული მორფოლოგიური ანალიზი; 3.ქართული სიტყვის შედგენა სიტყვის ძირისა და მორფოლოგიური კატეგორიების მიცემით; 4.ქართული სიტყვის დაშლა მორფემებად.

The present paper is devoted to the explicit solutions of the equilibrium boundary value problems (BVPs) for an elastic circle and for full plane with circular hole with a double-voids structure. The regular solution of the system of equations for an isotropic material with a double-voids structure is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The Dirichlet-type BVPs for a circle and for a plane with a circular hole are solved explicitly. The obtained solutions are presented as absolutely and uniformly convergent series.

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The present paper is devoted to the explicit solution of the Dirichlet type BVP for an elastic circle with microtemperatures. The regular solution of the system of equations for an isotropic materials with microtemperatures is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The Dirichlet type BVP for a circle is solved explicitly. The obtained solutions are presented as absolutely and uniformly convergent series.

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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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The object of the present paper is to construct explicit solutions of BVPs for an isotropic 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 the infinite strip.

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We construct a homogeneity test based on the kernel-type estimators of the distribution density and investigate its consistency.

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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. Here is shown that the direct application of CBM allows us to control FDR on the desired level. Theoretically it 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. The correctness of the obtained theoretical results is confirmed by computation results of concrete examples.

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A homogeneity test is constructed based on kernel-type estimators of a distribution density. The limit power of the test thus constructed is found for Pitman-type close alternatives.

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This paper represents the part of a cycle of works, dedicated to the solution of the Cauchy problem for ordinary differential equations with high order of accuracy.

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

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In this paper, 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 this paper, we consider boundary-value problems with shift and show that such problems are equivalent to boundary-value problems for generalized analytic functions. We interpret the shift as a change of the complex structure on the complex plane with a given closed curve.

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We discuss equilibrium configurations of the Coulomb potential of positive point charges with positions satisfying certain quadratic constraints in the plane and three-dimensional Euclidean space. The main attention is given to the case of three point charges satisfying a positive definite quadratic constraint in the form of equality or inequality. For a triple of points on the boundary of convex domain, we give a geometric criterion of the existence of positive point charges for which the given triple is an equilibrium configuration. Using this criterion, rather comprehensive results are obtained for three positive charges in the disc, ellipse, and three-dimensional ball. In the case of the circle, we strengthen these results by showing that any configuration consisting of an odd number of points on the circle can be realized as an equilibrium configuration of certain nonzero point charges and give a simple criterion for existence of positive charges with this property. Similar results are obtained for three point charges each of which belongs to one of the three concentric circles. Several related problems and possible generalizations are also discussed.

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In this paper we consider so called Beltrami parametrization of Riemann surfaces and show that the Riemann-Hilbert boundary value problem with shift is equivalent to classical Riemann-Hilbert boundary value problem with respect to the complex structures defined by Beltrami parametrization induced from shift operator.

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Equilibrium configurations of three equal point charges with Coulomb interaction confined to a system of concentric coplanar circles are discussed. For arbitrary values of radii of the given circles, it is shown that the number of equilibria is finite. The main results are concerned with detailed investigation of aligned equilibrium configurations. In particular, it is shown that, for genericvalues of radii, all aligned configurations are non-degenerate critical points of Coulomb potential, and explicit formulas for their Morse indices are given. It is also proven that, for certain non-generic values of radii, a pitchfork bifurcation happens at one of the aligned equilibrium configurations, which enables us to determine the exact number of equilibria for arbitrary values of radii of the given circles. Some related results and an application to Bohr’s 1913 model of lithium atom are also given.

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It is shown, that the Euler integral of the first kind (beta integral) in some area of its divergence is integrable in the sense of generalized functions. The equality of the mentioned integral and the Fourier transform of a singular exponential function is shown. The connection between the beta integral and the complex Dirac delta function is obtained. In addition, the analytical representation and the asymptotic behavior of the Euler beta functional are derived

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In the present article the dynamics of generation and propagation of planetary global weatherforming ultra-low frequency (ULF) electromagnetic wave structures in the dissipative ionosphere are given. These waves are stipulated by spatial inhomogeneous geomagnetic field. The large-scale waves are weakly damped. The waves generate the geomagnetic field from several tens to several hundreds nT and more. It is established, that planetary ULF electromagnetic waves, at their nonlinear interaction with the local shear winds, can self-localize in the form of nonlinear long-lived solitary vortices, moving along the latitude circles westward as well as eastward with velocity, different from phase velocity of corresponding linear waves. The vortex structures transfer the trapped particles of medium and also energy and heat. The nonlinear vortex structures represent can be the structural elements of strong macroturbulence of the ionosphere main drivers of the electromsgnetic weather at ionospheric level.

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In weakly collisional space plasmas, the turbulent cascade provides most of the energy that is dissipated at small scales by various kinetic processes. Understanding the characteristics of such dissipative mechanisms requires the accurate knowledge of the fluctuations that make energy available for conversion at small scales, as different dissipation processes are triggered by fluctuations of a different nature. The scaling properties of different energy channels are estimated here using a proxy of the local energy transfer, based on the third-order moment scaling law for magnetohydrodynamic turbulence. In particular, the sign-singularity analysis was used to explore the scaling properties of the alternating positive-negative energy fluxes, thus providing information on the structure and topology of such fluxes for each of the different type of fluctuations. The results show the highly complex geometrical nature of the flux, and that the local contributions associated with energy and cross-helicity non-linear transfer have similar scaling properties. Consequently, the fractal properties of current and vorticity structures are similar to those of the Alfvénic fluctuations.

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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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The main goal of the book is to obtain more complete information about extreme events in Georgia and provide more detailed climate forecast on a regional scale. The innovative methodology used in the book (a multi-model regional ensemble system (WRF, WRF chem/climate; RegCM, PRESIS, HYSPLIT) and modern computing and satellite information technologies (RCMES, meteorological data GDAS, MODIS, CALIPSO) is applied for a better understanding of the causes and problems associated with current climate change in regions with a complex orography, The book discuses and simulates the acceleration of extreme events in Georgia, some features of climate cooling in Western Georgia, as well as droughts (desertification) in Eastern Georgia, dust transfer from deserts to the South Caucasus and the dust effect in the formation of a regional climate Georgia. As well as future scenarios of extreme air temperatures prediction based on the statistical downscaling method are modelled and studied.

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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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We consider a boundary value problem for an infinite plate with a circular hole. The plate is the mixture of two isotropic elastic materials. 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 three-dimensional equations of equilibrium of an elastic binary mixture (Green–Naghdi–Steel model) by Vekua’s reduction method. The problem is solved analytically by the method of the theory of functions of a complex variable.

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In the paper we consider some plane boundary value problems of asymmetric theory of elasticity for perforated domains. The domain is the square with holes arranged in the definite manner. The formulated problems are solved approximately by using the method of fundamental solutions is used.

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The boundary value problems of elastostatics for a porous circular ring with voids are considered. The general solution of the system of equations is represented by harmonic, biharmonic and metaharmonic functions. Explicit solutions of problems are obtained in the form of series. The conditions are established that ensure absolute and uniform convergence of these series.

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The nonlinear dynamical model for one class of marketing relation with delay in control is given. For a corresponding optimization problem the existence theorem of optimal control and necessary optimality conditions are provided. In the linear case found all controls which are doubtful on optimality. An example is considered.

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The present monograph is concerned with the investigation and numerical solution of the initial-boundary value problems for some nonlinear partial differential and parabolic type integro-differential models. The models are based on the well-known system of Maxwell equations which describes the process of propagation of an electromagnetic field into a medium. The existence, uniqueness and asymptotic behavior of solutions, as time tends to infinity, for some types of initial-boundary value problems are studied. The examples of one-dimensional nonlinear systems and their analytical solutions are given which show that those systems do not, in general, have global solutions. Consequently, the case of a blow-up solution is observed. Linear stability of the stationary solution of the initial-boundary value problem for one nonlinear system is proved. The possibility of occurrence of the Hopf-type bifurcation is established. Semi-discrete and finite difference approximations are discussed. The splitting-up scheme with respect to physical processes for one-dimensional case as well as additive Rothe-type semi-discrete schemes for multi-dimensional cases are investigated. The stability and convergence properties for those schemes are studied. Algorithms for finding approximate solutions are constructed. Results of numerical experiments with tables and graphical illustrations are given. Their analysis is carried out.

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Nonlinear interaction of magnetized Rossby waves with sheared zonal flow in the Earth's ionospheric E-layer is investigated. It is shown that in case of weak nonlinearity 2D Charney vorticity equation can be reduced to the one-dimensional modified KdV equation.

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In this work we consider equations of equilibrium of the isotropic elastic plate. By means of Vekua’s method, the system of differential equations for plates is obtained (approximation N=1), when on upper and lower face surfaces displacements are assumed to be known. The general solution for approximations N=1 is constructed. The concrete problem is solved.

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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. The Dirichlet BVP is solved for a circle.

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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. By means of I. Vekua’s method the system of differential equations for the nonlinear theory of non-shallow shells is obtained. The general solutions of the approximation of Order N = 0, 1, 2, 3, 4 are obtained.

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In this paper the conditions for the existence of a neutral surface of elastic shells is consider, when the neutral surfaces are not the middle surface of the shell, it is the equidistant surface of the middle surface. Boundary value problems of the theory of generalized analytic functions are used for convex shells.

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Set Theory has experienced a rapid development in recent years, with major advances in forcing, point set theory, axiomatic set theory, inner models, large cardinals and descriptive set theory. All of three parts of the present book covers each of these areas, giving the reader an understanding of the ideas involved. It can be used for introductory students and is broad and deep enough to bring the reader near the boundaries of current research. Students and researchers in the field will find the book invaluable both as a study material and as a desktop reference

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It is shown how the Steinhaus property and ergodicity of a translation invariant extension μ of the Lebesgue measure depend on the measure-theoretic density of μ-measurable sets. Some connection of the Steinhaus property with almost convex sets is considered and a translation invariant extension of the Lebesgue measure is presented, for which the generalized Steinhaus property together with the mid-point convexity do not imply the almost convexity.

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A translation invariant measure on the real line R is constructed, which extends the Lebesgue measure on R and for which the Steinhaus property fails in a strong form.

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Semantic Web is a collection of different technologies, where most of them is already standardized. The main purpose of these technologies is to describe semantic content of the web, i.e. their meaning and sense, in the format understood by computers. As a consequence, computer programs will be able to use more (human) knowledge to do assigned tasks. In this paper we overview the ontology and logic layers of the semantic web stack. Although ontology languages are standardized by W3C, there are still many problems remaining, which are related to reasoning over the ontologies. On the logic layer of the semantic web stack are considered unranked languages, where function and predicate symbols do not have a fixed arity. Such languages can naturally model XML documents and operations on them. In this paper we present survey of reasoning methods over such unranked languages.

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Vibration problem of an antiplane strain (shear) of orthotropic non-homogeneous prismatic shell-like bodies is considered when the shear moduli depending on the body projection (i.e. on a domain lying in the plane of interest) variables may vanish on a part of the boundary of the projection.

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Some related questions concerning the measurability properties of real-valued functions with respect to a certain class of measures are discussed.

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In ZF, the existence of a Hamel basis does not yield a well–ordering of R.

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The problems of one of the basic branches of mathematical statistics - statistical hypotheses testing - are considered in this book. The intensive development of these methods began at the beginning of the last century. The basic results of modern theory of statistical hypotheses testing belong to the cohort of famous statisticians of this period: Fisher, Neyman-Pearson, Jeffreys and Wald (Fisher, 1925; Neyman and Pearson, 1928, 1933; Jeffreys, 1939; Wald, 1947a,b). Many other bright scientists have brought their invaluable contributions to the development of this theory and practice. As a result of their efforts, many brilliant methods for different suppositions about the character of random phenomena are under study, as well as their applications for solving very complicated and diverse modern problems. Since the mid-1970s, the author of this book has been engaged in the development of the methods of statistical hypotheses testing and their applications for solving practical problems from different spheres of human activity. As a result of this activity, a new approach to the solution of the considered problem has been developed, which was later named the Constrained Bayesian Methods (CBM) of statistical hypotheses testing. Decades were dedicated to the description, investigation and applications of these methods for solving different problems. The results obtained for the current century are collected in seven chapters and three appendices of this book. The short descriptions of existing basic methods of statistical hypotheses testing in relation to different CBM are examined in Chapter One. The formulations and solutions of conventional (unconstrained) and new (constrained) Bayesian problems of hypotheses testing are described in Chapter Two. The investigation of singularities of hypotheses acceptance regions in CBM and new opportunities in hypotheses testing are presented in Chapter Three. Chapter Four is devoted to the investigations for normal distribution. Sequential analysis approaches developed on the basis of CBM for different kinds of hypotheses are described in Chapter Five. The special software developed by the author for statistical hypotheses testing with CBM (along with other known methods) is described in Chapter Six. The detailed experimental investigation of the statistical hypotheses testing methods developed on the basis of CBM and the results of their comparison with other known methods are given in Chapter Seven. The formalizations of absolutely different problems of human activity such as hypotheses testing problems in the solution - of which the author was engaged in different periods of his life - and some additional information about CBM are given in the appendices. Finally, it should be noted that, for understanding the materials given in the book, the knowledge of the basics of the probability theory and mathematical statistics is necessary. I think that this book will be useful for undergraduate and postgraduate students in the field of mathematics, mathematical statistics, applied statistics and other subfields for studying the modern methods of statistics and their application in research. It will also be useful for researchers and practitioners in the areas of hypotheses testing, as well as the estimation theory who develop these new methods and apply them to the solutions of different problems.

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In this paper the 2D linear theory of steady vibrations of thermoelastic materials with voids is considered. The representation of common decision of the system of equations in the considered theory is obtained. The fundamental and some other matrices of singular solutions are constructed in terms of elementary (meta-harmonic) functions. Some basic properties of single-layer and double-layer potentials are also established.

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The nonlinear optimal control problem with several constant delays in the phase coordinates and controls is considered. The necessary conditions of optimality are obtained for the initial and final moments, for delays having in the phase coordinates and the initial vector, for the initial function and control.

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In the present paper we investigate the elastic sphere with voids and microtemperatures. Special representations of a general solution of a system of equations for a homogeneous isotropic thermoelastic medium with voids and microtemperatures are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The Neumann type boundary value problems for the sphere are solved explicitly. The obtained solutions are represented by absolutely and uniformly convergent series.

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Variation formulas of solutions for nonlinear controlled functional differential equations are proved which show the effect of perturbations of the initial moment, constant delays and also that of the continuous initial condition.

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For the controlled functional differential equation with several delay parameters with discontinuous initial condition the variation formulas of solutions are proved. In addition, the necessary conditions of optimality are proved for the optimization problems with several delays, general boundary conditions and functional

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For the controlled functional differential equation with several delay parameters in the phase coordinates and delays under the control function with the continuous initial condition the necessary condition optimality is obtained.

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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 extension(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 24 value problems by arbitrary order of accuracy depending only on order of smoothness of the desired solution.

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In this paper, we study the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of functions in L logL.

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In this paper we study the maximal operator for a class of subsequences of strong Nörlund logarithmic means of Walsh-Fourier series. For such a class we prove the almost everywhere strong summability for every integrable function f.

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In this paper we prove a BMO-estimate for rectangular partial sums of two-dimensional Walsh-Fourier series, and using this result we establish almost everywhere exponential summability of rectangular partial sums of double Walsh-Fourier series.

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The article presents an exact version of the boundary element method, in particular, the fictitious load method used to solve boundary value and boundary-contact problems of elasticity. The method is developed in the polar coordinate system. The circular boundary of the area limited with the coordinate axes of this system is divided not into small segments like in case of a standard boundary element method (BEM), but into small arcs, while the linear part of the boundary divides into small segments. In such a case, the considered area can be described more accurately than when it divides into small segments, and as a result, a more accurate solution of the problem is obtained. Two test boundary-contact problems were solved by using a boundary element method developed in the polar coordinate system (PCSBEM), and the obtained numerical values are presented as tables and graphs.

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In the present work are stated and solved non-classical elasticity problems for the homogeneous isotropic elastic half-space. The article consider the plane deformation. Namely, there considered non-classical problems, which formulate in the following way: what normal stress supposed to be applied to the part of the half-plane boundary to obtain the pre-given stress or displacement at the segment inside the body. The problems are solved with a boundary element method. There are test examples given showing the value of normal stress supposed to apply to the section of the half-plane boundary to obtain the pre-given stress or displacement at the segment inside the body. By using MATLAB software, the numerical results are obtained and corresponding graphs are constructed.

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The work considers the method used to present the solutions of the boundary value problems of elasticity for a confocal elliptic ring and its parts by using the analytical (exact) solutions of an ellipse and its corresponding internal and external problems. The analytical solutions are obtained by the method of separation of variables. The numerical results of the concrete problems are obtained and corresponding graphs are constructed.

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The present work states and analytically (exactly) solves, using the method of separation of variables, the external boundary value problems of elastic equilibrium of the homogeneous isotropic body bounded by the parabola, when on parabolic border are given normal or tangential stresses. Numerical results and corresponding graphs of some a mentioned problems 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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Finitary matching problems are those that have finitely many solutions. Pattern calculi generalize the lambda-calculus, replacing the abstraction over variables by an abstraction over terms that are called patterns. Consequently, reduction requires solving a pattern matching problem. The framework described in this paper considers the case when such problems are finitary. It is parametrized by the solving function, which is responsible for computing solutions to the matching problems. A concrete instance of the function gives a concrete version of the pattern calculus. We impose conditions on the solving function, obtaining a generic confluence proof for a class of pattern calculi with finitary matching. Instances of the solving function are presented.

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System of Maxwell equations is considered. Reduction to the integro-differential form is given. Existence, uniqueness and large time behavior of solutions of the initial-boundary value problem for integro-differential model with two-component and one-dimensional case are studied. Finite difference scheme is investigated. Wider class of nonlinearity is studied than one has been investigated before. FreeFem++ realization code and results of numerical experiments are given.

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Uniqueness of solution and finite difference scheme of corresponding initial-boundaryvalue problem for one nonlinear partial integro-differential averaged model with source termsare studied. Mentioned model is based on Maxwell system which describes electromagneticfield penetration into a substance. Mixed boundary condition is considered. Large time behaviorof solution is fixed too. Convergence of the fully discrete scheme is proved. Wider class ofnonlinearity is studied than one has been investigated before.

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In the present paper, we use the perturbation algorithm to reduce a purely implicit four-layer semi-discrete scheme for an abstract evolutionary equation to two-layer schemes. An approximate solution of the original problem is constructed using the solutions of these schemes. Estimates of the approximate solution error are proved in a Hilbert space.

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Properties and peculiarities of linear 3D-propagation of electromagnetic internal gravity waves in an ideally conducting incompressible medium embedded in uniform magnetic field are studied. Local and non-local (plane waves) approaches are applied. It is shown that ordinary internal gravity waves couple with Alfven waves. Associated partial differential equations and dispersion relations are obtained. New branches of oscillations are revealed. Obtained results are applicable to the Earth's ionosphere and solar atmosphere.

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Properties of certain families of subsets of Euclidean spaces are established. Using the established properties theorems concerning the structure of constituents of finite independent families of convex bodies in $R^2$ and $R^3$ spaces are proved

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In this paper, we study mixed and crack type boundary value problems of the linear theory of thermopiezoelectricity for homogeneous isotropic bodies possessing the inner structure and containing interior cracks. The model under consideration is based on the Green–Naghdi theory of thermopiezoelectricity without energy dissipation. This theory permits propagation of thermal waves at finite speed. Using the potential method and the theory of pseudodifferential equations on manifolds with boundary we prove existence and uniqueness of solutions and analyze their smoothness and asymptotic properties. We describe an efficient algorithm for finding the singularity exponents of the thermo-mechanical and electric fields near the crack edges and near the curves where different types of boundary conditions collide. By explicit calculations it is shown that the stress singularity exponents essentially depend on the material parameters, in general.

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In the present work the Cauchy problem for an abstract evolution equation with a Lipschitz-continuous operator is considered, where the main operator represents the sum of positive definite self-adjoint operators. The fourth-order accuracy decomposition scheme is constructed for an approximate solution of the problem. The theorem on the error estimate of an approximate solution is proved. Numerical calculations for different model problems are carried out using the constructed scheme. The obtained numerical results confirm the theoretical conclusions.

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Linear and nonlinear propagation of fast magnetohydrodynamic waves (or magnetoacoustic waves) are studied in homogeneous, magnetized and warm collisionless electron-positron (e-p) plasma by using two fluid magnetohydrodynamic model. In the linear limit, the wave dispersion relation is obtained and wave dispersion effect which appears through inertial length in e-p plasma system is also discussed. Using reductive perturbation method, the Korteweg-de Vries (KdV) equation for small but finite wave amplitude of magnetoacoustic waves is derived with appropriate boundary conditions. The cnoidal wave and soliton solutions are obtained using well known Sagdeev potential approach for magnetoacoustic waves in e-p plasmas propagating in the direction perpendicular to the external magnetic field. The phase portrait analysis and numerical illustration of magnetoacoustic cnoidal waves and solitons is also presented by using the parameters such as magnetic field intensity, plasma density and temperature of electron and positron fluids for astrophysical plasma situations exist in the literature.

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Lecture Notes in Engineering and Computer Science cover the frontier issues in the engineering and the computer science and their applications in business, industry and other subjects. The series reports scientific researches presented at many of the most important engineering and computer science meetings around the world. Publishing proceedings can provide quick access to valuable information for the research communities when compared with the traditional journal literature. The series is published with both online and print versions. All the papers in the online version are available freely with open access full-text content and permanent worldwide web link. The abstracts will be indexed and available at major academic databases.

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The present paper considers the equilibrium theory of thermo-microstretch elastic solids with microtemperatures. The method to solve the Neumann-type boundary value problem (BVP) for the whole space with spherical cavity is presented. The solution of this BVP in the form of absolutely and uniformly convergent series is obtained.

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The test of homogeneity is constructed by using kernel-type estimators of a distribution density. The limit power of the constructed test is found for close Pitman-type alternatives. The constructed test is compared with Pearson’s -square test.

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In the paper, 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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In the article, 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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Results of six completed research works shaped as application packages united by common subjects, ideology, methodological basis and execution procedure are given. Among them: AUTOMATED WATER (AIR) QUALITY CONTROL SYSTEM is a system functioning in real time and solving the problem of computer-aided operative monitoring of aqueous and aerial entities; AUTOMATIC DETECTION OF RIVER WATER EMERGENCY POLLUTION SOURCES SOLVES the problem of automatic identification of emergency discharge sources in rivers between two controlled ranges; APPLICATION PACKAGE FOR EXPERIMENTAL DATA PROCESSING and OPTIMAL DECISION-MAKING APPLICATION PACK-AGE are intended for numeric experimental data processing, practically, in any field of knowledge, including ecology; RESEARCHING OPERATOR WORKSTATION (ROW) and COMPUTER-AIDED ECOLOGIST’S WORKING PLACE (CAEWP) are designed for computer-aidec storage, retrieval and processing (by means of up-to-date mathematical methods) of ecological information.

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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 demonstrated by an example.

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The paper discusses the application of constrained Bayesian method (CBM) of testing the directional hypotheses. It is proved that decision rule of CBM restricts the mixed directional false discovery rate (mdFDR) and total Type III error rate as well.

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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 extension(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 24 value problems by arbitrary order of accuracy depending only on order of smoothness of the desired solution.

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The iteration algorithm of computation of effective estimators of the shape parameters of beta distributions using the unbiased estimators of the end point 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 computation results, realized on the basis of the simulation of the appropriate random samples, demonstrate the correctness of the obtained theoretical outcomes.

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This report represents the part of cycle of works dedicated to the solution of Cauchy problem for evolution equation by high order of accuracy. For the permission of this problem the necessary stage is the problem of approximate solution of boundary value problems for the systems of partial differential equations (PDEs) by high order accuracy. In this report we consider the cases when the object of creation represents 2-dim strong elliptic systems of PDEs in the square e with classical boundary conditions when as examples consider well-known typical DEs, refined theories with variable thickness, the stable hierarchical models corresponding to the thin-walled elastic structures. As mathematical apparatus we used the continuous analogy of Douglas-Rachford alternative direction and multipointing difference methods, operator factorization schemes.

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The present paper deals with the three-dimensional linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. In the half-space solutions of some basic boundary value problems are constructed in quadratures.

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The period of the formation of the theory of generalized analytic functions, nowadays known as the Bers-Vekua theory, meets the beginning of scientific activity of Bogdan Bojarski together with his supervisor Ilia Vekua in Moscow. Bojarski’s fundamental new approach to the solution of the system of elliptic partial differential equations on a plane, opened up a direct pathway to many important issues in the geometric theory of analytic functions and related boundary value problems. In this paper we give a short overview of important results of Bogdan Bojarski in the theory of generalized analytic functions and his point of view on the theory of boundary value problems.

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We present several results on equilibria of point charges in a line segment with charged end-points obtained in the framework of inverse problems approach to linear ion traps. In particular, we give a solution of inverse electrostatic problem for four and five point charges

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Equilibrium configurations of three mutually repelling point charges confined to a flexible contour of fixed length are discussed. For given values of charges and perimeter, we compute all possible equilibrium configurations and critical values of Coulomb energy. Moreover, for any triangle with the given perimeter, we compute the values of three charges such that this triangle is congruent to their equilibrium configuration in isoperimetric setting.

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Equilibrium configurations of point charges with Coulomb interaction on a circle, line segment, and a system of three concentric circles is discussed. A characterization of stable electrostatic configurations with a few points is obtained.

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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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We discuss on algebraic structure of solutions space of regular Carleman-Bers-Vekua equation. Using higher derivation in sense of Bers of generalized analytic functions we construct sequence of real vector spaces and such way we classify solutions space. Periodicity properties of first kind pseudo-analytic functions we extend on second kind pseudo-analytic functions and proved similar results for solutions of Beltrami equations. Consequently we obtain periodicity of complex structures on Riemann surfaces. We also give several related problems and conjectures are also discussed.

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In this paper we continue our investigation of vortex equation and related topics in framework of generalized analytic functions. We show that solution space of systems of vortex equations does not depend on location of zeros of Higgs field and in this way we obtain another proof of the well known Taubes’ theorem on description of solution space of vortex equation modulo gauge equivalence. It turns out that the first equation of this system is a particular case of Carleman-Bers-Vekua equation, and the second equation is a property of non dependence of the solution space of the first equation on complex structure of the noncompact Riemann surface, which is a Riemann sphere without zeros of the Higgs field.

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Theoretical explanation of the generation and intensification of low frequency (LF) internal gravity waves (IGW) is presented. The method used is based on generalizing results on shear flow phenomena from the hydrodynamicscommunity. In the 1990s, it was realized that fluctuation modes of spectrally stable nonuniform shearedflows are non-normal. That is, the linear operators of the flows modal analysis are non-normal and the corresponding eigenmodes are not orthogonal. The non-normality results in linear transient growth with bursts of the perturbations and the mode coupling, which causes the amplification of LF IG waves shear flow driven ionospheric plasma. Transient growth substantially exceeds the growth of the classical dissipative trapped-particle instability of the system. 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. With amplitude growth the nonlinear mechanism turns on and the process ends with self-organization of nonlinear solitary, strongly localized IGW vortex structures. Accumulation of these vortices in the ionosphere medium can create the strongly turbulent state.

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In the present article the dynamics of generation and propagation of planetary global weatherforming ultra-low frequency (ULF) electromagnetic wave structures in the dissipative ionosphere are given. These waves are stipulated by spatial inhomogeneous geomagnetic field. The large-scale waves are weakly damped. The waves generate the geomagnetic field from several tens to several hundreds nT and more. It is established, that planetary ULF electromagnetic waves, at their nonlinear interaction with the local shear winds, can self-localize in the form of nonlinear long-lived solitary vortices, moving along the latitude circles westward as well as eastward with velocity, different from phase velocity of corresponding linear waves. The vortex structures transfer the trapped particles of medium and also energy and heat. The nonlinear vortex structures represent can be the structural elements of strong macroturbulence of the ionosphere main drivers of the electromsgnetic weather at ionospheric level.

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The linear generation and intensification of internal gravity waves (IGW) in the ionosphere with non-uniform zonal wind (shear flow) is studied. On the basis of non-modal approach, the equations of dynamics and the energy transfer of IGW disturbances in the ionosphere with a shear flow is obtained. 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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Theoretical explanation intensification of low frequency (LF) internal gravity waves (IGW) is presented. The method used is based on generalizing results on shear flow phenomena from the hydrodynamics community. In the 1990s, it was realized that fluctuation modes of spectrally stable nonuniform sheared flows are non-normal. That is, the linear operators of the flows modal analysis are non-normal and the corresponding eigenmodes are not orthogonal. The non-normality results in linear transient growth with bursts of the perturbations and the mode coupling, which causes the amplification of LF IG waves shear flow driven ionospheric plasma and generation of the higher frequency oscillations. Transient growth substantially exceeds the growth of the classical dissipative trapped-particle instability of the system.

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The present note 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.

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In the present paper we consider the approximate solution issues for nonlinear boundary value problem for the Kirchhoff type static beam. The problem is reduced by means of Green’s function to a nonlinear integral equation. To solve this problem we use the Picard type iterative method. For both of these problems the new algorithms of approximate solutions are constructed and numerical experiments are executed. The results of computations are presented in tables and diagrams.

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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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The effect of the dust on the climate in the Caucasus region, with a specific focus on Georgia, was investigated with a Regional Climate Model RegCM interactively coupled with a dust model. For this purpose we have executed sets of 30 years simulations (1985–2014) with and without dust effects by RegCM4.7 model with 16.7 km resolution over the Caucasus domain and with 50 km resolution encompassing most of the Sahara, the Middle East, and the Great Caucasus with adjacent regions. Results of calculations have shown that the dust aerosol is an active player in the climate system of Georgia. Mineral dust aerosol influences on temperature and aerosol optical depth spatial and temporally inhomogeneous distribution on the territory of Georgia and generally has been agreed with MODIS satellite data. Results of numerical calculations have shown that dust radiative forcing inclusion has improved simulated summer temperature. The mean annual temperature increased across the whole territory of Georgia in simulations when dust direct effect was considered.

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System of Maxwell equation is considered. Reduction to the integro-differential form is done. Existence, uniqueness and large time behavior of solutions of the initial-boundary value problem is given. Finite difference scheme is investigated. Wider class of nonlinearity is studied than one has been investigated before.

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The present work is intended to provide some materials for participants of the forthcoming TICMI Advance Courses (September 22-25, 2019) on ”Mathematical Models of Piezoelectric Solids and Related Problems”. This work is oriented mainly on the lecture course of the same name ”Piezoelectric Viscoelastic Kelvin-Voigt Cusped Prismatic Shells”, foreseen in the prospective programme of the above-mentioned Advance Courses. It mainly contains unpublished results of the author concerning piezoelectrics. Some auxiliary materials, which make the work self-contained, are provided as well. The aim of the present work is also to draw the attention of scientists, particularly of young researchers, 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.

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In this paper we study skolemization for unranked logics with classical first-order semantics. Skolemization is a transformation on first-order logic formulae, which removes all existential quantifiers from a formula. This technique is vital in proof theory and automated reasoning, especially for refutation based calculi, like resolution, tableaux, etc. Here we extend skolemization procedure to unranked formulae and prove that the procedure is sound and complete.

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This book presents eleven peer-reviewed papers from the 3rd International Conference on Applications of Mathematics and Informatics in Natural Sciences and Engineering (AMINSE2017) held in Tbilisi, Georgia in December 2017. Written by researchers from the region (Georgia, Russia, Turkey) and from Western countries (France, Germany, Italy, Luxemburg, Spain, USA), it discusses key aspects of mathematics and informatics, and their applications in natural sciences and engineering. Featuring theoretical, practical and numerical contributions, the book appeals to scientists from various disciplines interested in applications of mathematics and informatics in natural sciences and engineering. G.Jaiani, D.Natroshvili (Eds.)

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One of the most important areas of artificial intelligence is computer vision-based objects recognition and perception of the environment in general. It is known that the artificial neural networks (ANN) training process is a rather complicated, delicate and time-consuming process. Collecting the data needed for training and giving it such form that will be required for a given moment makes this process even more complicated. The achievement of the desired goal requires a large amount of work and, accordingly, takes much time. Since the tasks related to the training process in the majority of cases must be performed by a human, and in the case of process automation, human intervention is to some extent still required. For this reason, we suggest a new approach implementing the software which makes possible to detect unknown object independently by computer vision, and then automatically carry out the steps necessary for the training process. In other words, by means of this algorithm, detection of unknown object in the frame occurs and then automatic search of materials necessary for training via the Internet (or elsewhere) and after collecting the desired amount of information, the training of artificial neural networks starts. As a result, we get a perfect file of Haar Cascade type format and the given object is added to the list of the known objects. In our opinion, this approach is convenient and gives artificial intelligence a greater degree of autonomy provided by computer vision.

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The article presents the pioneering study of a linear system of equilibrium equations of elastic bodies with double porosity when the rigid skeleton of the body is a mixture of two isotropic materials. The general solutions of this system of equations are represented by means of harmonic functions and a metaharmonic function. Based on the constructed general solution, the class of boundary value problems of porous elasticity for the rectangular parallelepiped is solved analytically using the method of separation of variables. The corresponding boundary-contact problems for the multilayer rectangular parallelepiped are also considered.

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We solve the static two-dimensional boundary value problems for an elastic porous circle with voids. Special representations of a general solution of a system of differential equations are constructed via elementary 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. The question pertaining to the uniqueness of regular solutions of the considered problems is investigated.

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In the present paper, the special representations of a general solution of a system of differential equations of the theory of elastic materials with voids is constructed by using 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 initial problems. These representations are used to solve problems for an elastic plane with circular hole and with voids. The solutions are written explicitly in the form of absolutely and uniformly converging series. The uniqueness of regular solutions of the considered problems is also investigated.

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For nonlinear controlled functional differential equations variation formulas of solution are proved, in which the effects of perturbations of the initial moment and constant delays, and also that of the continuous initial condition are detected.

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For the linear controlled neutral differential equation an inverse problem approximately is solved.

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Questions of the sensitivity analysis are investigated for the delay differential equation and the optimal problem with the mixed initial condition .

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Linear stability and Hoph bifurcation of a solution of the initial-boundary value problem as well as the finite difference scheme for one system of nonlinear partial differential equations are investigated. The blow up case is fixed. The mentioned system is based on the Maxwell equations which describe the process of electromagnetic field penetration into a substance. Numerous computer experiments are carried out and relying on the obtained results, some graphical illustrations are presented.

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In this article, the linear theory of binary thermoviscoelastic mixtures is considered and the basic boundary value problems (BVPs) of steady vibrations are investigated. Namely, the fundamental solution of the system of equations of steady vibrations is constructed explicitly and its basic properties are established. Green’s second and third identities are obtained and the uniqueness theorems for classical solutions of the internal and external basic BVPs of steady vibrations are proved. The surface and volume potentials are constructed and their basic properties are given. The determinants of symbolic matrices are calculated explicitly. The BVPs are reduced to the always solvable singular integral equations for which Fredholm’s theorems are valid. Finally, the existence theorems for classical solutions of the internal and external BVPs of steady vibrations are proved by the potential method and the theory of singular integral equations.

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In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with double porosity is considered. Some basic results on the solutions of the quasi-static and steady vibrations equations are obtained. Indeed, the fundamental solutions of the systems of equations of quasi-static and steady vibrations are constructed by elementary functions and their basic properties are established. Green’s formulae and the integral representation of regular solution in the considered theory are obtained. Finally, a wide class of the internal boundary value problems of quasi-static and steady vibrations is formulated and on the basis of Green’s formulae the uniqueness theorems for classical solutions of these problems are proved.

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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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The paper is concerned with some properties of the theory of elementary volume. It is shown that there exists an extension of the standard Jordan measure of R^2, which does not possess the strong uniqueness property in the class of π_2-volumes.

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In terms of a group G of isometries of Euclidean space, it is given a necessary and sufficient condition for the uniqueness of a G-measure on the Borel σ-algebra of this space.

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The Borel types of some classical small subsets of the real line are considered. In particular, under Martin’s axiom it is shown that there are at least c^+ pairwise incomparable Borel types of generalized Luzin sets (resp. of generalized Sierpiński sets), where c stands for the cardinality of the continuum.

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In this Note, we show that the notion of a basis of a finite-dimensional vector space could be introduced by an argument much weaker than Gauss’ reduction method. Our aim is to give a short proof of a simply formulated lemma, which in fact is equivalent to the theorem on frame extension, using only a simple notion of the kernel of a linear mapping, without any reference to special results, and derive the notions of basis and dimension in a quite intuitive and logically appropriate way, as well as obtain their basic properties, including a lucid proof of Steinitz’s theorem.

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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.

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For the controlled functional differential equation with the delay parameter in the phase coordinate, the variation formulas of solutions are proved.

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In the present paper the linear equilibrium theory of thermoelasticity with microtemperatures is considered. The explicit solutions of the Neumann type boundary value problems in the theory of thermoelasticity with microtemperatures for the sphere and for the whole space with a spherical cavity are constructed. The obtained solutions are represented by means of absolutely and uniformly convergent series.

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The two-dimensional system of nonlinear partial differential equations is considered. This system arises in process of vein formation of young leaves. Additive splitting and variable directions type finite difference schemes are used. Comparison of numerical calculations of the proposed methods are done

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The paper is devoted to the construction and study of the additive average semi-discrete scheme for two nonlinear multidimensional integro-differential equations of parabolic type. The studied equation is based on well-known Maxwell’s system arising in mathematical simulation of electromagnetic field penetration into a substance. Existence, uniqueness and long-time behavior of solutions of initial-boundary value problems for nonlinear systems of parabolic integro-differential equations are fixed too.

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The paper considers the boundary value problems of elasticity for an ellipse and ellipse with a crack when tangential stress is applied to the ellipse boundary. The mathematical models of these problems are obtained by setting the relevant problems for a semi-ellipse: a) the continuity conditions for the problem solution are given at the linear border, b) the continuity conditions for the problem solution are given at the portion of the linear boundary, beyond the focuses, with the tangential stresses given on the section between the focuses. So, a semi-ellipse can be bound as a whole ellipse, with the continuity conditions of the solution on the section between its focuses met in one case (when there is no crack) and not met in another case (when there is a crack, which is affected by the tangential stress). The problem solution for the cracked ellipse is reduced to the solution of the internal and external problems of elasticity, which are solved quite simply by the method of separation of variables.

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The paper considers boundary value problems of elasticity for semi-ellipse, when boundary conditions at the portion of the linear boundary between the focuses are nonzero and outside the focuses are zero. Thus, the continuity conditions for the problem solution are given at the portion of the linear boundary, therefore it is possible to bind the semi-ellipse as a whole ellipse, in which on the section between the focuses the condition of uninterrupted continuation of the problem solution not performed along this part, i.e. we have a crack on which, for example, the tangential stress acts. The problem solution for the cracked ellipse is reduced to the solution of the internal and external problems of elasticity, which are solved quite simply by the method of separation of variables.

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Contact problems have a broad range of applicability in building mechanics, mining mechanics, soil mechanics, engineering fields, such as wheel-rail contact in railway industry, investigation of friction and wear, ball bearing, etc. Such problems consider elastic contact between two bodies. When they are pressed together, an elastic field of stress, strain and displacement arises in each body. A contact area is formed where the two body surfaces coincide with each other. The stresses exerted on each body consist of normal stress and tangential stress. In the so called normal contact problems, we solve for the contact area and the normal stress on it. In so called tangential contact problems the research questions are to find the adhesion and slip areas, and the distribution of tangential stresses. 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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In the prezent work we consider the problem of finding an equally strong contour for a rectangular plate weakened by a rectilinear cut which ends are cut out by convex smooth arcs. It is assumed that absolutely smooth rigid punches are applied to every link of the rectangular. The punches are under the action of normal stretching forces with the given principal vectors and the internal part of the boundary is free from external forces. Our problem is to find an elastic equilibrium of the plate and analytic form of the unknown contour under the condition that the tangential normal stress on it takes constant value (the condition of the unknown contour full-strength). For solution of the problem using the method of complex analysis and Kolosov-Muskhelishvilis potentials and the equation of the equally strong contour are constructed effectively (analytically).

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In the present paper we consider the problem of finding a partially unknown boundary of the plane theory of elasticity for a rectangular domain which is weakened by an equally strong contour (the unknown part of the boundary). The unknown part of the boundary is assumed to be free from external force, and to the remaining part of the rectangular boundary are applied the same absolutely smooth rigid punches subjected to the action of external normal contractive forces with the given principal vectors. For solution of the problem using the method of complex analysis and Kolosov-Muskhelishvili’s potentials and the equation of the equally strong contour are constructed effectively (analytically).

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One nonlinear integro-differential system with source terms is considered. The model arises on mathematical simulation of the process of penetration of a magnetic field into a substance. Initial-boundary value problem with mixed boundary condition is investigated. Finite difference scheme is constructed and studied. Graphical illustrations of numerical experiments are given.

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The problem of finding an equally strong contour for a rectangular plate weakened by a rectilinear cut which ends are cut out by convex smooth arcs (we will call the set of these arcs the unknown part of the boundary) is considered. It is assumed that absolutely smooth rigid punches are applied to every link of the rectangular. The punches are under the action of normal stretching forces with the given principal vectors and the internal part of the boundary is free from external forces. Our problem is to find an elastic equilibrium of the plate and analytic form of the unknown contour under the condition that the tangential normal stress on it takes the constant value (the condition of the unknown contour full-strength). The problem is solved by the method of complex analysis. The complex potentials of N. Muskhelishvili and equations of an unknown contour are constructed effectively (analytically).

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In this paper we give a characterization of points at which the Marcinkiewicz-Fejér means of double Vilenkin-Fourier series converge.

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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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In this paper plane problems of elasticity for a circular ring with double porosity is considered. The solutions are represented by means of three analytic functions of a complex variable and one solution of the Helmholtz equation. The problems are solved when the components of the displacement vector is known on the boundary of the circular ring.

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The static equilibrium of porous elastic materials with triple-porosity is considered in the case of an elastic Cosserat medium. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of three analytic functions of a complex variable and three solutions of Helmholtz equations. Concrete problem are solved for the circle.

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In the paper it is shown that any regular surface can be imbedded in the 3-dimensional Riemannian manifold.

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In this paper differential boundary problem is considered for the system of second order differential equations of elliptic type in plane domains bounded by smooth curves. The scheme of reduction of the desired problem to the problem of Riemann-Hilbert type for generalized analytic vectors is given.

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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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Integrals with the Weierstrass kernel and their properties are considered. With those integrals some problems of hydrodynamics are connected. More precisely, nonlinear waves propagation in the infinite reservoir. By means of these integrals asymptotics of the waves at peaks is obtained.

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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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This paper describes PρLog: a tool that combines Prolog with the ρLog calculus. Such a combination brings strategy-controlled conditional transformation rules into logic programming. They operate on sequences of terms. Transformations may lead to several results, which can be explored by backtracking. Strategies provide a control on rule applications in a declarative way. They are programmable: Users can construct complex strategies from simpler ones by special combinators. Different types of first- and second-order variables provide flexible control on selecting parts from sequences or terms. As a result, the obtained code is usually pretty compact and declaratively clear. In programs, PρLogspecific code can be intermixed with the standard Prolog code. The tool is implemented and tested in SWI-Prolog

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Rewriting logic is a well-known logic that emerged as an adequate logical and semantic framework for the specification of languages and systems. ⇢Log is a calculus for rule-based programming with labeled rules. Its expressive power stems from the usage of a fragment of higher-order logic (e.g., sequence variables, and function variables) to express atomic formulas. Its adequacy as a computational model for rule-based programming is derived from theoretical results concerning E-unification and E-matching in the fragment of logic adopted by ⇢Log. In this paper we choose a fragment of the ⇢Log calculus and argue that it can be used to perform deduction in rewriting logic. More precisely, we define a mapping between the entailment systems of rewriting logic and ⇢Log for which the conservativity theorem holds. It implies that, like rewriting logic, ⇢Log also can be used as a logical and semantic framework

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Antiplane strain (shear) of orthotropic non-homogeneous prismatic shell-like bodies are 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. We study the dependence of the well-posedness of the boundary conditions (BCs) on the character of the vanishing of the shear modulus. The case of vibration is considered as well.

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First type initial-boundary value problem for one nonlinear parabolic integro-differential equation is considered. This model is based on Maxwell 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 not only power type that already were studied but more wide cases of nonlinearities. Existence, uniqueness and long-time behavior of solutions are fixed too.

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Generation of sheared zonal flow by low-frequency coupled electrostatic drift and ion-acoustic waves is presented. Primary waves of different (small, intermediate, and large) scales are considered, and the appropriate system of equations consisting of generalized Hasegawa-Mima equation for the electrostatic potential (involving both vector and scalar nonlinearities) and equation of parallel to magnetic field ions motion is obtained. It is shown that along with the mean poloidal flow with strong variation in minor radius mean sheared toroidal flow can also be generated. According to laboratory plasma experiments, main attention to large scale drift-ion-acoustic wave is given. Peculiarities of the Korteweg-de Vries type scalar nonlinearity due to the electrons temperature non-homogeneity in the formation of zonal flow by large-scale turbulence are widely discussed. Namely, it is observed that such type of flows need no generation condition and can be spontaneously excited.

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Two systems of nonlinear partial differential equations are considered. Both systems are obtained at mathematical modeling of process of electromagnetic field penetration in the substance. In the quasistationary approximation, this process, taking into account of Joule law is described by nonlinear well known system of Maxwell equations. Taking into account heat conductivity of the edium and again the Joule law, the different type nonlinear system of partial differential equations is obtained. Investigation and approximate solution of the initial-boundary value problems are studied for these type models. Linear stability of the stationary solution is studied. Blow-up is fixed. 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. Averaged additive semi-discrete models, finite difference schemes are constructed and theorems of convergence are given.

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The present work is devoted to the bending problems of prismatic shell with the thickness vanishing at infinity as an exponential function. The bending equation in the zero approximation of Vekua’s hierarchical models is considered. The problem is reduced to the Dirichlet boundary value problem for elliptic type partial differential equations on half-plane. The solution of the problem under consideration is constructed in the integral form.

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We consider some properties of functions, which have thick (or massive) graphs with respect to certain classes of measure and some applications of set-theoretical and algebraic methods to measurability of sets and function.

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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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In the paper, using the perturbation algorithm, purely implicit three-layer and four-layer semi-discrete schemes for an abstract evolutionary equation are reduced to two-layer schemes. The solutions of these two-layer schemes are used to construct an approximate solution of the initial problem. By using the associated polynomials the estimates for the approximate solution error are proved.

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In the paper the differential equation u⁽ⁿ⁾(t)+p(t)|u(τ(t))| (µ(t)) sign u(τ(t))=0, is considered. Here, we assume that n≥ 3, p∈ L loc (R₊; R₋), µ∈ C (R₊;(0,+∞)), τ∈C(R₊;R₊), τ (t)≤ t for t∈ R₊ and lim t τ (t)=+∞. In case µ (t)≡ const> 0, oscillatory properties of equation have been extensively studied, where as if µ (t)≢ const, to the extent of authors' knowledge, the analogous questions have not been examined. In this paper, new sufficient conditions for the equation (∗) to have Property B are established.

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

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In the paper, 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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In this paper the 2D linear theory of thermoelasticity for materials with double porosity is considered. There the fundamental and singular matrices of solutions are constructed in terms of elementary functions. The single and double layer potentials are obtained. Finally the basic properties of these potentials are established.

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The paper deals with the problem of creating the theory of distributions and is an attempt to show that Andrea Razmadze stood behind the creation of this theory. He was the first to have introduced the class of finite-jump functions, which are considered to be both native solutions (extremals) of some variational problems and foundations for creating the generalized functions.

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Computational science (also scientific computing or scientific computation (SC)) (see Wikipedia) is „a rapidly growing multidisciplinary field that uses advanced computing capabilities to understand and solve complex problems. Computational science fuses three distinct elements: • Algorithms (numerical and non-numerical) and modeling and simulation software developed to solve science (e.g., biological, physical, and social), engineering and, humanities problems • Computer and information science that develops and optimizes the advanced system hardware, software, networking, and data management components needed to solve computationally demanding problems • The computing infrastructure that supports both the science and engineering problem solving and the developmental computer and information science” . Among same works the book of A.Quarteroni and F. Saleri [1] which was published four times by Springer-Verlag from 2003 y., is defused and wide applicable. Unfortunately same phenomenon for the our educational processes remained without attention. Such type publications have an extensive auditorium with different education and a matter of taste (for this aim we cited [2]). Our works [3-4] would be been in this direction too, but the manuals same [1] kind have new important value especially for practice. In this connect on this conference we will present the materials which evidently extend and refined a corresponding well-known methodology for some class of boundary value problems for differential equations without narrowing admissible classes and recommended to users an optimal and logical lightly schemes with MATLAB and design. The possibility of generalization and improvement of some other parts of [1] follows immediately from [3-4]. The presenting in this report data and design calculated by Z.Vashakidze.

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Technological progress brought the civilization to the emergence of numerous artificial factors, the influence of which on the environment becomes in all perceptible. This stipulates the quantitative and qualitative change in the environment. In this connection, there arises an actual problem of studying and analyzing the current situation with the purpose of elaboration of the principles and facilities of saving the environment in the suitable condition of living. It is necessary to have adequate information about the quality of the environment for its study, analysis and management. The environment is characterized by an enormous number of physical, chemical and biological parameters. A lot of measurements are to be carried out for permanent control of these parameters. Therefore, the solution of the problems of control and management of the quality of the environment can be realized only by using automated, continuously operating pollution analyzers and automated environment control systems...

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The present paper is devoted to construction of hierarchical models for 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 boundary value problems are set.

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The profitability and efficiency of farming economies achieved by introduction of advanced agricultural methods duri ng two years, in particular, the increase in the productivity of soil with the use of advanced manure, treated in biogas facilities, the increase in the productivity of degraded and low-productiveagricultural lands (exchange of seeds, introduction of new cultures, drainage etc.) and soil erosion prevention techniques were investigated by application of the methods of mathematical statistics to the obtained results. High profitability, efficiency and economic justification of these methods are shown as a result of this investigation.

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A theoretical-numerical description of zonal flow generation in the turbulent ionosphere by controlled inhomogeneous background wind is given. The generalized Charney–Obukhov equation, which describes the nonlinear interaction of five different-scale modes (primary modes, relatively short-wave ultra-low frequency (ULF) magnetized Rossby waves (MRWs) (pumping waves), two satellites of these MRWs, long-wave zonal mode, and large-scale background shear flows (inhomogeneous wind)) is used. New features of energy transfer from relatively small-scale waves and the background shear flow into that of largescale zonal flows and nonlinear self-organization of the five-wave collective activity in the ionospheric medium are identified based on the numerical solution of the corresponding system of equations for perturbation amplitudes (generalized eigenvalue problems). It is shown that if there is the background shear flow with a moderate amplitude growth the modulation instability increment and intensifies the zonal flow generation, while a very strong shear flow significantly reduces the modulation instability increment and can even suppress the generation process.

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The present work is devoted to the one dimensional mathematical model studding gas flow in the inclined and branched pipeline. A simplified mathematical model governing the dynamics of gas non-stationary flow in the inclined, branched pipeline is constructed. Formula describing gas pressure distribution along the branched pipeline is presented.

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We discuss equilibrium configurations of point charges with Coulomb interaction confined to a circle or linear segment. Specifically, we aim at characterization of finite configurations of points which can serve as equilibrium configurations of repulsive point charges. We also investigate the stability of arising equilibrium configurations. For concyclic configurations, we present a few general results concerned with these two problems. For aligned configurations, the main results refer to three points in a linear segment with prescribed point charges at its ends. We also discuss the case of several concentric circles and connections with the mathematical theory of electrostatic ion traps.

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The note 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. These type of equations are natural generalizations. The studied equations are based on well-known Maxwell’s system arising in mathematical simulation of electromagnetic field penetration into a substance .

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In the paper the shells consisting of binary mixtures are considered. Based on I.Vekua’s works, the question of existence of neutral surfaces in such shells is studied. By neutral surface is called a surface which belongs to a shell but is not subject to tensions and compressions by the deformation of the elastic body.

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The purpose of this paper is to consider the linear theory of elasticity for solids with double porosity. From this system of the equations, using a method of a reduction of I. Vekua, we receive the equilibrium equations. Using the analytic functions of a complex variable and solutions of the Helmholtz equation. The Dirichlet boundary value problem are solved explicitly for approximation N = 1.

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The purpose of this paper is to consider the two-dimensional version of the linear theory of elasticity for solids with triple-porosity in the case of an elastic Cosserat medium. Using the analytic functions of a complex variable and solutions of the Helmholtz equation the Dirichlet boundary value problem are solved explicitly for the concentric circular ring.

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The static equilibrium of porous elastic materials with triple-porosity is considered in the case of an elastic Cosserat medium. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of three analytic functions of a complex variable and three solutions of Helmholtz equations. Concrete problem are solved for the circle.

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Nonlinear parabolic integro-differential model obtained by the reduction of well-known Maxwell system of partial differential equations is considered. Unique solvability of the stated initial-boundary value problem and asymptotic behavior of solution as t → ∞ are investigated. The semi-discrete and implicit finite-difference schemes are constructed. Stability and convergence of those schemes are given.

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Theorems on the continuous dependence of solutions on perturbations of the initial data and the right-hand side of equations are proved. Under initial data we imply the collection of an initial moment, delay and initial functions. Perturbations of the initial data are small in a standard norm and perturbations of the right-hand side of equation are small in the integral sense.

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For nonlinear functional differential equations with several constant delays, the theorems on the continuous dependence of solutions of the Cauchy problem on perturbations of the initial data and on the right-hand side of the equation are proved. Under the initial data we mean the collection of the initial moment, constant delays, initial vector and initial function. Perturbations of the initial data and of the right-hand side of the equation are small in a standard norm and in an integral sense, respectively. Variation formulas of a solution are derived for equations with a discontinuous initial and continuous initial conditions. In the variation formulas, the effects of perturbations of the initial moment and delays as well as the effects of continuous initial and discontinuous initial conditions are revealed. For the optimal control problems with delays, general boundary conditions and functional, the necessary conditions of optimality are obtained in the form of quality or inequality for the initial and final moments, for delays and an initial vector and also in the form of the integral maximum principle for the initial function and control.

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Present artcile studies semantics of the constraint logic programming built over sequences and contexts, called CLP(SC). Sequences and contexts are constructed over function symbols and function variables which do not have fixed arity, together with term, sequence, and context variables. For some function symbols, the order of the arguments matter (ordered symbols). For some others, this order is irrelevant (unordered symbols). Term variables stand for single terms, sequence variables for sequences, context variables for contexts, and function variables for function symbols. We have studied the semantics of CLP(SC) and showed its application in membrane computing.

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Theorems on the continuous dependence of solutions on perturbations of the initial data and the nonlinear term of the right-hand side are given for the neutral differential equation whose right-hand side is linear with respect to the concentrated prehistory of the phase velocity and nonlinear with respect to the distributed prehistory of the phase coordinates. Under the initial data we understand the collection of initial moment, of delay function and initial functions. Perturbations of the initial data and of the right-hand side of the equation are small in a standard norm and in the integral sense, respectively.

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In this article we show how access control policies can be expressed in PρLog, which is a system for programming with conditional transformation rules, controlled by strategies. PρLog combines the power of logic programming with rewriting, which makes it convenient to reason about the policies.

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In the present work, 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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We consider a boundary-value problem for the nonlinear integrodifferential equation, simulating the static state of the Kirchhoff beam. The problem is reduced to a nonlinear integral equation, which is solved by using the Picard iterative method. The convergence of the iterative process is established and the error is estimated.

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In this paper plane problems of elasticity for a circle with double porosity is considered. The solutions are represented by means of three analytic functions of a complex variable and one solution of the Helmholtz equation. The problems are solved when the components of the displacement vector is known on the boundary.

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We examine the third kind integral equations in Hölder class. The coefficients of the equations are piecewise strictly monotone functions having simple zeros. By singular integral equations theory, for solvability of considered equations, we give the necessary and sufficient conditions. Finding a solution is reduced to solving a regular integral equation of second kind.

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The purpose of this paper is to consider the two-dimensional version of the linear theory of elasticity for solids with triple-porosity in the case of an elastic Cosserat medium. Using the analytic functions of a complex variable and solutions of the Helmholtz equation basic boundary value problems are solved explicitly for the circle.

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In the present paper the linear theory of micropolar viscoelasticity is considered. The explicit expression of fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions and its basic properties are established. The Green's formulas in the considered theory are obtained. The formulas of integral representations of Somigliana-type of regular vector and regular (classical) solution are presented. The representation formulas of Galerkin-type solution of the system of nonhomogeneous equations and of the general solution of the system of homogeneous equations by means of eight metaharmonic functions are presented. The completeness of these solutions is proved.

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In the present paper, the linear theory of binary viscoelastic mixtures is considered. The basic properties of plane harmonic waves are established. Green’s first identity for 3D bounded and unbounded domains is obtained. On the basis of this identity the uniqueness theorems of regular (classical) solutions of the boundary value problems (BVPs) of steady vibrations are proved. Then these theorems are established in the quasi-static case. Finally, the uniqueness theorems for the first internal and external BVPs of steady vibrations in general and quasi-static cases are proved under weak condition on the viscoelastic constants.

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In this article, the linear theory of thermoviscoelasticity for Kelvin–Voigt materials with double porosity is considered. The fundamental solution of the system of equations of steady vibrations is constructed by elementary functions and its basic properties are established. The Green’s first identity in the considered theory is obtained. A wide class of the internal and external boundary value problems (BVPs) of steady vibrations is formulated. Finally, on the basis of the Green’s identity, the uniqueness theorems for regular (classical) solutions of these BVPs are proved.

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Strange Functions in Real Analysis, Third Edition differs from the previous editions in that it includes five new chapters as well as two appendices. More importantly, the entire text has been revised and contains more detailed explanations of the presented material. In doing so, the book explores a number of important examples and constructions of pathological functions. After introducing basic concepts, the author begins with Cantor and Peano-type functions, then moves effortlessly to functions whose constructions require what is essentially non-effective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum. Finally, the author considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms. On the whole, the book is devoted to strange functions (and point sets) in real analysis and their applications.

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For certain groups of isometric transformations of the Euclidean plane R^2, negligible and absolutely negligible subsets of this plane are considered and compared with each other.

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A characterization is given of all sets in an uncountable commutative (G,+) which contain at least one absolutely nonmeasurable subset of G.

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The purpose of this paper is to consider the two-dimensional version of the linear theory of elasticity for solids with triple-porosity in the case of an elastic Cosserat medium. Using the analytic functions of a complex variable and solutions of the Helmholtz equation the second fundamental problem for the infinite plane with a circular hole are solved explicitly.

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The new numerical algorithms for a two-point boundary value problem with a non-constant coefficient are proposed. The Green function of the given problem is represented as a nonlinear operator with respect to the coefficient. This operator is approximated by an operator interpolation polynomial of the Newton type. For the inverse operators approximate formulas of different types are derived. The numerical algorithms and results of calculation of test problems are given.

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http://www.rmi.ge/eng/QUALITDE-2016/Shavadze_workshop_2016.pdf

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Long-time behavior of solution and semi-discrete scheme for one nonlinear parabolic integro-differential equation are studied. Initial–boundary value problem with mixed boundary conditions are considered. Attention is paid to the investigation of more wide cases of nonlinearity than already were studied. Considered model is based on Maxwell’s system describing the process of the penetration of a magnetic field into a substance.

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Initial–boundary value problem with mixed boundary conditions for one nonlinear parabolic integro-differential equation is considered. The model is based on Maxwell system describing the process of the penetration of a electromagnetic field into a substance. Unique solvability and asymptotic behavior of solution are fixed. Main attention is paid to the convergence of the finite difference scheme. More wide cases of nonlinearity that already were studied are investigated.

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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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Functional logic programming is an extension of the functional programming style with two important capabilities: to define nondeterministic operations with overlapping rules, and to use logic variables in both defining rules and expressions to evaluate. A suitable model for functional logic programs are conditional constructor-based term rewrite systems (CB-CTRSs), which can be transformed into an equivalent program in a simpler class of rewrite systems (the core language) where computations can be performed more efficiently. Recently, Antoy and Hanus proposed a translation of CB-CTRSs into an equivalent class of programs where computation can be performed efficiently by mere rewriting. Their computational model has the limitation of computing only ground answer substitutions for equations with strict semantics interpreted as joinability to a value. We propose two adjustments of their computational models, which are capable to compute non-ground answers.

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Exact solution of two dimensional problems of elasticity are constructed in the parabolic coordinates in domain bounded by coordinate lines of the parabolic coordinate system. Here we represent internal boundary value problems of elastic equilibrium of the homogeneous isotropic body bounded by coordinate lines of the parabolic coordinate system, when on parabolic border normal or tangential stresses are given. Exact solutions are obtained using the method of separation of variables. Using the MATLAB software numerical results and constructed graphs of the mentioned boundary value problems are obtained.

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The basic two-dimensional boundary value problems of the fully coupled linear equilibrium theory of elasticity for solids with double porosity structure are reduced to the solvability of two types of a problem. The first one is the BVPs for the equations of classical elasticity of isotropic bodies, and the other is the BVPs for the equations of pore and fissure fluid pressures. The solutions of these equations are presented by means of elementary (harmonic, metaharmonic, and biharmonic) functions. On the basis of the gained results, we constructed an explicit solution of some basic BVPs for an ellipse in the form of absolutely uniformly convergent series.

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One-dimensional system of nonlinear partial differential equations based on Maxwell’s model is considered. The initial-boundary value problem with mixed type boundary conditions is discussed. It is proved that in some cases of nonlinearity there exists critical value $\psi_{c}$ of the boundary data, such that for $0< \psi < \psi_{c}$ the steady state solution of the studied problem is linearly stable, while for $\psi> \psi_{c}$ is unstable. It is shown that when $\psi$ passes through $\psi_{c}$ then the Hopf type bifurcation may take place. The finite difference scheme is constructed. Numerical experiments agree with theoretical investigations.

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In this article for detection of gas accidental escape localization in the branched gas pipelines two mathematical models are suggested. The first model is indented for leak detection and localization in the horizontal branched pipeline and second one for an inclined section of the main gas pipeline. The algorithm of leak localization in the branched pipeline is not demand on knowledge of corresponding initial hydraulic parameters at entrance and ending points of each sections of pipeline. For detection of the damaged section and then leak localization in this section special functions and equations are constructed. Some results of calculations for horizontal pipelines having two, four and five sections are presented. Also a method and formula for the leak localization in the inclined section of the main gas pipeline are suggested. Some results of numerical calculations for the inclined pipeline are presented too

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The present work is devoted to the analysis of some mathematical models describing a movement of subsoil waters into the soil having the non-homogeneous multilayer structure in the vertical direction. Namely the corresponding systems of two-dimensional differential equations in stationary and non-stationary cases are considered. For the first one the problem with classical and non-classical boundary conditions is stated. For numerical solution of the problem with nonlocal boundary conditions the iteration process is constructed, which allows one to reduce the solution of the initial problem to the solution of a sequence of classical Dirichlet problems. Some results of numerical calculations for the soil having two-layer structure are presented

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CLP(H) is an instantiation of the general constraint logic programming scheme with the constraint domain of hedges. Hedges are finite sequences of unranked terms, built over variadic function symbols and three kinds of variables: for terms, for hedges, and for function symbols. Constraints involve equations between unranked terms and atoms for regular hedge language membership. We study algebraic semantics of CLP(H) programs, define a sound, terminating, and incomplete constraint solver, investigate two fragments of constraints for which the solver returns a complete set of solutions, and describe classes of programs that generate such constraints.

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In this work we consider equations of equilibrium of the isotropic elastic shell. By means of Vekua’s method, the system of differential equations for thin and shallow shells is obtained, when on upper and lower face surfaces displacements are assumed to be known. The general solution for approximations N=1 is constructed. The concrete problem is solved.

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In the present paper we consider the problem of bending of a plate for a curvilinear quadrangular domain with a rectilinear cut. It is assumed that the external boundary of the domain composed of segments (parallel to the abscissa axis) and arcs of one and the same circumference. The internal boundary is the rectilinear cut (parallel to the Ox-axis). The plate is bent by normal moments applied to rectilinear segments of the boundary, the arcs of the boundary are free from external forces, while the cut edges are simply supported. The problem is solved by the methods of conformal mappings and boundary value problems of analytic functions. The sought complex potentials which determine the bending of the midsurface of the plate are constructed effectively (in the analytical form). Estimates are given of the behavior of these potentials in the neighborhood of the corner points.

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In this paper, the 3-D geometrically and physically nonlinear theories of non-shallow shells are considered. The isometrical system of coordinates is of special interest, since in this system we can obtain bases equations of the theory of shells in a complex form. This circumstance makes is possible to apply the methods developed by N. Muskhelishvili and his disciples by means of the theory of functions of a complex variable and integral equations.

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In the present paper we consider a plane problem of elasticity for a polygonal domain with a curvilinear hole, which is composed of the rectilinear segment (parallel to the abscissa axis) and arc of the circumference. The problem is solved by the methods of conformal mappings and boundary value problems of analytic functions. The sought complex potentials are constructed effectively (in the analytical form). Estimates of the obtained solutions are derived in the neighborhood of angular points.

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In the present paper we consider the geometrically nonlinear and non-shallow spherical shells, when components of the deformation tensor have nonlinear terms. By means of I. Vekua’s method the system of equilibrium equations in two variables is obtained. Using complex variable functions and the method of the small parameter approximate solutions are constructed for N = 2 in the hierarchy by I. Vekua. Concrete problem has been solved.

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We consider a plane problem of elasticity for double-connected domain bounded by polygons. The problem is solved by the methods of conformal mappings and boundary value problems of analytic functions. The sought complex potentials are constructed effectively (in the analytical form). Estimates of the obtained solutions are derived in the neighborhood of angular points.

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We consider the three-dimensional system of the equations of elastic static equilibrium of bodies with double porosity. From this system of the equations, using a method of a reduction of I. Vekua, we receive the equilibrium equations for the shallow shells having double porosity. Further we consider a case of plates of constant thickness in more detail. Namely, the system of the equations corresponding to approximations N=1 it is written down in a complex form and we express the general solution of these systems through analytic functions of complex variable and solutions of the Helmholtz equation. The received general representations of decisions give the opportunity to analytically solve boundary value problems about elastic equilibrium of plates with double porosity.

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I. Vekua constructed several versions of the refined linear theory of thin and shallow shells, containing, the regular processes by means of the method of reduction of 3-D problems of elasticity to 2-D ones. In the present paper, by means of Vekua’s method, the system of differential equations for the Geometrically nonlinear theory non-shallow shells is obtained.

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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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Nonlinear parabolic integro-differential model which is based on Maxwell system is considered. Large time behavior of solutions of the initial-boundary value problem with mixed boundary condition is given. Finite difference scheme is investigated. Wider class of nonlinearity is studied than one has been investigated before.

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The Dirichlet-type problem for one quasi-linear elliptic system is investigated.

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In the present paper hierarchical model for cusped, in general, elastic prismatic shells is considered, when on the face surfaces a normal to the projection of the prismatic shell component of a traction vector and parallel to the projection of the prismatic shell components of a displacement vector are known.

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Generation of large-scale zonal flows by the small-scale electrostatic drift wave turbulence in the magnetized plasma under the action of mean poloidal sheared flow is considered. Attention to large-scale (compared to the ion Larmor radius) drift structures is paid. To this end, the generalized Hasegawa-Mima equation containing both vector and scalar nonlinearities is derived, and the appropriate eigenvalue problem is solved numerically. Destabilizing role of the small amplitude mean shear flow and spatial inhomogeneity of electron temperature is shown.

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The Stokes nonlinear waves associated with the nonlinear problem of a free boundary with peaks in incompressible heavy fluid are studied in 2D. In the early works of the author the problem was reduced to the nonlinear integral equation with the weakly singular kernel. The approximate solution of this equation is obtained. The profile of the free boundary is plotted by means of Maple-12.

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The paper deals with the measurability properties of some classical subsets of the real line ℝ having an extra-ordinary descriptive structure: Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpiński sets. In particular, it is shown that there exists a translation invariant measure ???? on ℝ extending the Lebesgue measure and such that all Sierpiński sets are measurable with respect to ????

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In the recently published paper,1 the authors assert that they have investigated large-scale nonlinear dispersionless Alfven waves and have obtained appropriate vortex structures. First of all, this conclusion contradicts the generally accepted fact that in nonlinear plasma theory, waves without dispersion at the nonlinear stage should undergo steepening leading to break. The authors did not explain the physical mechanism of self-organization of dispersionless Alfven waves into stationary vortical structures.

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For the general functional differential equation, the sufficient conditions n –th order to have Property A (Property B) are established. As particular was, we consider almost linear ordinary differential equation deviating argument. The sufficient conditions are obtained for the solutions to be oscillatory. These criteria cover the wee-known results for the linear differential equations.

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In this work we consider the two-dimensional system of the differential equations describing the plane statical thermoelastic balance of homogenous isotropic elastic bodies, the microelements of which have microtemperature in addition to the classical displacement and a temperature field. It is construed the general solution of this system of the equations by means of analytic functions of complex variable and solutions of the equation of Helmholtz. The general representation of the solution obtained gives the opportunity to construct the analytical solutions of a number of plane boundary value problems of microthermoelasticity. As an example we consider the boundary value problem for a concentric circular ring.

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The purpose of this paper is to consider the two-dimensional version of the fully coupled theory of elasticity for solids with double porosity the and to solve explicitly some boundary value problems (BVPs) of statics for an elastic circle. The explicit solutions of this BVPs are constructed by means of absolutely and uniformly convergent series. The questions on the uniqueness of a solutions of the problems are investigated.

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In this paper, the fully coupled theory of elasticity for solids with double porosity is considered. The explicit solutions of the basic boundary value problems (BVPs) in the fully coupled linear equilibrium theory of elasticity for the space with double porosity and spherical cavity are constructed. The solutions of these BVPs are represented by means of absolutely and uniformly convergent series.

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The main goal of this paper is to consider the Dirichlet type boundary value problem (BVP) of the fully coupled equilibrium theory of elasticity for solids with double porosity and to construct explicitly the solution of BVP for a spherical layer in the form of absolutely and uniformly convergent series.

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The present article studies the equilibrium theory of thermomicrostretch elastic solids with microtemperatures. The general solution of the equations for a homogeneous isotropic microstretch thermoelastic sphere with microtemperatures is constructed and the solution of the Dirichlet-type boundary value problem for the sphere in the form of absolutely and uniformly convergent series is obtained.

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In the paper the test of homogeneity and goodness-of-fit for checking the hypotheses of equality distribution densities is constructed. The power asymptotics of the constructed test of homogeneity and goodness-of-fit for certain types of close alternatives is also studied.

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We construct new criteria for testing the hypotheses that p ≥ 2 independent samplings have identical densities of distribution (hypothesis of homogeneity) or the same well-defined densities of distribution (goodness-of-fit test). The limiting power of the constructed criteria is established for some local “close” alternatives.

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Estimation of a non linear integral functional of probability distribution density and its derivatives is studied. The truncated plug-in-estimator is taken for the estimation. The integrand function can be unlimited, but it cannot exceed polynomial growth. Consistency of the estimator is proved and the convergence order is established. Aversion of the central limit theorem is proved. As an example an extended Fisher information integral and generalized Shannon's entropy functional are considered.

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The paper deals with the problem of electronic verification of people on the basis of measurement information of a fingerprint reader and new approaches to its solution. The offered method guaranties the restriction of error probabilities of both type at the desired level at making a decision about permitting or rejecting the request on service in the system. On the basis of investigation of real data obtained in the real biometrical system, the choice of distribution laws is substantiated and the proper estimations of their parameters are obtained. Using chosen distribution laws, the normal distribution for measurement results of characteristics of the people having access to the system and the beta distribution for the people having no such access, the optimal rule based on the Constrained Bayesian Method (CBM) of making a decision about giving a permission of access to the users of the system is justified. The CBM, the Neyman–Pearson and classical Bayes methods are investigated and their good and negative points are examined. Computation results obtained by direct computation, by simulation and using real data completely confirm the suppositions made and the high quality of verification results obtained on their basis

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Helps the reader to understand some conjectures arising in the criticism of null hypothesis significance testing (NHST) Includes a special chapter that helps the reader to calculate infinite-dimensional Riemann integrals over infinite-dimensional rectangles in R8 Considers how to construct objective consistent estimates of an unknown parameter in a Polish group

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A huge literature is devoted to the study of cusped prismatic shells on the basis of the classical theory of elasticity. It was stimulated by the works of I. Vekua. I. Vekua considered very important to carry out investigations of boundary value and initial boundary value problems for such bodies, since they are connected with egenerate partial differential equations and, therefore, are not classical, in general. The present paper is devoted to cusped prismatic shells on the basis of the theory of micropolar elasticity. Namely, on the basis of the N = 0 approximation of hierarchical models for micropolar elastic cusped prismatic shells constructed by the I. Vekua dimension reduction method.

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It is considered a transmittion process of a useful signal in Ornstein-Uhlenbeck model in C[−l,l[ defined by the stochastic differential equation dΨ(t,x,ω)=∑n=02mAn∂n∂xnΨ(t,x,ω)dt+σdW(t,ω) with initial condition Ψ(0,x,ω)=Ψ0(x)∈FD(0)[−l,l[, where m≥1, (An)0≤n≤2m∈R+×R2m−1, ((t,x,ω)∈[0,+∞[×[−l,l[×Ω), σ∈R+, C[−l,l[ is Banach space of all real-valued bounded continuous functions on [−l,l[, FD(0)[−l,l[⊂C[−l,l[ is class of all real-valued bounded continuous functions on [−l,l[ whose Fourier series converges to himself everywhere on [−l,l[, (W(t,ω))t≥0 is a Wiener process and Ψ0(x) is a useful signal. By use a sequence of transformed signals (Zk)k∈N=(Ψ(t0,x,ωk))k∈N at moment t0>0, consistent and infinite-sample consistent estimations of the useful signal Ψ0 is constructed under assumption that parameters (An)0≤n≤2m and σ are known. Animation and simulation of the Ornstein-Uhlenbeck process in C[−l,l[ and an estimation of a useful signal are also presented.

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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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I. Vekua constructed hierarchical models for elastic prismatic shells, in particular, plates of variable thickness, when on the face surfaces either stresses or displacements are known. In the present paper other hierarchical models for cusped, in general, elastic isotropic and anisotropic prismatic shells are constructed and analyzed, namely, when on the face surfaces (i) a normal to the projection of the prismatic shell component of a stress vector and parallel to the projection of the prismatic shell components of a displacement vector, (ii) a normal to the projection of the prismatic shell component of the displacement vector and parallel to the projection of the prismatic shell components of the stress vector are prescribed. We construct also hierarchical models, when other mixed conditions are given on face surfaces. In the zero approximations of the models under consideration peculiarities (depending on sharpening geometry of the cusped edge) of correct setting boundary conditions at edges are investigated. In concrete cases some boundary value problems are solved in an explicit form. As an example of application of the constructed Vekua-type models to composite structures, an unidirectional lamina with fibers parallel to $x_2$-axis under shear strain is considered. Tension–compression is treated as well.

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A certain modified version of Kolmogorov’s strong law of large numbers is used for an extension of the result of C. Baxa and J. Schoißengeier (2002) to a maximal set of uniformly distributed sequences in (0,1) which strictly contains the set of all sequences having the form ({αn})n∈N for some irrational number α and having the full ℓ1∞-measure, where ℓ1∞ denotes the infinite power of the linear Lebesgue measure ℓ1 in (0,1).

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The paper deals with the constrained Bayesian Method (CBM) for testing composite hypotheses. It is shown that, similarly to the cases when CBM is optimal for testing simple and multiple hypotheses in parallel and sequential experiments, it keeps the optimal properties at testing composite hypotheses. In particular, it easily, without special efforts, overcomes the Lindley’s paradox arising when testing a simple hypothesis versus a composite one. The CBM is compared with Bayesian test in the classical case and when the a priori probabilities are chosen in a special manner for overcoming the Lindley’s paradox. Superiority of CBM against these tests is demonstrated by simulation. The justice of the theoretical judgment is supported by many computation results of different characteristics of the considered methods.

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In the present paper on the basis of the linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures the zeroth order approximation of hierarchical models of elastic prismatic shells with microtemperatures in the case of constant thickness (but, in general, with bent face surfaces) is considered. The existence and uniqueness of solutions of basic boundary value problems when the projections of the bodies under consideration are bounded and unbounded domains with closed contours are established. The ways of solving boundary value problems in explicit forms and of their numerical solution are indicated.

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We consider the Wiener process with drift dXt=μdt+σdWt with initial value problem X0=x0, where x0∈R, μ∈R and σ>0 are parameters. By use values (zk)k∈N of corresponding trajectories at a fixed positive moment t, the infinite-sample consistent estimates of each unknown parameter of the Wiener process with drift are constructed under assumption that all another parameters are known. Further, we propose a certain approach for estimation of unknown parameters x0,μ,σ of the Wiener process with drift by use the values (z(1)k)k∈N and (z(2)k)k∈N being the results of observations on the 2k-th and 2k+1-th trajectories of the Wiener process with drift at moments t1 and t2 , respectively.

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Statistical methods all are more widely used in all spheres of human activity. Their importance in medicine and biology especially intensively is developing and increasing since the latest decade of the previous century. The reason of this circumstance consists in especial complexity of the problems of these domains caused by complexity of their character, by the great number of the parameters included in them and of the factors influencing their. Many of the factors affecting the observation results used for investigation of the problems under study are random by their nature and, hence, the observation results are random. Therefore the study and solution of these problems require the application of the modern methods of probability and mathematical statistics...

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The present paper is devoted to a model for elastic layered prismatic shells which is constructed by means of a suggested in the paper approach which essentially differs from the known approaches for constructing models of laminated structures. Using Vekua’s dimension reduction method after appropriate modifications, hierarchical models for elastic layered prismatic shells are constructed. We get coupled governing systems for the whole structure in the projection of the structure. The advantage of this model consists in the fact that we solve boundary value problems separately for each ply. In addition, beginning with the second ply, we use a solution of a boundary value problem of the preceding ply. We indicate ways of investigating boundary value problems for the governing systems. For the sake of simplicity, we consider the case of two plies, in the zeroth approximation. However, we also make remarks concerning the cases when either the number of plies is more than two or higher-order approximations (hierarchical models) should be applied. As an example, we consider a special case of deformation and solve the corresponding boundary value problem in the explicit form.

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We consider Ornstein-Uhlenbeck process 0 (1 ) t t t x x e e         ( ) 0 , t t s s e dW       where 0 x ,   0,   ,  0 and Ws are Wiener process. By using the values ( ) k k N z  of the corresponding trajectories at a fixed positive moment t, the estimates Tn and ** Tn of unknown parameters 0 x and  are constructed, where 0 x is an underlying asset initial price and  is a rate by which these shocks dissipate and the variable reverts towards the mean in the Ornstein-Uhlenbeck’s stochastic process. By using Kolmogorov’s Strong Law of Large Numbers the consistence of estimates Tn and ** Tn are proved.

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We obtained and investigated consistent, unbiased and efficient estimators of the parameters of irregular right-angled triangular distribution on the basis of maximum likelihood estimators. Some computation results realized on the basis of simulation of the appropriate random samples demonstrate theoretical outcomes.

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A representation of the Dirac delta function in C(R∞) in terms of infinite-dimensional Lebesgue measures in R∞ is obtained and some it's properties are studied in this paper.

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The concept of uniform distribution in [0,1] is extended for a certain strictly separated maximal (in the sense of cardinality) family (λt)t∈[0,1] of invariant extensions of the linear Lebesgue measure λ in [0.1], and it is shown that the λ∞t measure of the set of all λt-uniformly distributed sequences is equal to 1, where λ∞t denotes the infinite power of the measure λt. This is an analogue of Hlawka's (1956) theorem for λt-uniformly distributed sequences. An analogy of Weyl's (1916) theorem is obtained in similar manner.

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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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In this paper we consider the vortex equation as a particular case of Carleman-Bers-Vekua Equation and analyzed solutions space of this equation from the point of view of generalized analytic functions.

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The paper contains a brief description of Yamasaki's remarkable investigation (1980) of the relationship between Moore-Yamasaki-Kharazishvili type measures and infinite powers of Borel diffused probability measures on ${\bf R}$. More precisely, we give Yamasaki's proof that no infinite power of the Borel probability measure with a strictly positive density function on $R$ has an equivalent Moore-Yamasaki-Kharazishvili type measure. A certain modification of Yamasaki's example is used for the construction of such a Moore-Yamasaki-Kharazishvili type measure that is equivalent to the product of a certain infinite family of Borel probability measures with a strictly positive density function on $R$. By virtue of the properties of equidistributed sequences on the real axis, it is demonstrated that an arbitrary family of infinite powers of Borel diffused probability measures with strictly positive density functions on $R$ is strongly separated and, accordingly, has an infinite-sample well-founded estimator of the unknown distribution function. This extends the main result established in [ Zerakidze Z., Pantsulaia G., Saatashvili G. On the separation problem for a family of Borel and Baire $G$-powers of shift-measures on $\mathbb{R}$ // Ukrainian Math. J. -2013.-65 (4).- P. 470--485 ].

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The notion of a Haar null set introduced by Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke, has been used, in the last two decades, in studying exceptional sets in diverse areas, including analysis, dynamic systems, group theory, and descriptive set theory. In the present paper, the notion of “prevalence” is used in studying the properties of some infinite sample statistics and in explaining why the null hypothesis is sometimes rejected for “almost every” infinite sample by some hypothesis testing of maximal reliability. To confirm that the conjectures of Jum Nunnally [17] and Jacob Cohen [5] fail for infinite samples, examples of the so called objective and strong objective infinite sample well-founded estimate of a useful signal in the linear one-dimensional stochastic model are constructed.

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In the paper [G.Pantsulaia, On Uniformly Distributed Sequences on [-1/2,1/2], Georg. Inter. J. Sci. Tech.,4(3) (2013), 21--27], it was shown that μ-almost every element of R∞ is uniformly distributed in [−12,12], where μ denotes Yamasaki-Kharazishvili measure in R∞ for which μ([−12,12]∞)=1. In the present paper the same set is studying from the point of view of shyness and it is demonstrated that it is shy in R∞. In Solovay model, the set of all real valued sequences uniformly distributed module 1 in [−12,12] is studied from the point of view of shyness and it is shown that it is the prevalent set in R∞.

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By using the notion of a Haar ambivalent set introduced by Balka, Buczolich and Elekes (2012), essentially new classes of statistical structures having objective and strong objective estimates of unknown parameters are introduced in a Polish non-locally-compact group admitting an invariant metric and relations between them are studied in this paper. An example of such a weakly separated statistical structure is constructed for which a question asking ”whether there exists a consistent estimate of an unknown parameter” is not solvable within the theory (ZF ) & (DC). A question asking ”whether there exists an objective consistent estimate of an unknown parameter for any statistical structure in a non-locally compact Polish group with an invariant metric when subjective one exists” is answered positively when there exists at least one such a parameter the pre-image of which under this subjective estimate is a prevalent. These results extend recent results of authors. Some examples of objective and strong objective consistent estimates in a compact Polish group {0; 1} are considered in this paper. Primary 62-02; secondary 62D05.

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We present the proof of a certain modified version of Kolmogorov's strong law of large numbers for calculation of Lebesgue Integrals by using uniformly distributed sequences in (0,1). We extend the result of C. Baxa and J. Schoiβengeier (cf.\cite{BaxSch2002}, Theorem 1, p. 271) to a maximal set of uniformly distributed (in (0,1)) sequences Sf⊂(0,1)∞ which strictly contains the set of sequences of the form ({αn})n∈N with irrational number α and for which ℓ∞1(Sf)=1, where ℓ∞1 denotes the infinite power of the linear Lebesgue measure ℓ1 in (0,1).

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By using main properties of uniformly distributed sequences of increasing finite sets in infinite-dimensional rectangles in R∞ described in [G.R. Pantsulaia, On uniformly distributed sequences of an increasing family of finite sets in infinite-dimensional rectangles, Real Anal. Exchange. 36 (2) (2010/2011), 325--340 ], a new approach for an infinite-dimensional Monte-Carlo integration is introduced and the validity of some infinite-dimensional Strong Law type theorems are obtained in this paper. In addition, by using properties of uniformly distributed sequences in a unite interval, a new proof of Kolmogorov's strong law of large numbers is obtained which essentially differs from its original proof.

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The paper is devoted to the existence and uniqueness of a solution of the initial-boundary problem for one nonlinear multi-dimensional integro-differential equation of parabolic type. Construction and study of the additive averaged Rothe’s type scheme is also given. The studied equation is based on well-known Maxwell’s system arising in mathematical simulation of electromagnetic field penetration into a substance.

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New computing algorithms for approximate solution of the two-point boundary value problem with variable coefficients are described in the paper. Green function of the given boundary value problem considered as a non-linear operator with respect to the variable coefficient is a approximated by means of operator interpolation polynomial of the Newton type. For approximation of the inverse operator two different types of formulae are constructed. Conventionally these formulas can be called direct and modified formulas. Consequently, for approximate solution of the two-point boundary value problem with variable coefficients direct and modified interpolation operator methods are used. Description of the algorithms for approximate solution are provided and the computation results of the test problems are given in tables.

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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.

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 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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Since the speech recognition system has been created, it has developed significantly, but it still has a lot of problems. As you know, any specific natural language may owns about tens accents. Despite the identical word phonemic composition, if it is pronounced in different accents, as a result, we will have sound waves, which are different from each other. Differences in pronunciation, in accent and intonation of speech in general, create one of the most common problems of speech recognition. If there are a lot of accents in language we should create the acoustic model for each separately. When the word is pronounced differently, then the software can become confused and misunderstand (perception) also correctly what is pronounced. The same can also occur, if the human speaks slowly or vice versa quickly, then the program expects. There are any partial decisions (solutions) but they don’t solve all problems. We have developed an approach, which is used to solve above mentioned problems and create more effective, improved speech recognition system of Georgian language and of languages, which are similar to Georgian language. In addition, by the realization of this method, it is available to solve the artificial intelligence issues, such as arrange sound dialogue between computer and human, independent from any accents of any languages.

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In this paper we present a goal-directed proof-search algorithm for formula schemata, which is based on a sequent calculus. Usually, sequent calculus inference rules can be applied freely, producing a redundant search space. The standard approaches are extended to formula schemata to get rid of redundancy in a proof-search. A formula schema is a finite representation of an infinite sequence of first-order formulas, thus complete automation of the process is not feasible. Still, there are some (not so trivial) subclasses, where the process can be fully automated.

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We present the proof of a certain version of Kolmogorov strong law of large numbers which differs from Kolmogorov’s original proof.

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In the paper we study proof construction methods for first-order unranked logic. Unranked languages have unranked alphabet, meaning that function and predicate symbols do not have a fixed arity. Such languages can model XML documents and operations over them, thus becoming more important in semantic web. We present a version of sequent calculus for first-order unranked logic and describe a proof construction algorithm under this calculus. We give implementation details of the algorithm. We believe that this work will be useful for the undergoing work on semantic web logic layer.

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Nonlinear Burger’s equation describes shock waves in liquid and gas. It can be also used to model vehicles density on motor roads. Burger’s equation connects the dissipative uux term with the convectional uux one (see below Eq. (10)). Using the straightforward method used by Tsamalashvili in [1] soliton like exact solutions are obtained for the 2D nonlinear modified Burger’s equation. Employing the special exp-function expansion method Mohyud-Din et al. [2] constructed exact traveling wave solutions for (2+1) - dimensional Burger’s equation. Unfortunately this paper contains numerous wrong results and our main purpose is to revise previously obtained solutions.

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The presented book is devoted to the certain combinatorial and set-theoretical aspects of the geometry of Euclidean space and consists of two parts. The material of this book is primarily devoted to various discrete geometric structures and, respectively, to certain constructions of algorithmic type which are associated with such structures. Typical questions of combinatorial, discrete and convex geometry are examined and discussed more or less thoroughly. There are indicated close relationships between the questions of geometry and other areas of discrete mathematics.

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We consider two-dimensional differential equations of the theory of binary mixtures in case of double porosity. The general solution of this system is represented by five analytic functions of a complex variable and solution of the Helmholtz equation. The general representation of the solution gives the opportunity to construct the analytical solutions of static boundary value problems.

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In this paper we consider some statically definable problems for a cylindrical shell with constant thickness. The expand of middle surface of the shell on the plane is a rectangle. The shell is so thin that Hooke’s law does not apply. It means that, the body of the transverse stress field is assigned beforehand and for the tangential stress components the system of equations is obtained. This system of equations is based on the physical boundary conditions and the problems are solved analytically.

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A simple algorithm for construction of the approximate solution of some classical and nonlocal boundary value problems of the mathematical physics is considered. The efficiency of the offered algorithm for construction of the approximate solutions of problems is shown on the examples of two-dimensional classical and nonlocal boundary value problems of the theory of elasticity and for two-dimensional equations of Laplace and Helmholtz.

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We consider the basic two-dimensional differential equations of static equilibrium poroelastic materials with double porosity. We construct the general solution of this system of equations by means of three analytic functions of a complex variable and solution of the Helmholtz equation. On the basis of the constructed general solution we have defined the effect caused by pressures in a porous medium which is similar to temperature effect of Muskhelishvili.

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The monograph is dedicated to the theoretical investigation of basic, mixed, and crack type three-dimensional initial-boundary value problems of the generalized thermo-electro-magnetoelasticity theory associated with Green–Lindsay’s model. The essential feature of the generalized model under consideration is that heat propagation has a finite speed. We investigate the uniqueness of solutions to the dynamical initial-boundary value problems and analyse the corresponding boundary value problems of pseudo-oscillations which are obtained form the dynamical problems by the Laplace transform. The solvability of the boundary value problems under consideration are analyzed by the potential method in appropriate Sobolev–Slobodetskii (W^s_ p ), Bessel potential (H^s_p), and Besov (B^s_{p;q}) spaces. The smoothness properties and singularities of thermo-mechanical and electro-magnetic fields are investigated near the crack edges and the curves where the different types of boundary conditions collide.

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The static equilibrium of porous elastic materials with double porosity is considered in the case of an elastic Cosserat medium. The corresponding three-dimensional system of differential equations is derived. Detailed consideration is given to the case of plane deformation. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of three analytic functions of a complex variable and two solutions of Helmholtz equations. The constructed general solution enables one to solve analytically a sufficiently wide class of plane boundary value problems of the elastic equilibrium of porous Cosserat media with double porosity. A concrete boundary value problem for a concentric ring is solved.

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We consider a two-dimensional system of differential equations of the moment theory of elasticity. The general solution of this system is represented by two arbitrary harmonic functions and solution of the Helmholtz equation. Based on the general solution, an algorithm of constructing approximate solutions of boundary value problems is developed. Using the proposed method, the approximate solutions of some problems on stress concentration on the contours of holes are constructed. The values of stress concentration coefficients obtained in the case of moment elasticity and for the classical elastic medium are compared. In the final part of the paper, we construct the approximate solution of a nonlocal problem whose exact solution is already known and compare our approximate solution with the exact one. Supposedly, the proposed method makes it possible to construct approximate solutions of quite a wide class of boundary value problems.

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In this work we consider the two-dimensional system of the differential equations describing the plane statical thermoelastic balance of homogenous isotropic elastic bodies, the microelements of which have microtemperature in addition to the classical displacement and a temperature field. It is construed the general solution of this system of the equations by means of analytic functions of complex variable and solutions of the equation of Helmholtz. The general representation of the solution obtained gives the opportunity to construct the analytical solutions of a number of plane boundary value problems of microthermoelasticity. As an example we consider the boundary value problem for a concentric circular ring.

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A three-dimensional system of differential equations is considered that describes a thermoelastic equilibrium of homogeneous isotropic elastic materials, microelements of which, in addition to classical displacements and thermal fields, are also characterized by microtemperatures. In the Cartesian system of coordinates the general solution of this system of equations is constructed using harmonic and metaharmonic functions. Some boundary value micro-thermoelasticity problems are stated for the rectangular parallelepiped. An analytical solution of this class of boundary value problems is constructed using the above-mentioned general solution. When the coefficients characterizing microthermal effects are zero, the obtained solutions lead to the solutions of corresponding classical boundary value thermoelasticity problems, the majority of which have been solved for the first time. It should be noted that the aim of the given work is to construct an effective (analytical) solution for a class of boundary vale problems rather than to investigate the validity or applicability of the involved theory.

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In the present paper we consider the two-dimensional system of differential equations describing plane thermoelastic equilibrium for elastic bodies of Cosserat with microtemperature. The general solution of this system of equations is constructed using analytical functions of a complex variable and solutions of the Helmholtz equation.

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Continuity of the minimum of a general functional is proved with respect to perturbations of the initial data and right-hand side of the equation with variable distributed and concentrated delays. Under the initial data, we understand the collection of initial moment, of variable delays, and initial function. Perturbations of the right-hand side of the equation are small in the integral sense.

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Theorems on the continuous dependence of the solution on perturbations of the initial data and the right-hand side of equation are proved. Under initial data we understand the collection of initial moment, of delay function and initial function. Perturbations of the right-hand side of equation are small in the integral sense.

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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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In this paper, we present a general procedure for solving of homogeneous equations that describe penetration and diffusion of X-rays in plane geometry. Starting from Van Kampen’s and Case’s observation that it suffices that “solutions” be distributions, elementary solutions of a homogeneous equation are found. We also prove that general solutions can be obtained by superposition of elementary solutions.

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In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with double porosity is considered. Some basic properties of plane harmonic waves are established and the boundary value problems (BVPs) of steady vibrations are investigated. Indeed on the basis of this theory three longitudinal and two transverse plane harmonic waves propagate through a Kelvin–Voigt material with double porosity and these waves are attenuated. The basic properties of the singular integral operators and potentials (surface and volume) are presented. The uniqueness and existence theorems for regular (classical) solutions of the BVPs of steady vibrations are proved by using the potential method (boundary integral equations method) and the theory of singular integral equations.

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The transformation of the original characteristic equation of the multivelocity linear transport theory was carried out by expanding the scattering function for the problem to be solved as a spectral integral over a complete set of eigenfunctions for the previously solved transport problem. The obtained equation represents a singular integral equation containing a spectral integral over the spectrum of the solved problem, whose kernel depends on the difference between the scattering of the problem to be solved and that of the already solved problem. We consider also the examples illustrating the validity of such a transformation. M. Kanal and J. Davies made a similar transformation of the characteristic equation of the one-velocity transport theory.

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The aim of this paper is to construct Green’s function in an infinite medium for the light scattering equation. To this end the method of spectral resolution of the solutions by the eigenfunctions of the corresponding characteristic equation is used.

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In this work we consider the geometrically nonlinear and non-shallow spherical shells for I.N. Vekua N=1 approximation. Concrete problem using complex variable functions and the method of the small parameter has been solved.

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The images of absolute null sets (spaces) under bijective continuous mappings are studied. It is shown that, in general, these images do not possess regularity properties from the viewpoint of topological measure theory.

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I. Vekua obtained the conditions for the existence of the neutral surface of a shell, when the neutral surface is the middle surface. In this paper the neutral surface is considered as any equidistant surfaces of the shell.

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The ensemble of humidity processes (fogs, layered clouds) has been simulated on the basis of the numerical model of a non-stationary mesoscale boundary layer of atmosphere (MBLA) developed by us. In this work the accent becomes on interaction and interconversion of humidity processes in the above-stated ensemble. Genesis of Foehns is in detail investigated. They are classified on dryadiabatic, mostadiabatic and most-dryadiabatic Foehns. It is stated a problem about numerical modelling of Foehns in frame of a flat, two-dimensional mesoscale boundary layer. The problem is at a stage of numerical realisation. The first encouraging results are received

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It is proved that every uncountable solvable group contains two negligible sets whose union is an absolutely nonmeasurable subset of the same group.

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

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Under Martin’s Axiom, it is proved that there exists an absolute null subset of the Euclidean plane R^2, the orthogonal projections of which on all straight lines in R^2 are absolutely nonmeasurable. A similar but weaker result holds true within the framework of ZFC set theory.

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It is shown that the difference between a Vitali–Bernstein selector and a partial Vitali–Bernstein selector can be of Lebesgue measure zero and of first Baire category. This result gives an answer to a question posed by G. Lazou.

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We consider strong at-subsets of the Euclidean space Rn and estimate from below the growth of the maximal cardinality of such subsets (our method essentially differs from that of [6]). We then apply some properties of strong at-sets to the illumination problem.

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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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In the present work, the generation of large-scale zonal flows and magnetic field by short-scale collisionless 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 on the basis of multi-scale expansion. 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. The conditions for the instability development and possibility of the generation of large-scale structures are determined. The nonlinear increment of this interaction substantially depends on the wave vector of Alfven pumping and on the characteristic scale of the generated zonal structures. This means that the instability pumps the energy of primarily small-scale Alfven waves into that of the large-scale zonal structures which is typical for an inverse turbulent cascade. The increment of energy pumping into the large-scale region noticeably depends also on the width of the pumping wave spectrum and with an increase of the width of the initial wave spectrum the instability can be suppressed.

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By using the method developed in the paper [G.Pantsulaia, G.Giorgadze, On some applications of infinite-dimensional cellular matrices, Georg. Inter. J. Sci. Tech., Nova Science Publishers, Volume 3, Issue 1 (2011), 107-129], it is obtained a representation in an explicit form of the particular solution of the linear non-homogeneous ordinary differential equation of the higher order whose coefficients are real-valued simple functions.

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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 present work, the generation of large-scale zonal flows and magnetic field by short-scale collisionless 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 on the basis of multi-scale expansion. 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. The conditions for the instability development and possibility of the generation of large-scale structures are determined. The nonlinear increment of this interaction substantially depends on the wave vector of Alfven pumping and on the characteristic scale of the generated zonal structures. This means that the instability pumps the energy of primarily small-scale Alfven waves into that of the large-scale zonal structures which is typical for an inverse turbulent cascade. The increment of energy pumping into the large-scale region noticeably depends also on the width of the pumping wave spectrum and with an increase of the width of the initial wave spectrum the instability can be suppressed.

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Plasma vortices are often detected by spacecraft in the geospace (atmosphere, ionosphere, magnetosphere) environment, for instance in the magnetosheath and in the magnetotail region. 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, turbulent state of plasma medium (especially, ionosphere) is overviewed. Experimental observation data from THEMIS mission (Keiling et al., 2009) is investigated and numerical simulations are carried out. By analyzing the THEMIS data for that event, we find that several vortices in the magnetotail are detected together with the main one and these vortices constitute a vortex chain. Such vortices can cause the strong turbulent state in the different media. The strong magnetic turbulence is investigated in the ionsophere as an ensemble of such strongly localized (weakly interacting) vortices. Characteristics of power spectral densities are estimated for the observed and analytical stationary dipole structures. These characteristics give good description of the vortex structures.

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In this paper we introduce λ R : 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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Simple models of mass movement and seismic processes are important for understanding the mechanisms for their observed behavior. In the present paper, we analyze the dynamics of a singleblock and Burridge-Knopoff model on horizontal and inclined slope with Dieterich–Ruina and Carlson friction laws. In our experiments, the slip events are distinguished by acoustic emission bursts, which are generated by slider displacement. Also acceleration was recorded on each sliding plate using attached accelerometer. In the case of the inclined slope experimental model a seismic vibrator, which produces low periodic impact (forcing) was attached to the sliding plate. This was a numerical simulation of dynamic processes occurring at one- and four-plate Burridge-Knopoff system

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In this paper stress-deformed state for some ”bridge-form” multystructures studied having difficult geometry. Particularly the boundary-contacted problem is considered. Two rectangle (particularly a square) form membranes are connected by a string; We consider classic linear boundary problems for membranes (Poisson’s equation), but for string nonlinear Kirchhoff type integro-differential equation. The account program in MATLAB is created and numerical experiments are made.

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Algorithms were always an important part of many branches in the sciences. In many manuals and handbooks, algorithms of problems of computational mathematics are focused on the manual performance or by means of a calculator. In this book, descriptions of algorithms, their solutions and main characteristics are discussed. The present work is the outcome of many years of the authors' work on solving different problems and tasks from domains of instruction making, metrology, system analysis, ecology, data analysis from ecology, agriculture, medicine and creation of corresponding universal computer packages and systems.

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The estimation of the increment of solution is obtained with respect to small parameter for nonlinear delay functional differential equation with the continuous initial condition. Moreover, value of the increment is calculated at the initial moment. This estimation plays an important role in proving the variation formulas of solution.

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In the paper the two-dimensional version of steady vibration in the fully coupled linear theory of elasticity for solids with double porosity is considered. Using the Fourier integrals, some basic boundary value problems are solved explicitly (in quadratures) for the half-plane.

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The system of two-dimensional nonlinear partial differential equations is considered. This system describes the vein formation in meristematic tissues of young leaves. Variable directions difference scheme is constructed and investigated. Absolute stability regarding space and time steps of scheme is shown. The convergence statement for the constructed scheme is proved. Rate of convergence is given. Various numerical experiments are carried out and results of some of them are considered in this paper. Comparison of numerical experiments with the results of the theoretical investigation is given too.

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In this paper 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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Recent satellite and ground-based observations prove that during the formative period of earthquakes VLF/LF and ULF electromagnetic emissions are observed in seismogenic areas. This work offers an original model of self-generated electromagnetic oscillations of local segments of the lithospheric origins of the emissions. In the paper, the seismogenic area is considered to be an oscillatory-distributed system. This model simplifies physical analyses of the nonlinear effects and qualitatively explains the mechanisms that generate very low frequency electromagnetic waves in the period prior to an earthquake.

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Nonlinear simulations of electromagnetic Rossby and Khantadze planetary waves in the presence of a shearless and sheared zonal flows in the weakly ionized ionospheric E-layer are carried out. The simulations show that the nonlinear action of the vortex structures keeps the solitary character in the presence of shearless zonal winds as well as the ideal solutions of solitary vortex in the absence of zonal winds. In the presence of sheared zonal winds, the zonal flows result in breaking into separate multiple smaller pieces. A passively convected scalar field is shown to clarify the transport associated with the vortices. The work shows that the zonal shear flows provide an energy source into the vortex structure according to the shear rate of the zonal winds.

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> We review the generation of zonal flow and magnetic field by coupled electromagnetic ultra-low-frequency waves in the Earth's ionospheric E-layer. It is shown that, under typical ionospheric E-layer conditions, different planetary low-frequency waves can couple with each other. Propagation of coupled internal-gravity-Alfvén, coupled Rossby-Khantadze and coupled Rossby-Alfvén-Khantadze waves is revealed and studied. A set of appropriate equations describing the nonlinear interaction of such waves with sheared zonal flow is derived. The conclusion on the instability of short-wavelength turbulence of such coupled waves with respect to the excitation of low-frequency and large-scale perturbation of the sheared zonal flow and sheared magnetic field is deduced. The nonlinear mechanism of the instability is based on the parametric triple interaction of finite-amplitude coupled waves leading to the inverse energy cascade towards longer wavelength. The possibility of generation of an intense mean magnetic field is shown. Obtained growth rates are discussed for each case of the considered coupled waves.

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In this paper stress-deformed state for some “bridge-form” multistructures studied having difficult geometry. Particularly the boundary-contacted problem is considered. Two rectangle (particularly a square) form plates are connected by a beam; We consider classic linear boundary problems for plates (biharmonic equation), but for beam nonlinear Kirchoff type integro-differential equation. The account program in MATLAB is created and numerical experiments are made.

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One-dimensional parabolic system of nonlinear partial differential equations is considered. The model is based on Maxwell’s system which arises at describing penetration of a magnetic field into a substance. Semi-discrete scheme is constructed for the first type initial-boundary value problem. Graphs of numerical experiments are given.

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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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Two classes of nonlinear partial differential models are considered. The first is based on well known Maxwell’s system describing electromagnetic field penetration in the substance. The second is some generalization of the biological model arising in the simulation of vein formation in meristematic tissues of young leaves. Decomposition algorithm with respect to physical processes for the first system and averaged model of sum approximation for the second one are stated. The convergence theorem is also given for the second model.

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Linear stationary multilayer flows of a viscous incompressible fluid in tubes bounded by coordinate surfaces of generalized cylindrical coordinates and circular flows of multilayer liquids in a circular cylindrical system of coordinates are investigated. In other words, multilayer flows are studied in rectilinear tubes of rectangular, circular, elliptic and parabolic cross-sections and in circular tubes of rectangular cross-section. Layers of flowing fluids of different viscosity are arranged along one of the coordinates. Related boundary-value contact problems of hydromechanics are stated and their effective solutions are found. The obtained results are used in studies of blood microcirculation.

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In the present paper gas pressure and flow rate distribution along the main branched pipeline is investigated. The study is based on the 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

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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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Soliton-like solutions of the 2D nonlinear Burger’s equation are obtained. Revision of the previously received solutions is carried out.

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უცხოელი ავტორები - Zvonkin,M., Fan J., & DrewniakJ. L.

Dynamical problem of the antiplane strain (shear) of an isotropic 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 vanishes either on a part or on the entire boundary of the projection.

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This book describes three classes of nonlinear partial integro-differential equations. These models arise in electromagnetic diffusion processes and heat flow in materials with memory. Mathematical modeling of these processes is briefly described in the first chapter of the book. Investigations of the described equations include theoretical as well as approximation properties. Qualitative and quantitative properties of solutions of initial-boundary value problems are performed therafter. All statements are given with easy understandable proofs. For approximate solution of problems different varieties of numerical methods are investigated. Comparison analyses of those methods are carried out. For theoretical results the corresponding graphical illustrations are included in the book. At the end of each chapter topical bibliographies are provided. Investigations of the described equations include theoretical as well as approximation properties; Detailed references enable further independent study; Easily understandable proofs describe real-world processes with mathematical rigor.

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The present paper is devoted to the system of degenerate partial differential equations arise from the investigation of elastic two layered prismatic shells.

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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 processes did not split and for this aim we cited von K´arm´an type system without variety of ad hoc assumptions since,in the classical form of this system of DEs one of them represents the conditionof compatibility but it is not an equilibrium equation.

In a recent paper, Lima, Panario and Wang have provided a new method to multiply polynomials expressed in Chebyshev basis which reduces the total number of multiplication for small degree polynomials. Although their method uses Karatsuba’s multiplication, a quadratic number of operations is still needed. In this paper, we extend their result by providing a complete reduction to polynomial multiplication in monomial basis, which therefore offers many subquadratic methods. Our reduction scheme does not rely on basis conversions and we demonstrate that it is efficient in practice. Finally, we show a linear time equivalence between the polynomial multiplication problem under monomial basis and under Chebyshev basis.

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

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In this paper we study a question of almost everywhere strong convergence of the quadratic partial sums of two-dimensional Walsh-Fourier series. Specifically, we prove that the asymptotic relation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tfrac{1} {n}\sum\limits_{m = 0}^{n - 1} {|S_{mm} f - f|^p \to 0} $\end{document} as n→∞ holds a.e. for every function of two variables belonging to L logL and for 0 < p ≤ 2. Then using a theorem by Getsadze [6] we infer that the space L log L can not be enlarged by preserving this strong summability property.

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A probability density functional (nonlinear and unbounded, generally speaking) is considered. The consistency and asymptotic normality conditions are established for the plug-in-estimator. A convergence order estimator is obtained.

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We develop the method of maximal likelihood for infinite-dimensional Hilbert spaces and prove several theorems about consistency and asymptotic normality.

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The paper introduces a new concept of Λ-variation of multivariable functions and studies its relationship with the convergence of multidimensional Fourier series.

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We establish the limit distribution of the square-integrable deviation of two nonparametric nuclear-type estimations for the Bernoulli regression functions. A criterion is proposed for the verification of the hypothesis of equality of two Bernoulli regression functions. We study the problem of verification and, for some “close” alternatives, investigate the asymptotics of the power.

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The maximal Orlicz space such that the mixed logarithmic means of multiple Fourier series for the functions from this space converge in L1 -norm is found

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The paper considers the following differential equation  ,0,0))(()()(' 1   ttxtptxm iii where p  i  ∈ L  loc  (R  + ;R  + ), τ i ∈ C(R  + ;R), τ  i  (t)≤ t  and i=1,...,m, ,)(lim tit i=1,...,m. New oscillation criteria of all solutions for this equation are established.

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Algorithms were always an important part of many branches in the sciences. In many manuals and handbooks, algorithms of problems of computational mathematics are focused on the manual performance or by means of a calculator. In this book, descriptions of algorithms, their solutions and main characteristics are discussed. The present work is the outcome of many years of the authors' work on solving different problems and tasks from domains of instruction making, metrology, system analysis, ecology, data analysis from ecology, agriculture, medicine and creation of corresponding universal computer packages and systems.

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Consider the first order linear differential equation with several retarded arguments In this paper the state the art on the oscillation of all solutions to these equations is reviewer and new sufficient conditions to the oscillation are established specially in the case of nonmonotone argument. Example illustrating the result are given.

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We have developed an original, simple and convenient software for testing statistical hypotheses concerning the parameters of probability distribution laws. It is intended for the users who are not professionals in the field of applied statistics and computer science because it is very simple and convenient for use, and the results of application of the methods realized in the package are given as simple for understanding texts in outcomes of the programs. The problems and the algorithms (of only original) realized in the package, as well as the features and the opportunities of their application are briefly described. Those features that distinguish favorably this package from other similar products are emphasized. Some examples showing singularities and efficiency of the algorithms realized in the package are cited.

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A sequential method based on constrained Bayesian methods is developed for testing multiple hypotheses. It controls the family-wise error rate and the family-wise power in a more accurate form than the Bonferroni or intersection scheme using the ideas of step-up and step-down methods for multiple comparisons of sequential designs. The new method surpasses the existing testing methods proposed earlier in a substantial reduction of the expected sample size.

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In the present paper we consider the following Satisfaction Problem of Consumers Demands (SPCD): {\it The supplier must supply the measurable system of the measure mk to the k-th consumer at time tk for 1≤k≤n. The measure of the supplied measurable system is changed under action of some dynamical system, What is a minimal measure of measurable system which must take the supplier at the initial time t=0 to satisfy demands of all consumers ?} In this paper we consider Satisfaction Problem of Consumers Demands measured by ordinary "Lebesgue measures" in R∞ for various dynamical systems in R∞. In order to solve this problem we use Liouville type theorems for them which describes the dependence between initial and resulting measures of the entire system.

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It is introduced a certain approach for equipment of an arbitrary set of the cardinality of the continuum by structures of Polish groups and two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki's certain question(2012) {\it what are the real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure.} It is demonstrated that for each diffused Borel probability measure μ defined in a Polish space (G,ρ,Bρ(G)) without isolated points there exist a metric ρ1 and a group operation ⊙ in G such that Bρ(G)=Bρ1(G) and (G,ρ1,Bρ1(G),⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ, where Bρ(G) and Bρ1(G) denote Borel σ algebras of subsets of G generated by metrics ρ and ρ1, respectively. Similar result is obtained for construction of locally compact non-compact or non-locally compact Polish groups equipped with two-sided (left or right) invariant quasi-finite Borel measures.

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In the paper [G.Pantsulaia, On Uniformly Distributed Sequences on [-1/2,1/2], Georg. Inter. J. Sci. Tech.,4(3) (2013), 21--27], it was shown that μ-almost every element of R∞ is uniformly distributed in [−12,12], where μ denotes Yamasaki-Kharazishvili measure in R∞ for which μ([−12,12]∞)=1. In the present paper the same set is studying from the point of view of shyness and it is demonstrated that it is shy in R∞. In Solovay model, the set of all real valued sequences uniformly distributed module 1 in [−12,12] is studied from the point of view of shyness and it is shown that it is the prevalent set in R∞.

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By using the properties of the uniformly distributed sequences of real numbers on (0,1), a short proof of a certain version of Kolmogorov strong law of large numbers is presented which essentially differs from Kolmogorov's original proof.

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In Solovay model it is shown that the duality principle between the measure and the Baire category holds true with respect to the sentence - "The domain of an arbitrary generalized integral for a vector-function is of first category."

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The paper contains a brief description of Yamasaki's remarkable investigation (1980) of the relationship between Moore-Yamasaki-Kharazishvili type measures and infinite powers of Borel diffused probability measures on ${\bf R}$. More precisely, we give Yamasaki's proof that no infinite power of the Borel probability measure with a strictly positive density function on $R$ has an equivalent Moore-Yamasaki-Kharazishvili type measure. A certain modification of Yamasaki's example is used for the construction of such a Moore-Yamasaki-Kharazishvili type measure that is equivalent to the product of a certain infinite family of Borel probability measures with a strictly positive density function on $R$. By virtue of the properties of equidistributed sequences on the real axis, it is demonstrated that an arbitrary family of infinite powers of Borel diffused probability measures with strictly positive density functions on $R$ is strongly separated and, accordingly, has an infinite-sample well-founded estimator of the unknown distribution function. This extends the main result established in [ Zerakidze Z., Pantsulaia G., Saatashvili G. On the separation problem for a family of Borel and Baire $G$-powers of shift-measures on $\mathbb{R}$ // Ukrainian Math. J. -2013.-65 (4).- P. 470--485 ].

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By using the method developed in the paper [G.Pantsulaia, G.Giorgadze, On some applications of infinite-dimensional cellular matrices, {\it Georg. Inter. J. Sci. Tech., Nova Science Publishers,} Volume 3, Issue 1 (2011), 107-129], it is obtained a representation in an explicit form of the weak solution of a linear partial differential equation of the higher order in two variables with initial condition whose coefficients are real-valued simple step functions

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By using the notion of a Haar ambivalent set introduced by Balka, Buczolich and Elekes (2012), essentially new classes of statistical structures having objective and strong objective estimates of unknown parameters are introduced in a Polish non-locally-compact group admitting an invariant metric and relations between them are studied in this paper. An example of such a weakly separated statistical structure is constructed for which a question asking "{\it whether there exists a consistent estimate of an unknown parameter}" is not solvable within the theory (ZF) & (DC). A question asking "{\it whether there exists an objective consistent estimate of an unknown parameter for any statistical structure in a non-locally compact Polish group with an invariant metric when subjective one exists}" is answered positively when there exists at least one such a parameter the pre-image of which under this subjective estimate is a prevalent. These results extend recent results of authors. Some examples of objective and strong objective consistent estimates in a compact Polish group {0;1}N are considered in this paper.

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By using properties of Markov homogeneous chains and Banach measure in N, it is proved that a relative frequency of even numbers in the sequence of n-th coordinates of all Collatz sequences is equal to the number 23+(−1)n+13×2n+1. It is shown also that an analogous numerical characteristic for numbers of the form 3m+1 is equal to the number 35+(−1)n+115×22(n−1). By using these formulas it is proved that under Collatz conjecture the Collatz mapping has no an asymptotic mixing property (mod3). It is constructed also an example of a real-valued function on the cartesian product N2 of the set of all natural numbers N such that an equality its repeated integrals (with respect to Banach measure in N) implies that Collatz conjecture fails. In addition, it is demonstrated that Collatz conjecture fails for supernatural numbers.

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Antiplane strain (shear) of an isotropic 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 vanishes 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 of the shear modulus is studied. When the above-mentioned domain is either the half-plane or the half-disk and the shear modulus is a power function with respect to the variable along the perpendicular to the linear boundary, the basic boundary value problems are solved explicitly in quadratures.

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Considered a stationary boundary layer of non-Newtonian fluid. Obtained selfsimilar solutions tasks of free convection a non-Newtonian fluid when variable conduction. The problem is solved by the integral method. Is shown that by choosing the parameters can be controlled surface friction.

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The isolation and nondegeneracy of constrained extrema arising in geometric problems and mathematical models of electrostatics are studied. In particular, it is proved that a convex concyclic configuration of polygonal linkages is a nondegenerate maximum of the oriented area. Geometric properties of equilibrium configurations of point charges with Coulomb interaction on convex curves are considered, and methods for constructing them are presented. It is shown that any configuration of an odd number of points on a circle is an equilibrium point for the Coulomb potential of nonzero point charges. The stability of the equilibrium configurations under consideration is discussed.

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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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We discuss a natural problem concerned with equilibrium configurations of Coulomb potential of three positive point charges constrained to a system of nested circles in the plane. After describing our approach in general setting, several concrete problems of such type are studied in detail. First, we consider a system of three concentric circles each of which contains exactly one charge, and give a complete description of configurations, which can serve as equilibria of three positive charges. Next, we give explicit formulae for the sought charges and obtain a geometric characterization of those configurations, which can serve as stable equilibria of three positive charges. Moreover, we obtain similar results in the case of three nested circles, which are not necessarily concentric and describe the topology of the set of equilibrium configurations. Several related problems and conjectures are also presented.

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

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Работа является продолжением работы [1]. В нём определены [2] безранговые кванторы существования и всеобщности и сформулированы некоторые их свойства [3,4]. ∃n+1xA1 ...AnA — (τx ∧n+1 A1 ...AnA/x) ∧n+1 A1 ...AnA Читается: ”существует такой x свойствA1,...,An, который имеет свойство A, n = 0, 1,.... ∀n+1xA1 ...AnA — ¬∃n+1xA1 ...An¬A Читается: каждый x свойств A1,...,An имеет свойство A,n = 0, 1,.... Заметим, что если с вышеопределенных операторов убрать верхние индексы, то получим безранговые кванторы. (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] Если символ x не имеет свободное вхождение в A, то (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 ...AnB Список литературы [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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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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The present work is dedicated to the investigation and approximate resolution of the initial-boundary value problem with first type boundary conditions for one nonlinear integro-differential parabolic equation.

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In the paper the gallstones formation process is considered from the mathematical point of view. The process is described by the nonlinear reaction-diffusion equation with the appropriate initial-boundary conditions. The solutions are obtained in the explicit form. Numerical examples are given.

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In the paper the free radical oxidation process in physiology is considered from the mathematical point of view. The process is described by the nonlinear Semionov equation with the appropriate initial conditions. The solutions are obtained in the explicit form. Numerical examples are given.

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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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For an arbitrary infinite additive group G and for an uncountable compact Hausdorff topological group H with card(H) = card(Hℵ0 ) = card(HG), H-valued measurable Gprocesses are constructed on the group HG and some set-theoretical characteristics of their various F ∗ (HG)-invariant extensions are calculated, where F ∗ (HG) denotes a group of transformations of HG generated by the eventually neutral sequences and all permutations of G. More precisely, an orthogonal family of F ∗ (HG)-invariant extensions of the left-invariant probability Haar-Baire measure on HG is constructed such that topological weights of metric spaces associated with such extensions are maximal. In addition, for such a family of measures in HG, the F ∗ (HG)-invariant measure extension problem is studied.

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Study of the spatial-temporal propagation of the air flow generated by the action of highpower phenomenon has great theoretical and practical value, especially for the mountainous territories because even the low hills slow down the velocity of flow motion and change its direction, sometimes even to the opposite direction. In the present paper the air flow generated by high-power pulse and its spatialtemporal propagation in the atmosphere above the uniform and non-uniform terrains are investigated. Some results of theoretical and numerical investigations are given. Received results can be useful in military and mining operations, especially in the process of open career works in populated places or near to them

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In the present article an anti-plane problem of the elasticity theory for a composite (piece-wise homogeneous) orthotropic body weakened by cracks intersecting the interface or reaching it in a right angle is studied. The studied problem is reduced to the singular integral equation (when crack reaches the interface) and system (pair) of singular integral equations (when crack intersects the interface) containing an immovable singularity with respect to the unknown characteristic function of the crack 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 or reaches 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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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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In this paper some results of numerical investigations in the troposphere of airflow above the territory of Georgia are presented. The numerical model is based on a 3-D hydrostatic non-stationary numerical model for the meso-scale atmospheric processes taking into consideration realrelief of Caucasus. 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 in case of western atmosphere currents invasion on the territory of Georgia have been performed. Some results of numerical calculations are presented and analyzed

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In this paper 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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The purpose of this paper is to consider the basic boundary value problems of the fully coupled equilibrium theory of elasticity for solids with double porosity and explicitly solve the BVPs of statics in the fully coupled theory for a sphere. The explicit solutions of these BVPs are represented by means of absolutely and uniformly convergent series.

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Our goal was to consider the two-dimensional version of the full coupled linear equilibrium theory of elasticity for materials with double porosity and to construct explicitly the solutions of BVPs, in the form of absolutely and uniformly convergent series that is useful in engineering practice. In this paper, the Neumann-type BVPs of statics for an elastic circle and for a plane with circular hole are considered. The uniqueness theorems of the considered boundary value problems are proved.

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In the present report, the notion of a Haar ambivalent introduced by Balka, Buczolich and Elekes in 2012 is used in studying the properties of some infinite sample statistics and in explaining why the null hypothesis is sometimes rejected for ”almost every”(in the sense of Christensen) infinite sample by some hypothesis testing of maximal reliability.The aims of the present report are: a) to apply the ”almost every”(in the sense of Christensen) approach to the study the properties of infinite sample statistics; b) to introduce concepts of ”subjective” and ”objective” infinite sample well-founded estimates of a useful signal in a linear one-dimensional stochastic model; c) to show that each infinite sample well-founded estimate of a useful signal in a linear one-dimensional stochastic model is ”subjective” or ”objective” d) to show that the conjectures of Jum Nunnally and Jacob Cohen hold only for subjective infinite sample well-founded estimates;

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In this paper 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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By using the notion of a Haar ambivalent set, introduced by Balka, Buczolich and Elekes in 2012, essentially new classes of statistical structures having objective and strong objective estimates of an unknown parameter are considered in a Polish non-locally-compact group admitting an invariant metric, and relations between them are studied. An example of such a weakly separated statistical structure is constructed for which a question whether there exists a consistent estimate of an unknown parameter remains unsolvable within the theory (ZE) & (DC). These results extend those obtained recently by Pantsulaia and Kintsurashvili in 2014.

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We establish condition which guarantees convergence in measure of logarithmic means of the two-dimensional Fourier series.

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Variation formulas of solution are obtained for linear with respect to prehistory of the phase velocity (quasi-linear) neutral functional differential equations with variable delays. In the variation formulas, the effect of perturbation of the delay function appearing in the phase coordinates is stated.

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In the paper we consider geometrically nonlinear equations of elastic balance for binary mixture of two isotropic materials. In the literature the considered model is called Green-Naghdi-Steel’s model. The main three-dimensional equations of static balance corresponding to the considered model are recorded in any curvilinear system of coordinates. The main two-dimensional relations for the shallow shells consisting of binary mixture are obtained from these equations using I. Vekua’s reduction method and basing on T. Meunargia’s works.

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In the paper the following inverse problem is considered: find such initial functions that the value of corresponding solution at given moment is equal to a fixed vector. On the basis of necessary

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In this paper stress-deformed state for some “ridge-form” multistructures having difficult geometry is studied. Particularly the boundary-contacted problem is considered. Two rectangle (particularly a square) form plates are connected by a beam. We consider classic linear boundary problems for plates (biharmonic equation), but for the beam nonlinear Kirchhoff type integro-differential equation is studied. In the equation of a beam we consider physical nonlinearity along with mathematical nonlinearity. The program in MATLAB is created and numerical experiments are made.

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Antiplane problems of the elasticity theory for composite (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 (the characteristic function of the cracks expansion). In the present article the system of singular integral equations is solved by a collocation method, in particular, by discrete singular method in cases of uniform located knots. The corresponding algorithms are composed and realized. The results of theoretical and numerical investigations are presented.

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In the present paper we consider the geometrically nonlinear and non-shallow spherical shells, when components of the deformation tensor have nonlinear terms. By means of I. Vekua’s method the system of equilibrium equations in two variables is obtained. Using complex variable functions and the method of the small parameter approximate solutions are constructed for N=1 in the hierarchy by I. Vekua. Concrete problem has been solved.

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We consider a plane problem of elasticity for a rectangular domain with a curvilinear quadrangular hole, which is composed of rectilinear segments (parallel to the abscissa axis) and arcs of one and the same circumference. The problem is solved by the methods of conformal mappings and boundary value problems of analytic functions. The sought complex potentials are constructed effectively (in the analytical form). Estimates of the obtained solutions are derived in the neighborhood of angular points.

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In this paper we consider non-shallow shells. By means of I. Vekua’s method of normed moments we get the approximate expression of the stress tensor which is compatible with boundary data on face surfaces.

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In the present paper, by means of Vekua’s method, the system of differential equations for the Geometrically and Physically nonlinear theory non-shallow shells is obtained.

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The aim of this paper is to study, in the class of H¨older functions, a linear integral equation with coefficient having two simple zeros in the interval under consideration. Using the theory of singular integral equations, we give the necessary and sufficient conditions for the solvability of this equation under some assumptions on their kernel. Finding a solution is reduced to solving a regular integral equation of the second kind.

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In this paper, we study, in the class of H¨older functions, linear two-dimensional integral equations with coefficients t that have zeros in the interval under consideration. Using the theory of singular integral equations, necessary and sufficient conditions for the solvability of these equations under some assumption on their kernels are given.

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Set Theoretical Aspects of Real Analysis is built around a number of questions in real analysis and classical measure theory, which are of a set theoretic flavor. Accessible to graduate students, and researchers the beginning of the book presents introductory topics on real analysis and Lebesgue measure theory. These topics highlight the boundary between fundamental concepts of measurability and nonmeasurability for point sets and functions. The remainder of the book deals with more specialized material on set theoretical real analysis. The book focuses on certain logical and set theoretical aspects of real analysis. It is expected that the first eleven chapters can be used in a course on Lebesque measure theory that highlights the fundamental concepts of measurability and non-measurability for point sets and functions. Provided in the book are problems of varying difficulty that range from simple observations to advanced results. Relatively difficult exercises are marked by asterisks and hints are included with additional explanation. Five appendices are included to supply additional background information that can be read alongside, before, or after the chapters. Dealing with classical concepts, the book highlights material not often found in analysis courses. It lays out, in a logical, systematic manner, the foundations of set theory providing a readable treatment accessible to graduate students and researchers.

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For any three-coloring of the plane, it is shown that there are continuum many triangles of a prescribed type (acute-angled, right-angled, obtuse-angled, isosceles) such that the vertices of each of them carry all three colors.

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t is proved that every uncountable solvable group (G, ·) admits a partition into three G-negligible sets.

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A set X in a real Hilbert space H is called an at-set if every three-element subset of X forms either an acute-angled triangle or a right-angled triangle. The maximal cardinality of an at-set in an infinite-dimensional H is found. Furthermore, the number of right angles in the unit cube [0,1]n is calculated. As an application, a simple solution of a well-known problem is given, concerning the maximal cardinality of a strong at-set in the Euclidean space ℝn.

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This report contains several remarks about inscribed and circumscribed polyhedra.

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In the paper the problem of existence of neutral surfaces for non-shallow shell is considered. By neutral surface is meant a surface which is not subject to tensions and compression under the deformation on the shell. It means that the neutral surfaces may be subject only to bendings or, in particular, may remain rigid. I. Vekua obtained the conditions for the existence of neutral surface of a shell, when the neutral surface is the middle surface. In this paper the neutral surface is considered as any equidistant surfaces.

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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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Some classical pathological subsets of the real line are considered and their descriptive properties are investigated from the measure-theoretical view-point. In addition, various combinations of such subsets are presented

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The Bitsadze-Samarskii nonlocal boundary value problem is considered. Variational formulation is done. The domain decomposition and Schwarz-type iterative methods are used. The parallel algorithm as well as sequential ones is investigated.

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In the present paper 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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Sequent calculus is widely used for formalizing proofs. However, due to the proliferation of data, understanding the proofs of even simple mathematical arguments soon becomes impossible. Graphical user interfaces help in this matter, but since they normally utilize Gentzen's original notation, some of the problems persist. In this paper, we introduce a number of criteria for proof visualization which we have found out to be crucial for analyzing proofs. We then evaluate recent developments in tree visualization with regard to these criteria and propose the Sunburst Tree layout as a complement to the traditional tree structure. This layout constructs inferences as concentric circle arcs around the root inference, allowing the user to focus on the proof's structural content. Finally, we describe its integration into ProofTool and explain how it interacts with the Gentzen layout.

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Языки, в которых функциональные или предикатные символы не имеют фиксированной арности (местности), в последние годы стали предметом интенсивного изучения по причине довольно широкой сферы их применимости [1]. Обычно встречаются переменные двух типов: предметные переменные, которые можно заменить одним термом, и последовательные переменные (далее мы назовем их «предметными последовательными переменными»), заменить которые можно конечной последовательностью термов. В отличии от вышеупомянутых языков, в изученном нами языке безранговой эгалитарной теории встречаются два типа последовательностей переменных: а) переменные предметной последовательности, представить которые можно конечной последовательностью термов и б) переменные пропозиционной последовательности, заменить которые можно конечной последовательностью формул. Кроме того, область операторов этой теории –  ,  ,  , x , x , x не зафиксирована – они безранговые операторы. Определение этих операторов происходит в рамках рациональных правил введения производных операторов Шалвы Пхакадзе [2]. На их основании в 216 безранговой эгалитарной теории были доказаны аналоги результатов, полученных в эгалитарной теории Н. Бурбаки [3]. писок используемых источников 1. Kutsia T. Theorem Proving with Sequence Variables and Flexible Arity Symbols / T. Kutsia // 9th International Conference, LPAR 2002 Tbilisi, Georgia, October 14–18, 2002 Proceedings. – 2002. – P. 278-291. 2. Пхакадзе Ш.С. Некоторые вопросы теории обозначений / Ш.С. Пхакадзе. – Тбилиси: Изд-во Тбилисского университета, 1977. – 195 с. 3. Бурбаки Н. Теория Множеств / Н. Бурбаки. – Москва: Мир, 1965. – 456 с. 222 4. Rukhaia Kh. One Method of Constructing a Formal System / Kh. Rukhaia, L. Tibua, G. Chankvetadze, B. Dundua // Applied Mathematics, Informatics and Mechanics. – 2006. – V. 11, N. 2. – P. 81-89.

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The effect of the wind shear on the roll structures of nonlinear internal gravity waves (IGWs) in the Earth's atmosphere with the finite vertical temperature gradients is investigated. A closed system of equations is derived for the nonlinear dynamics of the IGWs in the presence of temperature gradients and sheared wind. The solution in the form of rolls has been obtained. The new condition for the existence of such structures was found by taking into account the roll spatial scale, the horizontal speed and wind shear parameters. We have shown that the roll structures can exist in a dynamically unstable atmosphere.

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Electrostatic ion-acoustic periodic (cnoidal) waves and solitons in unmagnetized electron-positron-ion (EPI) plasmas with warm ions and kappa distributed electrons and positrons are investigated. Using the reductive perturbation method, the Korteweg-de Vries (KdV) equation is derived with appropriate boundary conditions for periodic waves. The corresponding analytical and various numerical solutions are presented with Sagdeev potential approach. Differences between the results caused by the kappa and Maxwell distributions are emphasized. It is revealed that only hump (compressive) structures of the cnoidal waves and solitons are formed. It is shown that amplitudes of the cnoidal waves and solitons are reduced in an EPI plasma case in comparison with the ordinary electron-ion plasmas. The effects caused by the temperature variations of the warm ions are also discussed. It is obtained that the amplitude of the cnoidal waves and solitons decreases for a kappa distributed (nonthermal) electrons and positrons plasma case in comparison with the Maxwellian distributed (thermal) electrons and positrons EPI plasmas. The existence of kappa distributed particles leads to decreasing of ion-acoustic frequency up to thermal ions frequency.

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A system of equations describing the nonlinear interaction of coupled Rossby-Khantadze electromagnetic waves with a sheared zonal flow in the Earth's ionospheric E-layer is obtained. For the linear regime the corresponding region of phase velocities is analyzed and the appropriate stability condition of zonal flow is deduced. It is shown that the sheared zonal flow may excite solitary vortical structures in the form of a row of counter-rotating vortices whose amplitudes decrease with the increase of the zonal flow parameter. This conclusion is consistent with the stabilizing idea of a sheared zonal flow. The possibility of an intense magnetic-field generation is shown.

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The nonlinear interaction of the electromagnetic ion cyclotron (EMIC) frequency waves with plasma particles in the inner magnetosphere is studied. The emission is considered to be circularly polarized electromagnetic waves propagating along the almost constant dipole geomagnetic field in the equatorial region of the inner magnetosphere. Under the action of the ion cyclotron ponderomotive force excitation of the magnetosonic waves through the amplitude modulation of the EMIC waves is investigated. Two dimensional nonlinear Schrodinger equation for the EMIC waves is derived. In the stationary case two solutions of the nonlinear Schrodinger equation with distinct natures are found. The generation of both vortices and of a quasistatic magnetic field across the geomagnetic field lines is discussed.

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Recent satellite and ground-based observations proved that during an earthquake preparation period VLF/LF and ULF electromagnetic emissions are observed in the seismogenic area. The present work offers possible physical bases of earth electromagnetic emission generation detected in the process of earthquake preparation. According to the authors of the present paper electromagnetic emission in radiodiapason is more universal and reliable than other earthquake indicators and VLF/LF electromagnetic emission might be declared as the main precursor of earthquake. It is expected that in the period before earthquake namely earth electromagnetic emission offers us the possibility to resolve the problem of earthquake forecasting by definite precision and to govern coupling processes going on in lithosphere-atmosphere-ionosphere (LAI) system.

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A nonlocal contact boundary problem for two-dimensional linear elliptic equations is stated and investigated. The uniqueness of the solution is proved. The iteration process is constructed, which allows one not only to prove the existence of a regular solution of the problem, but also to develop an approximate algorithm of its solution. The solution of a nonlocal contact problem is reduced to the solution of classical boundary value problems, in particular to the solution of Dirichlet problems

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One nonlinear averaged integro-differential system with source terms is considered. The model arises on mathematical simulation of the process of penetration of a magnetic field into a substance. Semi-discrete difference scheme is studied.

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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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One nonlinear partial integro-differential equation with source term is considered. The model arises at describing penetration of a magnetic field into a substance and is based on the Maxwell system. Large time behavior of solution of the initial-boundary value problem as well as semi-discrete finite scheme are studied. More wide class of nonlinearity is considered than one has been already investigated.

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In the Cartesian system of coordinates, thermoelastic equilibrium of an isotropic homogeneous rectangular parallelepiped is considered. On the lateral faces of a parallelepiped either symmetry or anti-symmetry conditions are defined while the top and bottom faces are free of stress. The problem is that to define the temperature on the top and bottom faces of a parallelepiped so that the normal displacement or the tangential displacements would take a priori fixed values on some two planes parallel to the bases. The problems are solved analytically using the method of separation of variables. The problems are non-classical, but they differ from other non-classical problems known in literature and are of a practical importance.

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Static thermoelastic equilibrium is considered for an N-layer along the radial coordinate body bounded by coordinate surfaces of a spherical system of coordinates. Each layer is isotropic and homogeneous and some of the layers may be composed of an incompressible elastic material. On the spherical surfaces of the involved body changes in the temperature or its normal derivative, stresses, displacements or their combinations are defined while on the remaining part of the boundary special type of homogeneous conditions are given. The stated problems are analytically solved using the method of separation of variables, the general solution being represented by means of harmonic functions. Problem solution is reduced to the solution of systems of algebraic equations with block diagonal matrices.

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The paper deals with a static thermoelastic equilibrium of an N-layer rectangular parallelepiped. The layers of the considered body are made of an isotropic homogeneous elastic material. A case when some of the layers consist of incompressible elastic materials, which are also assumed to be isotropic and homogeneous, is considered as well. Boundary conditions of symmetric or antisymmetric continuous extension of the solution are imposed on the lateral facets of the parallelepiped. Between the layers contact conditions of rigid, sliding or other type of contact can be defined. On the upper and lower facets of the parallelepiped, arbitrary boundary conditions are defined. Solution of the stated problems is made analytically using the method of separation of variables. The solution of the problems is reduced to the solution of systems of linear algebraic equations with block diagonal matrices. In the conclusion, a practical example establishing the elastic equilibrium of a three-layer rectangular parallelepiped is given.

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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 discussed. 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 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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Static thermoelastic equilibrium is considered for an N-layer along the radial coordinate body bounded by coordinate surfaces of a circular cylindrical system of coordinates. Each layer is isotropic and homogeneous and some of the layers may be composed of an incompressible elastic material. On the flat boundaries of the cylindrical body boundary conditions of either symmetrical or anti-symmetrical continuous extension of the solution are imposed. Between the layers contact conditions of rigid, sliding or other type of contact may be defined. The stated problems are solved using the method of separation of variables,the general solution being represented by means of harmonic functions. The solution of the problems is reduced to the solution of systems of algebraic equations with block diagonal matrices. At the end of the paper an application example is given which illustrates the applied approach for an analytical solution of problems.

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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 present work offers model of earth electromagnetic emission generation detected in the process of earthquake preparation on the basis of electrodynamics.Besides, scheme of the methodology of earthquake forecasting is created based on avalanche-like unstable model of fault formation and an analogous model of electromagnetic contour, synthesis of which, is rather harmonious. According to the authors of the work electromagnetic emission in radiodiapason is more universal and reliable that other anomalous variations geophysical phenomena in earthquake preparation period. Besides, VLF/LF electromagnetic emission might be declared as the main precursor of earthquake because it might turn out very useful with the view of prediction of large (M>5) inland earthquakes and to govern processes going on in lithosphere-atmosphere-ionosphere coupling (LAIC) system.

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The present paper 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.

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In this paper hierarchical models of biofilms occupying a thin prismatic domain are considered

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Some analytic methods for calculating of prismatic shells with two cusped edges are given.

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There exists nonzero, σ-finite, diffused Borel measure χ on RN, which is invariant with respect to an everywhere dense vector subspace of RN and, in addition, is metrical transitive (i. e., ergodic) with respect to the same subspace. We discuss relative measurability of real-valued functions with respect to some measures in the space R^N

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The limiting distribution of an integral square deviation between two kernel type estimators of Bernoulli regression functions is established in the case of two independent samples. The criterion of testing is constructed for both simple and composite hypotheses of equality of two Bernoulli regression functions. The question of consistency is studied. The asymptotics of behavior of the power of test is investigated for some close alternatives.

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It is shown that the class of all non-separable extensions of a nonzero σ-finite Borel measure in the topological vector space ℝℕ, which are invariant under some everywhere dense continual subgroup of ℝℕ and which possess the uniqueness property, has maximal cardinality 22c. Some related questions concerning the measurability properties of real-valued functions with respect to the class of non-separable measures are also discussed.

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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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The limiting distribution of an integral square deviation between two kernel type estimators of Bernoulli regression functions is established in the case of two independent samples. The criterion of testing is constructed for both simple and composite hypotheses of equality of two Bernoulli regression functions. The question of consistency is studied. The asymptotics of behavior of the power of test is investigated for some close alternatives.

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The limiting distribution of an integral square deviation between two kernel type estimators of Bernoulli regression functions is established in the case of two independent samples. The criterion of testing is constructed for both simple and composite hypotheses of equality of two Bernoulli regression functions. The question of consistency is studied. The asymptotics of behavior of the power of test is investigated for some close alternatives.

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In the article theorems are given which are related to the reconstruction of coefficients of a $d$-multiple Rademacher series (where $d$ is any natural number such that $d\geq 1$) by means of values of the sum of this series at appropriately chosen $2^{d}$ points. Well known theorems connected with Rademacher series as direct consequences of these theorems are considered.

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The convergence of partial sums and Cesáro means of negative order of double Walsh-Fourier series of functions of bounded generalized variation is investigated.

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The maximal Orlicz space such that the mixed logarithmic means of rectangular partial sums of multiple Fourier series for the functions from this space converge in measure is found.

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Nörlund strong logarithmic means of double Fourier series acting from space L log L (T2) into space L p (T2), 0 < p < 1, are studied. The maximal Orlicz space such that the Nörlund strong logarithmic means of double Fourier series for the functions from this space converge in two-dimensional measure is found.

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In the present paper two-dimensional model of prismatic shell is constructed. Existence and uniqueness of the solution of corresponding boundary value problem are proved, the rate of approximation of the solution of original problem by vector-function restored from the solution of two-dimensional problem is estimated.

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The theorem proving techniques are divided into two parts, goal-directed and refutational. In this paper 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 paper a planar classical quantum billiard in the hexagonal type areas with the hard wall conditions is considered. The process is described by the Helmholtz Equation in the hexagon and hexagonal rug with the homogeneous boundary conditions. By means of the conformal mapping method the problem is reduced to the elliptic partial differential equation in the rectangle with the homogeneous boundary condition. It is assumed that one parameter of mapping is sufficiently small. In this case the equation is simplified and analyzed. The asymptotic solutions are obtained. The spectrum and the corresponding eigenfunctions are found near the boundary of the hexagon. The wave functions are found in terms of the Bessel’s functions. The results are applied for the estimation of the energy levels of electrons in graphene.

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In the article methods of summability with a variable order are presented and theorems related to orthogonal series divergent by these methods are formulated.

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Consider the first order differential and difference equations. Sufficient oscillation conditions are presented for this equations

Considered on asymptotic behavior of solutions of generalized Emden-Fowler differential equations with delay argument In the case μ(t) ≡ const > 0, the oscillatory properties of given equation are extensively studied, where as for μ(t) ≢ const, to the best of authors’ know ledge, problems of this kind were not investigated at all. We also establish new sufficient conditions for the equation to have Property B.

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We consider a differential equation We say that the equation is almost linear if the condition lim inf t μ(t) =1 is satisfied. At the same time, if lim sup t μ(t)  ≠ 1 or lim inf t μ(t) ≠ 1, then Eq. is called an essentially nonlinear differential equation. The oscillatory properties of almost linear differential equations have been extensively studied. In the paper, new sufficient (necessary and sufficient) conditions are established for a general class of essentially nonlinear functional differential equations to have Property A.

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In this paper the 2D full coupled theory of steady vibrations of poroelasticity for materials with double porosity is considered. There the fundamental and singular matrixes of solutions are constructed in terms of elementary functions. Using the fundamental matrix we will construct the simple and double layer potentials and study their properties near the boundary.

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Genesis of Foehns is in detail investigated. They are classified on dryadiabatic, mostadiabatic and most-dryadiabatic Foehns. A problem about numerical modeling of Foehns in frame of a at, two-dimensional mesoscale boundary layer is stated. The problem is at a stage of numerical realisation. The first encouraging results are received.

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The process of wet washing down of aerosol particles in the atmosphere is examined. For the specially homogenous dispersive system containing aerosol particles and droplets (crystals), the analytic solution of coagulation kinetic equation is obtained under the conditions of constant generation of aerosol. The source in proportional to the initial distribution of particles. Using the solution the efficiency of wet washing down is assessed for different types of liquid precipitation (the relaxation time of aerosol particles). Microphysical lows of wet washing down in case of gravity coagulation are obtained as well.

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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 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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The specific features of hypotheses testing regions of the Berger’s $T*$ test and CBM (see Part I of this paper), namely, the existence of the no-decision region in the $T*$ test and the existence of regions of impossibility of making a unique or any decision in CBM give the opportunities to develop the sequential tests on their basis. Using the concrete example taken from [5], below these tests are compared among themselves and with the Wald sequential test [55]. For clarity, let us briefly describe these tests.

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New sequential methods of multiple testing problems based on special properties of hypoNew sequential methods of multiple testing problems based on special properties of hypotheses acceptance regions in the constrained Bayesian tasks of testing hypotheses are offered. Results of an investigation on the properties of one of these methods are given. They show the consistency, simplicity, and optimality of the results obtained in the sense of the chosen criterion. The essence of the criterion is to restrict from above the probability of the error of one type and to minimize the probability of the error of the second type. The facts of the validity of the suitable properties of the method are proved. Examples of testing of hypotheses for the sequentially obtained independent samples from the multivariate normal distribution with correlated components are cited. They show the high quality of the proffered methods. The results of the Wald sequential method are given for the examples with two hypotheses and compared with the results obtained by the proffered method theses acceptance regions in the constrained Bayesian tasks of testing hypotheses are offered. Results of an investigation on the properties of one of these methods are given. They show the consistency, simplicity, and optimality of the results obtained in the sense of the chosen criterion. The essence of the criterion is to restrict from above the probability of the error of one type and to minimize the probability of the error of the second type. The facts of the validity of the suitable properties of the method are proved. Examples of testing of hypotheses for the sequentially obtained independent samples from the multivariate normal distribution with correlated components are cited. They show the high quality of the proffered methods. The results of the Wald sequential method are given for the examples with two hypotheses and compared with the results obtained by the proffered method.

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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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Formulae for computation of probability of errors in sequential method of Bayesian type are offered. In particular, some relations between the errors of the first and the second kinds in constrained Bayesian task and in sequential method of Bayesian type depending on the divergence between the tested hypotheses are given. Dependencies of the Lagrange multiplier and the risk function on the probability of incorrectly accepted hypotheses are also presented. Theses results are necessary for computation of errors of made decisions attesting multiple hypotheses using the offered new sequential methods of testing hypotheses. Computation results of some examples confirm the rightness of theoretical analysis.

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Quantum computations can be implemented not only by the action of quantum circuits, but by the adiabatic evolution of a system’s Hamiltonian. Quantum adiabatic statement allows to solve some classically non algorithmic problems. Our reasoning favor of this argument.

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This article presents main results of investigations of the authors which were obtained during the last five years by the partially support on the Shota Rustaveli National Science Foundation (Grant no. 31–24). These results are Liouville-type theorems and describe the behavior of various phase motions in terms of ordinary and standard “Lebesgue measures” in R∞. In this context, the following three problems are discussed in this paper: Problem 1. An existence and uniqueness of partial analogs of the Lebesgue measure in various function spaces; Problem 2. A construction of various dynamical systems with domain in function spaces defined by various partial differential equations; Problem 3. To establish the validity of Liouville-type theorems for various dynamical systems with domains in function spaces in terms of partial analogs of the Lebesgue measure.

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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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Some classes of real-valued functions defined on a metric space V equipped with a nonzero sigma-finite diffused Borel measure µ were introduced and relationships between them (in the sense of inclusion) are studied. In particular, it is shown that when V is a Polish metric space then the properties of µ-massiveness along trajectories of all continuous functions on V and of µ-massiveness along trajectories of all measurable functions on V coincide. It is demonstrated also that relationships between these classes are rather different and surprising if (V, ρ) is not separable.

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By using an infinite-dimensional "Lebesgue measure" in an infinite-dimensional separable Banach space B with Schauder basis a solution of a heat equation with initial value problem on B is constructed. Properties of uniformly distributed real-valued sequences in an interval of the real axis are used for a construction of a certain algorithm which gives an approximation of corresponding solutions.

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It is shown that for every translation-invariant quasi-finite diffused Borel measure µ in an uncountable non-locally compact Polish group G that is dense in himself there does not exist a positive constant c such that each Borel set E with the µ-measure bigger than c contains three points such that an area defined by that points is equal to one. This answers negatively to a certain modification of P. Erd¨os problem [P. Erd¨os, Set-theoretic, measure-theoretic, combinatorial, and number- theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange, 4(2), (1978/79), 113–138] stated by us for a translation-invariant quasi-finite diffused Borel measure in an uncountable non-locally compact Polish group that is dense in himself.

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It is proved an existence of maximal ”small” plane sets in R^2 which contain only the vertices of a triangle of area less than one. It is shown also that the closing of each maximal ”small” plane set in R^2 contains the vertices of a triangle of area one.

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By using the technique of "Fourier differential operator" in R ∞ and Laplace transforms, a representation in a multiple trigonometric series of the solution of a certain generalized heat equation of many variables is obtained.

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Genesis of Foehns is in detail investigated. They are classified on dryadiabatic, mostadiabatic and mostdryadiabatic Foehns. It is stated a problem about numerical modelling of Foehns in frame of a flat, twodimensional mesoscale boundary layer. The problem is at a stage of numerical realisation. The first encouraging results are received.

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For a group Γ of all rotations of the plane R^sup 2^ about it's origin, by using the technique developed in a paper [Kharazishvili A. B., Small sets in uncountable abelian groups. Acta Univ. Lodz. Folia Math. No. 7 (1995), 31-39] it is proved an existence of a partition of the plane R^sup 2^ into absolutely Γ-negligible subsets of R^sup 2^ for which an intersection of every element of the partition with each beam leaving the origin of R^sup 2^ includes exactly one line segment of length 1. By the method developed in the monograph [Pantsulaia G.R., Invariant and quasiinvariant measures in infinite-dimensional topological vector spaces, Nova Science Publishers, Inc., New York, 2007. xii+234] it is shown that in Solovay's model an arbitrary non-trivial closed ball in an infinite dimensional non-separable Banach space l^sup ∞^ is an infinite-dimensionally Haar null set. This answers positively on 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 Banach space l^sup ∞^.

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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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In present report the peculiarities of the hydro-dynamical flows in a narrow canals with small slope bottom ,at low velocities of the stream , have been studied. It has been shown that the velocity and power of the currents are inversely proportional to the square of the parameter characterized the special features of the canal’s bottom. In the Earth atmosphere there are often observed non-periodical. 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” eventsIn 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. Obtained results are new and have as theoretical as well practical values

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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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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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In the present paper the mathematical model of the linear 2D dynamical theory of thermoelasticity with microtemperatures is considered. The representation of regular solution, the fundamental and singular solutions for a governing system of equations of this theory in the Laplace transform space are constructed. Finally, the single-layer, double-layer and volume potentials are presented

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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 to solve explicitly the Dirichlet and Neumann BVPs of statics in the full coupled theory for an elastic plane with a circular hole. The explicit solutions of these BVPs are represented by means of absolutely and uniformly convergent series. The questions on the uniqueness of a solutions of the problems are established.

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It is shown that limTfn := infn supm≥n Tfm and limTfn := supn infm≥n Tfm are objective infinite sample well-founded estimates of a useful signal θ in the linear one-dimensional stochastic model ξk = θ + ∆k (k ∈ N), where #(·) denotes a counting measure, ∆k is a sequence of independent identically distributed random variables on R with strictly increasing continuous distribution function F, expectation of ∆1 does not exist and Tn : R N → R (n ∈ N) is defined by Tn((xk)k∈N) = −F −1 (n −1#({x1, · · · , xn} ∩ (−∞; 0])) for (xk)k∈N ∈ R N.

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For the quasi-linear neutral functional differential equation the continuous dependence of a solution of the Cauchy problem on the initial data and on the nonlinear term in the right-hand side of that equation is investigated, where the perturbation nonlinear term in the right-hand side and initial data are small in the integral and standard sense, respectively. Variation formulas of a solution are derived, in which the effect of perturbations of the initial moment and the delay function, and also that of the discontinuous initial condition are detected. For initial data optimization problems the necessary conditions of optimality are obtained. The existence theorem for optimal initial data is proved.

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Variation formulas of solution are obtained for linear with respect to prehistory of the phase velocity (quasi-linear) controlled neutral functional-differential equation with variable delays. The effects of delay function perturbation and continuous initial condition are detected in the variation formulas

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For an optimal control problem involving neutral differential equation, 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 and initial functions, initial moment and vector, control and finally moment

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Theorems on the continuous dependence of solution on perturbations of the initial data and the right-hand side of equation are proved. Under initial data we imply the collection of initial moment, variable delays, initial vector and initial function.

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Variation formulas of solution are obtained for linear with respect to prehistory of the phase velocity neutral functional-differential equations with variable delays and with the discontinuous initial condition. In the variation formulas are detected the effects of perturbation of delay function entering in the phase coordinates and the discontinuous initial condition. The variation formula of solution plays the basic role in proving of the necessary conditions of optimality and under sensitivity analysis of mathematical models. Discontinuity of the initial condition means that the values of the initial function and the trajectory, in general, do not coincide at the initial moment.

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It is shown that for the vector space R^ N (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a σ-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each σ-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete σ-finite Borel measure is solved positively. The uniqueness problem for non-σ-finite semi-finite translation invariant Borel measures on a Banach space X which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure µ_0 on X, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of µ_0 and whose values on non-degenerate rectangles coincide with their volumes.

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Necessary optimality conditions are obtained for initial data of linear with respect to prehistory of the phase velocity (quasi-linear) neutral functional differential equation. Here initial data implies the collection of initial moment and vector, delay function entering in the phase coordinates and initial function. In this paper, the essential novelty are optimality conditions of the initial moment and delay function. Discontinuity of the initial condition means that the values of the initial function and the trajectory, in general, do not coincide at the initial moment.

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Variation formulas of solution for a nonlinear functional differential equation with variable delay and continuous initial condition are proved. The effects of delay function perturbation and continuous initial condition are detected in the variation formulas. The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.

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This book explores a number of new applications of invariant quasi-finite diffused Borel measures in Polish groups for a solution of various problems stated by famous mathematicians (for example, Carmichael, Erdos, Fremlin, Darji and so on). By using natural Borel embeddings of an infinite-dimensional function space into the standard topological vector space of all real-valued sequences, (endowed with the Tychonoff topology) a new approach for the construction of different translation-invariant quasi-finite diffused Borel measures with suitable properties and for their applications in a solution of various partial differential equations in an entire vector space is proposed.

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Variation formulas for solution are proved for a nonlinear differential equation with constant delays. In this work, the essential novelty is an effect of delay perturbation in the variation formulas. The mixed initial condition means that at the initial moment, some coordinates of the trajectory do not coincide with the corresponding coordinates of the initial function, whereas the others coincide. Variation formulas are used in the proof of necessary optimality conditions.

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The notion of a Haar null set introduced by Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke, has been used, in the last two decades, in studying exceptional sets in diverse areas, including analysis, dynamic systems, group theory, and descriptive set theory. In the present paper, the notion of “prevalence” is used in studying the properties of some infinite sample statistics and in explaining why the null hypothesis is sometimes rejected for “almost every” infinite sample by some hypothesis testing of maximal reliability. To confirm that the conjectures of Jum Nunnally [17] and Jacob Cohen [5] fail for infinite samples, examples of the so called objective and strong objective infinite sample well-founded estimate of a useful signal in the linear one-dimensional stochastic model are constructed.

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Cubic nonlinear Schrödinger type equation with specific initial-boundary conditions in the infinite domain is considered. The equation is reduced to an equivalent system of partial differential equations and studied in the case of solitary waves. The system is modified by introducing new functions, one of which belongs to the class of functions of negligible fifth order and vanishing at infinity exponentially. For this class of functions the system is reduced to a nonlinear elliptic equation which can be solved analytically, thereby allowing us to present nontrivial approximated solutions of nonlinear Schrödinger equation. These solutions describe a new class of symmetric solitary waves. Graphics of modulus of the corresponding wave function are constructed by using Maple.

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In this work the algorithm of the approximate solution of two-dimensional boundary value problems of thermoelasticity is offered for transversal isotropic body. The offered algorithm is based on use of representation of the general solution of system of the equations of balance by means of harmonic functions.

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In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with voids is considered and some basic results of the classical theory of elasticity are generalized. Indeed, the basic properties of plane harmonic waves are established. The explicit expression of 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 boundary value problems (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 Sommerfeld-Kupradze type radiation conditions are established.

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In this article, the linear theory of thermoviscoelasticity for Kelvin–Voigt materials with voids is considered. The fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions and its basic properties are established. The representation of a Galerkin-type solution is obtained. The formula of representation of the general solution for the system of homogeneous equations of steady vibrations in terms of six metaharmonic functions is established. The completeness of these representations of solutions is proved.

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In the present article, the linear theory of thermoviscoelasticity for Kelvin–Voigt materials with voids is considered. The Sommerfeld-Kupradze type radiation conditions are established. The uniqueness theorems of the internal and external basic boundary value problems (BVPs) of steady vibrations are proved. The Green's formulas and integral representations of Somigliana type of regular vector and classical solution are obtained. The basic properties of thermoelastopotentials and singular integral operators are given. Finally, the existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (boundary integral method) and the theory of singular integral equations.

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It is shown that under some appropriate set-theoretical assumptions there exists an absolutely nonmeasurable function acting from [0, 1] into [0, 1], whose graph is a projective subset of [0, 1]2 .

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In this report three classical constructions of Lebesgue nonmeasurable sets on the real line R are envisaged from the point of view of the thickness of those sets with respect to the standard Lebesgue measure λ on R.

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In the present paper by means of the I. Vekua method the system of differential equations for shallow spherical shells is obtained, when on upper and lower face surface displacement are assumed to be known. Using the method of the small parameter approximate solutions of I. Vekua’s equations for approximations N = 0 is constructed. The small parameter ε = 2h/R, where 2h is the thickness of the shell, R is the radius of the sphere.

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We envisage Bernstein subsets of the real line R from the point of view of their measurability with respect to certain classes of measures on R. In particular, it is shown that there exists a Bernstein set absolutely nonmeasurable with respect to the class of all nonzero σ-finite translation quasi-invariant measures on R, and that there exist countably many Bernstein sets which collectively cover R and are absolutely negligible with respect to the same class of measures.

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

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Three classical constructions of Lebesgue nonmeasurable sets on the real line R are envisaged from the point of view of the thickness of those sets. It is also 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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One characterization of an uncountable set E is given in terms of mappings f of E into itself and “large” subsets of E which are invariant with respect to f.

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We analyze Sierpiński's example of a real-valued Lebesgue measurable function on ℝ which is not bounded from above by any real-valued Borel function. In this context, some real-valued step-functions with analogous properties are discussed.

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For any σ -finite G-quasiinvariant measure μ given in a G-space, which is G-ergodic and possesses the Steinhaus property, it is shown that every nontrivial countable μ-almost G-invariant partition of the G-space has a μ-nonmeasurable member.

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I. Vekua has constructed several versions of the refined theory of thin and shallow shells. Using the reduction methods of I. Vekua, the 2-D system of equations for geometrically and physically nonlinear theory of non-shallow shells is obtained.

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In the present paper, using the method of I. Vekua, the three dimensional problems of the nonlinear theory of elasticity are reduced to the two dimensional problems of non-shallow spherical shells. Using the method of the small parameter, approximate solutions of these equations are constructed. One boundary value problems are solved for the approximation of order N=0.

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In the present paper, by means of Vekua’s method, the system of differential equations for thin and shallow shells is obtained, when on upper and lower face surfaces displacements are assumed to be known.

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This work is devoted to study of transient growth and further linear and nonlinear dynamics of planetary electromagnetic (EM) ultra-low-frequency internal waves (ULFW) in the rotating dissipative ionosphere due to non-normal mechanism, stipulated by 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. 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. Thus the structures under study may represent the ULF electromagnetic wave macro turbulence structural element in the ionosphere.

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The generation and further nonlinear dynamics of internal gravity wave (IGW) structures in a dissipative ionosphere in the presence of an inhomogeneous zonal wind (shear flow) have been studied. The effectiveness of the IGW amplification mechanism during the interaction with an inhomogeneous zonal wind is analyzed based on the corresponding model system of nonlinear dynamic equations constructed in (Aburjania et al., 2013). It has been indicated that IGWs effectively obtain the shear flow energy at the initial linear evolution stage and substantially (by an order of magnitude) increase their amplitude and, correspondingly, energy. The nonlinear self-localization mechanism starts operating with increasing amplitude, and the process terminates with the self-organization of nonlinear solitary strongly localized vortex structures. A new degree of system freedom and the disturbance evolution trend in a medium with a shear flow appear in such a way. Nonlinear IGW 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.

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The linear mechanism by which internal gravity waves (IGWs) are generated and subsequently intensified in a stably stratified dissipative ionosphere in the presence of an inhomogeneous zonal wind (shear flow) has been studied. In the case of shear flows, the operators of linear problems are nonselfadjoint and the corresponding eigenfunctions are nonorthogonal; a canonical approach can hardly be used to study such motions. It is more adequate to apply the socalled nonmodal calculation. Dynamic equations and equations of energy transfer of IGW disturbances in the ionosphere with a shear flow have been obtained based on a nonmodal approach. Exact analytical solutions for the constructed dynamic equations have been found. The growth rate of the IGW shear instability has been determined. It has been established that IGW disturbances are intensified in an algebraically power manner rather than exponentially in the course of time. The effec tiveness of the linear mechanism by which IGWs are intensified when interacting with an inhomogeneous zonal wind is analyzed. It has been indicated that IGWs effectively obtain the shear flow energy during the linear evolution stage and substantially increase (by an order of magnitude) their amplitude and energy. The frequency and the wave vector of generated IGW modes depend on time; therefore, a wide spectrum of wave like disturbances, depending on the linear, rather than nonlinear, turbulent effects, is formed in the iono sphere with a shear flow. Thereby, a new degree of freedom appears, and the turbulent state of atmospheric– ionospheric layers can be formed on IGW disturbances

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In this paper random measures and their nonlinear transformations are considered. The conditions of absolute continuity for this measures are obtained in case of nonlinear and random transformation of a space. There is given explicit formula for RadonNikodym derivative. The notion of measurable functional is used and the logarithmic derivative technique of measures is developed.

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The specific features of the generation and intensification of internal gravity wave structures in dif ferent atmospheric–ionospheric regions, caused by zonal local nonuniform winds (shear flows), are studied. The model of the medium has been explained and an initial closed system of equations has been obtained in order to study the linear and nonlinear dynamics of internal gravity waves (IGWs) when they interact with the geomagnetic field in a dissipative ionosphere (for the D, E, and F regions)

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The generation and further dynamics of planetary ULF waves are investigated in the rotating dissipative ionosphere in the presence of a smooth inhomogeneous zonal wind (shear flow). Planetary ULF waves appear as a result of the interaction of the medium with the spatially inhomogeneous geomagnetic field. An effective linear mechanism responsible for the intensification and mutual transformation of large scale magnetized Rossby type and small scale inertial waves is found. For shear flows, the operators of the linear problem are not self-adjoint, and therefore the eigenfunctions of the problem maybe non-orthogonal and can hardly be studied by the canonical modal approach. Hence it becomes necessary to use the so-called nonmodal mathematical analysis. The nonmodal approach shows that the transformation of wave disturbances in shear flows is due to the non-orthogonality of eigenfunctions of the problem in the conditions of linear dynamics. Using numerical modeling, it is illustrated the peculiar features of the interaction of waves with the background flow as well as the mutual transformation of wave disturbances in the ionosphere.

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The dynamic pattern calculus described in this paper integrates the functional mechanism of the lambda-calculus and the capabilities of pattern matching with hedge variables, i.e., variables that can be instantiated by any finite sequence of terms. We propose a generic confluence proof, where the way pattern abstractions are applied in a non-deterministic calculus is axiomatized.

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Bitsadze-Samarskii nonlocal boundary value problem for two-dimensional second order elliptic equa-tions is considered. The domain decomposition sequential as well as parallel type algorithms are studied. Varia-tional formulation is done.

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Existence, uniqueness, long-time behavior of solutions and algorithm of numerical resolution of initial-boundary value problem for one integro-differential system are studied. Attention is paid to construction and analysis of decomposition algorithms with respect to physical processes for one-dimensional nonlinear partial differential model based on Maxwell’s system. Semi-discrete averaged models are constructed and investigated for this system. Finite difference schemes are studied. Investigated systems arise in modeling of process of the penetration of a magnetic field in a substance.

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The manual is based on documents published by OMG (Object Management Group) Business Process Model and Notation (BPMN), Version 1.2 and BPMN Modeler for Visio. It contains the BPMN terms, designations, terms and conditions of use and perception rules to automate the process of describing a business process. The manual also contains the rules, capabilities and performance sequence of the Interfacing BPMN Modeler for Visio, a special software product created for this purpose. For easy comprehension of the presented material, it has a large number of graphic material attached.

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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 the present paper 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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Nonlinear vortex propagation of electromagnetic coupled Rossby and Khantadze planetary waves in the weakly ionized ionospheric E-layer is investigated with numerical simulations. Large scale, finite amplitude vortex structures are launched as initial conditions at low, mid, and high latitudes. For each k-vector the linear dispersion relation has two eigenmodes corresponding to the slow magnetized Rossby wave and the fast magnetic Khantadze wave. Both waves propagate westward with local speeds of the order of 10–20 m/s for the slow wave and of the order of 500–1000 km/s for the fast wave. We show that for finite amplitudes there are dipole solitary structures emitted from the initial conditions. These structures are neutrally stable, nonlinear states that avoid radiating waves by propagating faster than the corresponding linear wave speeds. The condition for these coherent structures to occur is that their amplitudes are such that the nonlinear convection around the core of the disturbance is faster than the linear wave speed for the corresponding dominant Fourier components of the initial disturbance. The presence of the solitary vortex states is indicative of an initial strong disturbance such as that from a solar storm or a tectonic plate movement. We show that for generic, large amplitude initial disturbances both slow and fast vortex structures propagate out of the initial structure.

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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 . 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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There are considered the alternating to perturbation technique convergence method by which the solution of operator equation (1) is same as to invert the operator L of simpler structure than L+εM and apply operator M (both processes-each) at N times. This scheme is realized for some one and two dim BVPs for DEs, for integral eq. with an irremovable singularity. For 2dim problems when mathematical models are Refined Theories in wide sense for thinwalled elastic structures in case of technical domains (rectangular, quadrant, band) there are developed special variational-projective methods by which there are solved approximately some concrete problems. Last part of this report are dedicated to the applications of Variational-Projective and Schwarz Alternative Methods for technical regions of complete geometry such as finite or unbounded crosses.

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It is shown that in the Earth's weakly ionized ionospheric E-layer with the dominant Hall conductivity, a new type of coupled Rossby–Alfvén–Khantadze (CRAK) electromagnetic (EM) planetary waves, attributable by the latitudinal inhomogeneity of both the Earth's Coriolis parameter and the geomagnetic field, can exist. Under such coupling, a new type of dispersive Alfvén waves is revealed. The generation of a sheared zonal flow and a magnetic field by CRAK EM planetary waves is investigated. The nonlinear mechanism of the instability is based on the parametric excitation of a zonal flow by interacting four waves, leading to the inverse energy cascade in the direction of a longer wavelength. A three-dimensional (3D) set of coupled equations describing the nonlinear interaction of pumping CRAK waves and zonal flow is derived. The growth rate of the corresponding instability and the conditions for driving them are determined. It is found that the growth rate is mainly stipulated by Rossby waves but the generation of the intense mean magnetic field is caused by Alfvén waves.

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Boundary value problems for ordinary differential equations with a small parameter in the leading derivative are solved by a high accuracy order multipoint method. The numerical scheme for a two-point boundary value problem for the second order ordinary differential equation is presented. The estimates for the derivatives of the solution are obtained with the use of the asymptotic expansions. The remainders of the difference scheme are evaluated. Numerical examples illustrate the efficiency of the proposed techniques.

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Planetary-scale waves and vortices in the weakly ionized ionospheric E layer are dominated by the Hall conductivity that couples the Rossby and Alfvén dynamics giving rise to what are called Rossby-Alfvén-Khantadze electromagnetic structures. At finite amplitudes we show that the nonlinearities in the dynamics generate sheared zonal-flow velocities and zonal magnetic field fluctuations. The zonal-flow mechanism is based on the parametric excitation of the zonal variations through three-wave mode coupling in the planetary-scale waves. The coupled dynamics of the nonlinear 3-D incompressible flows and the magnetic field fluctuations are derived and used to derive the structure and growth rates for the zonal flows and zonal magnetic fields. Large-amplitude planetary waves are shown to drive up magnetic fluctuations up to 100 nT.

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We investigated the problem of decision to construct the coefficients and eigenvalues of Secular Equation (SE). In this aim we develop the reliable technology for calculating coefficients of SE in the range of 1200 scientific digits based on our methodology defining classical orthogonal polynomials of degree of order 106 with 1200 decimal digits and assumption that the exact multiplication operations of two matrices with integer elements in the range 1016 approximately for 600 digits is possible.

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It is shown that in the Earth's weakly ionized ionospheric E-layer with the dominant Hall conductivity, a new type of coupled Rossby–Alfvén–Khantadze (CRAK) electromagnetic (EM) planetary waves, attributable by the latitudinal inhomogeneity of both the Earth's Coriolis parameter and the geomagnetic field, can exist. Under such coupling, a new type of dispersive Alfvén waves is revealed. The generation of a sheared zonal flow and a magnetic field by CRAK EM planetary waves is investigated. The nonlinear mechanism of the instability is based on the parametric excitation of a zonal flow by interacting four waves, leading to the inverse energy cascade in the direction of a longer wavelength. A three-dimensional (3D) set of coupled equations describing the nonlinear interaction of pumping CRAK waves and zonal flow is derived. The growth rate of the corresponding instability and the conditions for driving them are determined. It is found that the growth rate is mainly stipulated by Rossby waves but the generation of the intense mean magnetic field is caused by Alfvén waves.

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In this paper a boundary value problem for thermoelastic prismatic shell with microtemperatures is considered. A hierarchy of two-dimensional models for a static threedimensional model for prismatic shell with surface force, the normal component of heat flux and the first heat flux moment given on the upper and the lower faces of the prismatic shell is constructed. The two-dimensional boundary value problems corresponding to the hierarchical models are investigated in suitable function spaces. The convergence of the sequence of vector-functions of three space variables, restored from the solutions of the two-dimensional boundary value problems of the constructed hierarchy to the exact solution of the original three-dimensional problem is proved and the rate of approximation is estimated provided that the solution satisfies additional regularity conditions.

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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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One nonlinear integro-differential system is considered. The model describes penetration of a magnetic field into a substance. Semi-discrete difference scheme with respect to space variable is studied.

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In this paper, in full analogy with the theory of generalized analytic functions, are given formulas for a general representation of regular solutions of the matrix elliptic system, the so-called generalized analytic vectors. On this basis, the boundary-value problems of Riemann-Hilbert and linear conjugation in the case of Holder-continuous coefficients are considered.

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The elastic equilibrium of a multi-layer confocal elliptic ring is studied. The ring consists of steel, rubber and celluloid layers which differ in thickness and in the order in which they are placed relative to one another. Using the solutions of the considered problems, the following delocalization problem is solved: for a three-layer elliptic body, the external elliptic boundary of which is loaded by normal point force and the internal boundary is stress-free and the layers of which are in rigid or sliding contact with one another, by an appropriate choice of layer thickness and arrangement of the layers relative to one another we can obtain a sufficiently uniform distribution of normal displacements on the internal elliptic boundary. Numerical solutions are obtained by the boundary element method and the related graphs are constructed. For the two-layer ellipse, exact and approximate solutions of the same problem are obtained respectively by the method of separation of variables and by the boundary element method. The results obtained by both methods are compared and the conclusion as to the reliability of the numerical boundary element method is made.

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he main purpose of this work is construction of the mathematical theory of elastic plates and shells, by means of which the investigation of basic boundary value problems of the spatial theory of elasticity in the case of cylindrical do­ mains reduces to the study of two-dimensional boundary value problems (BVP) of comparatively simple structure. In this respect in sections 2-5 after the introductory material, methods of re­ duction, known in the literature as usually being based on simplifying hypotheses, are studied. Here, in contradiction to classical methods, the problems, connected with construction of refined theories of anisotropic nonhomogeneous plates with variable thickness without the assumption of any physical and geometrical re­ strictions, are investigated. The comparative analysis of such reduction methods was carried out, and, in particular, in section 5, the following fact was established: the error transition, occuring with substitution of a two-dimensional model for the initial problem on the class of assumed solutions is restricted from below. Further, in section 6, Vekua's method of reduction, containing regular pro­ cess of study of three-dimensional problem, is investigated. In this direction, the problems, connected with solvability, convergence of processes, and construction of effective algorithms of approximate solutions are studied.

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One nonlinear integro-differential system with source terms is considered. The model arises at describing penetration of a magnetic field into a substance. Large time behavior of solution of the initial-boundary value problem is given. Corresponding semi-discrete finite difference scheme is studied as well.

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In this paper, by using projective-variational discrete method, we solved approxiamately some BVPs for thin walled elastic structures corresponding to justifying mathematical models of Kirchhoff-von Kármán-Reissner-Midlin type refined theories.

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his article is dedicated to approximate solution of two-point boundary value problems for linear and nonlinear normal systems of ordinary differential equations. We study problems connected with solvability, construction of high order finite difference and finite sums schemes, error estimation and investigate the order of arithmetic operations for finding approximate solutions. Corresponding results refined and generalized well-known classical achievements in this field.

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In this article elastic cusped symmetric prismatic shells (i.e., plates of variable thickness with cusped edges) in the zero approximation of I.Vekua's hierarchical models is considered. 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 in the case of harmonic vibration. The approach works also for non-symmetric prismatic shells word for word.

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The convergence of a finite element scheme approximating a nonlinear system of integro-differential equations is proven. This system arises in mathematical modeling of the process of a magnetic field penetrating into a substance. Properties of existence, uniqueness and asymptotic behavior of the solutions are briefly described. The decay of the numerical solution is compared with both the theoretical and finite difference results.

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.In this paper elastic cusped symmetric prismatic shells (i.e., plates of variable thickness with cusped edges) in the N-th approximation of Vekua’s hierarchical models are considered. 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 in the case of harmonic vibration. The classical and weak setting of the BVPs in the case of the N-th approximation of hierarchical models 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 and its connection with weighted Sobolev spaces is established.

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In this article we study solution spaces of Carleman–Bers–Vekua equations with coefficients from the space L_p^{loc}(C), p > 2. In particular, we investigate such solutions which satisfy certain additional asymptotic conditions and calculate the dimensions of corresponding subspaces of these solution spaces.

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This paper deals with analytic solutions for a sufficiently wide class of systems of partial differential equations degenerate in one point.

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In this paper, the functional spaces of generalized analytic functions induced by irregular Carleman–Bers–Vekua equations and related boundary-value problems are studied.

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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 the transport sector in Tbilisi have been investigated. Using mathematical simulation, the concentration distribution of harmful substances, NOx, at Rustavely Avenue, the crossroad of King David Agmashenebeli and King Tamar Avenue, where traffic is congested, have been studied. Some results from the numerical calculations are presented

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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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The limiting distribution of the integral square deviation of a kernel-type nonparametric estimator of the Bernoulli regression function is established. The criterion of testing the hypothesis about the Bernoulli regression function is constructed. The question as to its consistency is studied. The power asymptotics of the constructed test is also studied for certain types of close alternatives.

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The nonparametric estimation of the Bernoulli regression function is studied. The uniform consistency conditions are established and the limit theorems are proved for continuous functionals on C[a, 1 − a], 0 < a < 1/2.

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The study of non-separable extensions of Borel measures in infinite-dimensional topological groups or in topological vector spaces is of special interest. There is a rather developed methodology which allows to investigate different aspects of the above-mentioned topic. Here we would like to consider some types of non-separable σ-finite measures from the point of view of the concept of measurability of real-valued functions with respect to certain classes of measures.

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The difference equation is considered. Necessary conditions are established for the above equation to have a positive solution. In addition oscillation criteria of new type are obtained.

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In this paper some aspects related to the crystal growth are considered. This process is described by non-linear reaction-diffusion equation with the specific initial-boundary conditions. Consequently, the first boundary-initial problem for non-linear reaction-diffusion equation is investigated analytically in the small time-interval by means of integral equations and finite-difference schemes. The approximate solutions are given by means of new absolutely stable explicit finite-difference schemes. The cases of cylindrical, cubical and hexagonal type single crystal growth are considered. Also, in some special cases the effective solutions are obtained. This result is applied to the description of diamond crystal growth. For the non-linearity of the second order we have introduced some special parameters and obtained new types of the approximate solutions for the pyramidal type crystal growth. For the case of nanowires (1D nanocrystals) growth we consider the linear reaction-diffusion equation with the appropriate initial-boundary conditions. The approximate solutions are given by means of new absolutely stable explicit finite-difference schemes. In the case of nanoneedles the effective solutions are obtained. As an example for growth of germanium nitric nanocrystals is considered.

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The present paper is devoted to construction of differential hierarchical models for elastic prismatic shells of variable thickness with microtemperatures. To this end, Vekua's dimension reduction method, based on the Fourier-Legendre expansions, is applied to basic equations of the linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures. The special emphasis is placed on cusped prismatic shells.

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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 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. Besides we construct hierarchical models, when on the one face surface conditions (i) and on the other one conditions (ii) are known and also models, when on the upper face surface stress vector and on the lower face surface displacements and vice versa 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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In the present work, using absolutely and uniformly convergent series, the 2D boundary value problems of statics of the linear theory of thermoelasticity with microtemperatures for an elastic circle are solved explicitly. The question on the uniqueness of a solution of the problem is investigated.

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In this article we consider the unsteady flow of viscous incompressible electrically conducting fluid in an infinitely long pipe placed in an external uniform magnetic field perpendicular to the pipe axis. It is stated that the motion is created by applied at the initial time in constant longitudinal pressure fall. The exact general solution of problem is obtained.

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Conditional Bayesian task of testing many hypotheses is stated and solved. The concept of conditionality is used for the designation of the fact that the Bayesian task is stated as a conditional optimization problem where the probability of one-type error is restricted and, under such a condition, the probability of second-type error is minimized. The offered statement gives the decision rule which allows us not to accept any hypothesis if, on the basis of the available information, it is impossible to make a decision with the set significance level. In such a case, it is necessary to ensure the additional information in the form of additional observation results or a change in the significant level of hypotheses testing. These properties make our statement more general than the usual statement of the Bayesian problem which is a special case of the one offered and improve the reliability of the made decision. The calculation results completely confirm the results of theoretical investigations.

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The present paper is devoted to a model for elastic layered prismatic shells. Using I. Vekua’s dimension reduction method hierarchical models for elastic layered prismatic shells are constructed. For each layer we construct hierarchical models assuming to be known stress vector components on the face surfaces of the layered body (structure) under consideration, while we calculate the values of $X_{ij}$ and $u_i$ on the interfaces from their FourierLegendre expansions there. So, we get coupled governing systems for the whole structure in the projection of the structure. Analogously, hierarchical models can be constructed, when on the face surfaces of the prismatic body displacements or mixed stress and displacement vector components are assumed to be known. For the sake of simplicity we consider the case of two plies, in the zeroth approximation (hierarchical model).

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In this paper we consider the relations between the Riemann–Hilbert monodromy problem and the matrix Riemann–Hilbert boundary-value problem with piecewise continuous coefficient and construct the so-called canonical matrix for the boundary-value problem for a piecewise continuous matrix-function. The formula for the calculation of the index is also obtained. D

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In this paper, the main properties of the partial indices of the Riemann boundary-value problem are considered. This important invariant point of view gives a modern approach to two central problems of complex analysis.

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In this article we study solution spaces of Carleman–Bers–Vekua equations with coefficients from the space, $L_p^loc(C)p > 2$. In particular, we investigate such solutions which satisfy certain additional asymptotic conditions and calculate the dimensions of corresponding subspaces of these solution spaces.

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Formulas of propositional logic are proved as algorithmic processes.It is estabilished that all thesprocesses can be completed and their final valies are one of the following halting constants. T(a valid formula), F(an incconsistent and S (an indefinite formula )

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The present paper is devoted to 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 results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulas in the case under consideration are obtained, basic boundary value problems are formulated, and uniqueness theorems are proved. The fundamental matrix of solutions for the governing system of the model and the corresponding single and double layer thermoelastopotentials are constructed. Properties of the potentials are studied. Applying the potential method, for the first and second boundary value problems, we construct singular integral equations of the second kind and prove the existence theorems of solutions for the bounded and unbounded domains.

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In the paper the turbulent diffusion equation in the axi-symmetric case with the appropriate initial condition is considered. The approximate solution is obtained by means of the stable finite-difference schemes. The numerical example is given.

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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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In the present paper we consider the following Satisfaction Problem of Consumers Demands (SPCD): The supplier must supply the measurable system of the measure mk to the k-th consumer at time tk for 1 ≤ k ≤ n. The measure of the supplied measurable system is changed under action of some dynamical system; What is a minimal measure of measurable system which must take the supplier at the initial time t = 0 to satisfy demands of all consumers ? In this paper we consider Satisfaction Problem of Consumers Demands measured by ordinary “Lebesgue measures” in R∞ for various dynamical systems in R∞. In order to solve this problem we use Liouville type theorems for them which describes the dependence between initial and resulting measures of the entire system.

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It is proved that the duality principle between the measure and category is valid with respect to the sentence P defined by: For every two Polish groups G_1 and G_2 , and for every Haar null set Y ⊂ G_1 we have (∀X)(X ⊆ G_2 → Y × X is Haar null in G_1 × G_2). By using that approach which differs from the approach of M. P. Cohen (2012), it is shown that every infinite product of unimodular Polish groups that are not compact decomposes into the disjoint union of a Haar null set and a meagre set, which gives a partial positive answer to a question of Darji (2011). We show that Baker's (2004) product of Haar measures defined on the product of unimodular Polish groups that are not compact is concentrated on a meagre set which is not covered by a countable family of compact sets. It is shown that a question from David Fremlins problem list (2012) asking whether one can simplify the definition of a Haar null set in R by leaving out the Borel set is independent of ZF + DC theory. Similar result is obtained for a question of M. Elekes and J. Stepráns (2012) asking whether there exists an atomless singular Borel probability measure in R which reflects the positive measure of every subset in R. By using Erdös-Sierpiski duality principle, it proved that dual of the Sierpiski set answers negatively on a certain topological analogue of M. Elekes and J. Stepráns (2012). As consequence, we get that Bartoszynski's(2002) and Burke-Miller's (2005) result is independent of ZF + DC theory.

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By using results from a paper [G.R. Pantsulaia, On ordinary and standard Lebesgue measures on R ∞ , Bull. Pol. Acad. Sci. Math. 57 (3-4) (2009), 209–222] and an approach based in a paper [T. Gill, A.Kirtadze, G.Pantsulaia , A.Plichko, The existence and uniqueness of translation invariant measures in separable Banach spaces, Functiones et Approximatio, Commentarii Mathematici, 16 pages, to appear ], a new class of translation-invariant quasi-finite Borel measures (the so called, ordinary and standard "Lebesgue Measures") in an infinite-dimensional separable Banach space X is constructed and some their properties are studied in the present paper. Also, various interesting examples of generators of two-sided (left or right) shy sets with domain in non-locally compact Polish Groups are considered.

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By using Martin Axiom, we prove that σ-ideals of Preiss-Tiser generalized shy sets in a Polish topological vector space, Mankiewicz generalized shy sets and Baker gen-eralized shy sets in the topological vector space of all real-valued sequences equipped with Tychonoff topology are closed under an operation taking a union fewer than c elements of the above mentioned σ-ideals of null sets.

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We consider one-dimensional families of bicentric polygons with the fixed incircle and circumcircle. The main attention is paid to the topology of moduli spaces of associated linkages and to the extremal values of area of bicentric n-gons. For even n we establish that moduli spaces of bicentric ngons have singular points of quadratic type and give an exact upper estimate for the number of singular points. We also indicate certain restrictions on the possible values of Euler characteristics of moduli spaces and discuss its possible changes in families of bicentric polygons. For n=6, 8 we give an estimate for the number of critical points of area in a family of bicentric n-gons and describe the shape of extremal polygons. Moreover, we calculate the mean value of area for n=3. A number of the other results in a few concrete cases are also established and two plausible conjectures are formulated

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We consider one-dimensional families of bicentric polygons with the fixed incircle and circumcircle. The main attention is paid to the topology of moduli spaces of associated linkages and to the extremal values of area of bicentric n-gons. For even n we establish that moduli spaces of bicentric ngons have singular points of quadratic type and give an exact upper estimate for the number of singular points. We also indicate certain restrictions on the possible values of Euler characteristics of moduli spaces and discuss its possible changes in families of bicentric polygons. For n=6, 8 we give an estimate for the number of critical points of area in a family of bicentric n-gons and describe the shape of extremal polygons. Moreover, we calculate the mean value of area for n=3. A number of the other results in a few concrete cases are also established and two plausible conjectures are formulated

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Properties of some nonlinear partial differential and integro-differential diffusion models are studied.

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Machine translation from any language into another requires composition of a word-form by giving its unchanged part and morphological categories. Besides this, for distance studying Natural language and make assistant computer mean for teaching the morphology of some languages to secondary school pupils , it is necessary by using unchanged part of given word to get by the computer all grammatically right word-forms. For solving all these problems, we have developed the software, by which it is possible to solve mentioned problems for languages like Georgian. Besides this, for solving such problems of artificial intelligence, which requires composing of natural language’s word-form by using the information defining this word-form, it is convenient to use the software developed by us.

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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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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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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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This paper is concerned with the linear theory of thermoelasticity with microtemperatures for homogeneous and isotropic solids. We consider the problem of equilibrium of a spherical ring and establish the solution of the Dirichlet boundary value problem (BVP).

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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.

Variation formulas of solution are obtained for a nonlinear controlled delay functional-differential equation with respect to perturbations of initial moment, constant delays, initial vector, initial functions and control function. The effects of delay perturbations and the mixed initial condition are discovered in the variation formulas.

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The theorems of the optimal element existence are proved for a two-stage nonlinear optimal problem with constant delays in phase coordinates and with general boundary conditions and a functional. An element implies the collection of initial and switching moments, an initial function and vector, delay parameters, a control function and a final moment.

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The paper presents the numerical investigation of some orographic effects in the troposphere taking place in conditions of nonstationarity of large-scale undisturbed background air flow. With this purpose a 3-D hydrostatic nonstationary model of meso-scale atmospheric processes is used. The upper boundary of the calculated domain is simulated by the free surface and on the lower boundary the condition of slipping of air particles has been used along the relief. The problem is solved numerically by the two-step Lax-Wendroff method. Performed numerical experiments in case of both model and real relief of Georgia have promoted some regularities of orographic effects caused by the non-stationary character of the background flow

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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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One nonlinear integro-differential equation with source terms is considered. The model arises at describing penetration of a magnetic field into a substance and is based on Maxwell’s system. Existence, uniqueness and large time behavior of solutions of the initial-boundary value problem as well as semi-discrete scheme is studied. More wide class of nonlinearity is considered than one has been already investigated in construction of semidiscrete analogue.

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The present article deals with special thermoelastic equilibrium of the rectangular parallelepiped; some non-classical thermoelasticity problems are stated and analytically solved. In particular, in the Cartesian system of coordinates thermoelastic equilibrium of an isotropic homogeneous rectangular parallelepiped is considered. Symmetry or antisymmetry conditions are defined on four lateral facets of the parallelepiped, while the remaining upper and lower facets are free of stress. The problem is to define the temperature on the upper and lower facets of the parallelepiped, so that normal displacements or tangential displacements on these facets would take a priori defined value (note that since zero values have been already defined on the upper and lower facets, on each of these facets instead of three conditions four or five conditions should be satisfied). It should be emphasized that at the end of the paper a three-dimensional thermal effect is stated, similar to Muskhelishvili's two-dimensional thermal effect.

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In the paper 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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The Stokes equation connected with the fluid flow over the axisymmetric bodies in a cylindrical area is considered. The equation is studied in a moving coordinate system with the appropriate boundary conditions. Effective formulas for the velocity components are obtained. The graphs of the velocity components and velocity profile are plotted.

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We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (G, +) for which the quotient group G/G0 is uncountable, where G0 denotes the torsion subgroup of G.

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ne of the central problems in measure theory is concerned with proper extensions of measures. This problem has various aspects: purely set theoretical, algebraic, topological. The present article is devoted to some open questions which are closely connected with the general measure extension problem and with the existence of sets nonmeasurable with respect to nonzero σ-finite invariant (quasiinvariant) measures

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Some additive properties of the following three families of “pathological” functions are briefly discussed: continuous nowhere differentiable functions, Sierpiński-Zygmund functions, and absolutely nonmeasurable functions.

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It is shown that well-orderings are either negligible or nonmeasurable with respect to the completions of σ-finite product measures. Several consequences of this fact are discussed in the light of some classical problems formulated by Hilbert, Lebesgue and Luzin.

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Assuming Martin's Axiom, it is proved that every function acting from the real line ℝ into itself is representable as a sum of two absolutely nonmeasurable injective functions. A similar result is obtained for any endomorphism of the additive group ℝ.

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There is discussed some pathological homomorphisms of uncountable commutative groups, more precisely we discuss this question weather there are ultimately bad homomorphisms from an uncountable commutative group (G, +) into R and describe all those commutative groups (G, +) for which such homomorphisms exist.

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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 imultaneously 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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In the present paper 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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In this present paper is suggested the method of a small parameter for I. Vekua’s and Koiter-Naghdi’s non-shallow shells.

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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 case of the shear flows the operators of linear problem are non-selfadjoint, and the corresponding Eigen functions – nonorthogonal. Thus, canonical – modal approach is of less use studying such motions. Non-modal mathematical analysis becomes more adequate for such problems. On the basis of non-modal approach, the equations of dynamics and the energy transfer of IGW disturbances in the ionosphere with a shear flow is obtained. Exact analytical solutions of the linear as well as the nonlinear dynamic equations of the problem are built. The increment of shear instability of IGW is defined. It is revealed that the transient amplification of IGW disturbances due time does not flow exponentially, but in algebraic – power law manner. 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 an order of magnitude) own amplitude and energy. With amplitude growth the nonlinear mechanism turns on and the process ends with self-organization of nonlinear solitary, strongly localized IGW vortex structures (the monopole vortex, the transverse vortex chain or the longitudinal vortex street). Accumulation of these vortices in the ionospheric medium can create the strongly turbulent state.

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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 case of the shear flows the operators of linear problem are non-selfadjoint, and the corresponding Eigen functions - nonorthogonal. Thus, canonical - modal approach is of less use studying such motions. Non-modal mathematical analysis becomes more adequate for such problems. On the basis of non-modal approach, the equations of dynamics and the energy transfer of IGW disturbances in the ionosphere with a shear flow is obtained. Necessary conditions of instability of the considered shear flows are obtained. The increment of shear instability of IGW is defined. Exact analytical solutions of the linear as well as the nonlinear dynamic equations of the problem are built. 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. With amplitude growth the nonlinear mechanism of self-localization turns on and the process ends with self-organization of nonlinear solitary, strongly localized IGW vortex structures. Therefore, a new degree of freedom of the system and accordingly, the path of evolution of disturbances appear in a medium with shear flow. Depending on the type of shear flow velocity profile the nonlinear IGW structures can be the pure monopole vortices, the transverse vortex chain or the longitudinal vortex street in the background of non-uniform zonal wind. Accumulation of these vortices in the ionosphere medium can create the strongly turbulent state.

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A corresponding model system of nonlinear dynamic equations for the lower ionosphere has been constructed in order to study the generation and further nonlinear dynamics of internal gravity wave (IGW) structures in a dissipative ionosphere in the presence of a nonuniform zonal wind (shear flow). The criterion for the development of the IGW shear instability in the ionosphere has been obtained.

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Let D be an infinite Diophantine set in the Euclidean space R^n, where n≥ 1. Then all points of D are collinear.

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Some properties of discrete point-sets in Euclidean space are studied and their connections with certain combinatorial invariants of associated continuous geometric images are indicated.

Software tools developed for computer realization of syntactic, semantic, and morphological models of natural language texts, using rule based programming. The tools are efficient for a language, which has free order of words and developed morphological structure like Georgian. For instance, a Georgian verb has several thousand verb-forms. It is very difficult to express rules of morphological analysis by finite automaton and it will be inefficient as well. Resolution of some problems of full morphological analysis of Georgian words is impossible by finite automaton. Splitting of some Georgian verb-forms into morphemes requires non-deterministic search algorithm, which needs many backtrackings. To minimize backtrackings, it is necessary to put constraints, which exist among morphemes and verify them as soon as possible to avoid false directions of search. Software tool for syntactic analysis has means to reduce rules, which have the same members in different order. We used the tool for semantic analysis as well.Thus, proposed software tools have many means to construct efficient parser, test and correct it. We realized morphological and syntactic analysis of Georgian texts by these tools. In presented article, we describe the software tools and its application for Georgian language .

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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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 one basic boundary value problem (BVP) 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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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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Electrostatic acoustic nonlinear periodic (cnoidal) waves and solitons are investigated in unmagnetized pair-ion plasmas consisting of the same mass ion species with different temperatures. It is found that the temperature difference between negatively and positively charged ions appropriates the dispersion property to linear acoustic waves and this difference has a decisive role in nonlinear dynamics as well. Using a reductive perturbation method and appropriate boundary conditions the Korteweg–de Vries equation is derived. Both cnoidal wave and soliton solutions are discussed in detail. In the special case, it is revealed that the amplitude of a soliton may become larger than what is allowed by the nonlinear stationary wave theory, which is equal to the quantum tunneling by a particle through a potential barrier effect. The serious flaw in the results obtained for ion acoustic nonlinear periodic waves by Yadav et al (1995 Phys. Rev. E 52 3045) in two-electron temperature plasmas and Chawla and Misra (2010 Phys. Plasmas 17 102315) in electron–positron–ion plasmas is also pointed out.

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The Uniform convergence of double Fourier-Legendre series of function of bounded Harmonic variation and bounded partial Λ-variation are investigated.

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Nonlinear dynamics of coupled internal-gravity (IG) and alfvén electromagnetic planetary waves in the weakly ionized ionospheric E-layer is investigated. Under such coupling new type of alfvén waves is revealed. It is shown that such short wavelength turbulence of IG and alfvén waves is unstable with respect to the excitation of low-frequency and large-scale perturbations of the zonal flow and magnetic field. A set of coupled equations describing the nonlinear interaction of coupled IG and alfvén waves with zonal flows is derived. The nonlinear mechanism of the instability is driven by the advection of vorticity and is based on the parametric excitation of convective cells by finite-amplitude coupled IG and alfvén waves leading to the inverse energy cascade toward the longer wavelength. The growth rates of the corresponding instability and the conditions for driving them are determined. The possibility of generation of the intense mean magnetic field is shown.

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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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In the present work we consider non-classical mathematical models describing pollution transfer and diffusion of various mixtures in the water streams, and other problems dealing with engineering, physics, and ecology. The model considered presents pluri-parabolic equations with classical boundary and nonlocal initial conditions. The uniqueness issue of solution is studied. The iteration method reducing calculation of a solution of a stated non-classical problem on that of a classical one is developed for calculation of a solution. Convergence of iteration process is evaluated. © 2012 by Nova Science Publishers, Inc. All rights reserved.

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The mathematical model describing the oil pollution transfer process in the seas is considered in the present work. The model consists of advective transference and turbulent diffusion non-stationary linear equation with initial and non-classical (nonlocal) boundary conditions. Correctness of mathematical model is studied. It is shown, that solution of mentioned non-classical initial-boundary value problem can be reduced on the solution of Dirichlet problem with classical boundary conditions via certain iteration algorithm. Using the decomposition method the averaged additive difference scheme of parallel calculation is constructed for numerical realization of developed algorithm on the computer; stability and convergence issues are studied.

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Parabolic regularization of the one-dimensional analog of Maxwell’s system which describes process of penetration of the magnetic field into a substance is studied. Results of numerical experiments based on finite difference schemes are given.

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Galerkin finite element method for the approximation of a system of nonlinear integro-differential equations describing the process of penetrating of a magnetic field into a substance is studied. Initial-boundary value problem with mixed boundary conditions is investigated. The convergence of the finite element scheme is proved. The rate of convergence is given too.

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In the present paper the distributions of petroleum and mineral oil into the soils in case of their emergency spilling at accidents pipelines failure is studied by numerical modelling. Numerical model describing mineral oil distributions into the soils was elaborated. The behavior of the infiltration and diffusion parameters were studied. Experimental results, obtained by numerical calculations of the model of the distributions of petroleum and mineral oil into the soils are presented. Science Publishers, Inc. All rights reserved.

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In the present article finite-difference solution of antiplane 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 numerical 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 approach with the results of theoretical investigations

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The present paper devoted to investigation of special decomposition methods for stationary and nonstationary problems in the case of partial differential equations. Based on the proposed decomposition method are constructed parallel computing algorithms. We consider also the parallel version of the Schwarz alternating method, based on area decomposition. The independent problem is the solution of difference problems representing itself the system of linear or nonlinear algebraic equations. In this paper is considered both synchronous and asynchronous parallel iterative methods for the numerical solution of nonlinear equations and systems of equations.

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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Systematic investigation of Elliptic systems on Riemann surface and new results from the theory of generalized analytic functions and vectors are presented. Complete analysis of the boundary value problem of linear conjugation and Riemann-Hilbert monodromy problem for the Carleman-Bers-Vekua irregular systems are given.

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n the present paper one-dimensional hierarchical model for general elliptic system defined on three-dimensional domain is constructed. The reduced one-dimensional boundary value problem is studied and the uniqueness and existence of its solution in suitable weighted Sobolev spaces is proved. Moreover, we prove convergence of the sequence of vector-functions of three variables restored from the solutions of the constructed one-dimensional boundary value problems to the exact solution of the original problem and estimate the rate of convergence.

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Some special nonclassical problems of thermoelasticity are formulated and solved in the generalized cylindrical coordinate system (Cartesian, circular cylindrical, cylindrical elliptic, cylindrical parabolic and cylindrical bipolar coordinate systems). The nonclassical formulation of a thermoelastic problem means that, given zero stresses on the upper and lower plane boundaries of an elastic body and a temperature disturbance on the lower boundary, it is required to choose a temperature value on the upper boundary such that normal displacements to the plane boundaries would obey certain conditions within the body.

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A two-dimensional boundary value problem of elastic equilibrium of a plane-deformed infinite body with an elliptical opening is studied. A part of the opening is fixed and from some points of the unfixed part of the cylindrical boundary there come curve line 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.

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We investigate a three-dimensional mixed initial-boundary value problem arising in the dynamical solid–fluid interaction theory. A 3D domain occupied by an incompressible and viscous Stokes fluid may be bounded or unbounded, while a domain occupied by an elastic body immersed in the fluid is assumed to be bounded. On the basis of the results obtained for an elastic inclusion of an arbitrary geometrical shape, we derive a special model and analyze in detail the case when an elastic inclusion is a thin prismatic shell, in particular a plate of variable thickness. Here, we apply I. Vekua’s dimension reduction method in the elastic part which reduces 3D solid–3D fluid interaction problems to the 2D solid–3D fluid interaction problems and which is important from the practical point of view since it takes into account intrinsic differences of the dimensions of solid and fluids part. The main goal of the paper was to study the strain–stress state of the elastic part under the action of the Stokes flow. The corresponding mechanical model is described mathematically as a transmission problem for the linear Stokes system and the dynamical Lamé equations in the corresponding domains with appropriate initial conditions along with the boundary and interface conditions. For 3D solid–3D fluid dynamical interaction problems, we prove the uniqueness and existence theorem. Further, considering the case when the elastic inclusion is a thin prismatic shell of variable thickness, we apply the N = 0 approximation of Vekua’s hierarchical model for the elastic field in the solid part. In contrast to the usual classical streamline conditions, in the case under consideration, on the cut surface, there appear non-local boundary conditions. We prove unique solvability of the non-classical boundary value problem that leads to the existence results for the solid–fluid interaction problem with a thin elastic inclusion.

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