: We prove that homogeneous problem for PDE of second order in time variable, and generally infinite order in spatial variables with local two-point conditions with respect to time variable, has only trivial solution in the case when the characteristic determinant of the problem is nonzero. In another, opposite case, we prove the existence of nontrivial solutions of the problem, and we propose a differential-symbol method of constructing them.

Розробленно лiнгвометричний метод алгоритмiчного забезпечення процесiв контент-монiторiнгу для розв'язання задачi автоматичного визначення автора україномовного текстового контенту на основi технологiї статистичного аналiзу коефiцiєнтiв мовної рiзноманiтностi. Проведено декомпозицiю методу визначення автора на основi аналiзу таких коефiцiєнтiв мовлення як лексична рiзноманiтнiсть, ступiнь (мiра) синтаксичної складностi, зв'язнiсть мовлення, iндекси винятковостi та концентрацiї тексту. Проаналiзованi також параметри авторського стилю як кiлькiсть слiв у певному текстi, загальна кiлькiсть слiв цього тексту, кiлькiсть речень, кiлькiсть прийменникiв, кiлькiсть сполучникiв, кiлькiсть слiв iз частотою 1, та кiлькiсть слiв iз частотою 10 та бiльше. Особливостями розробленого є адаптацiя морфологiчного та синтаксичного аналiзу лексичних одиниць до особливостей конструкцiй україномовних слiв/текстiв. Тобто при аналiзi лiнгвiстичних одиниць типу слiв враховувалась належнiсть до частини мови та вiдмiнювання в межах цiєї частини мови. Для цього провадився аналiз флексiй цих слiв для класифiкацiї, видiлення основи для формування вiдповiдних алфавiтно-частотних словникiв. Наповнення цих словникiв в подальшому враховувалися на наступних кроках визначення авторства тексту як розрахунок параметрiв та коефiцiєнтiв авторського мовлення. Для iндивiдуального стилю письменника показовими є саме службовi (стоповi або опорнi) слова, оскiльки вони нiяк не пов'язанi з темою i змiстом публiкацiї. Проведено порiвняння результатiв на множинi 200 одноосiбних робiт технiчного спрямування бiля 100 рiзних авторiв за перiод 2001-2017 рр. для визначення, чи змiнюються i як коефiцiєнти рiзноманiтностi тексту цих авторiв в рiзнi промiжки часу. Виявлено, що для обраної експериментальної бази з понад 200 робiт найкращих результатiв за критерiєм щiльностi досягає метод аналiзу статтi без початкової обов'язкової iнформацiї як анотацiї та ключовi слова рiзними мовами, а також списку лiтератури Ключовi слова: NLP, контент-монiторiнг, стоп-слова, контент-аналiз, статистийний лiнгвiстичний аналiз, квантитативна лiнгвi стика

Асимптотичний метод дослідження резонАнсних режимів неліній них згинАльних коливАнь пружного вАлА Purpose. To develop a method for determining resonant modes of industrial equipment of elastic shaft type, which is widely used in the mining industry, through the study of mathematical model of nonlinear oscillations. Mathematical models of oscillatory systems previously were studied in the literature mainly based on the numerical and experimental approaches. This paper proposes using a combination of the wave theory of motion and asymptotic methods of nonlinear mechanics using special apparatus of periodic functions to investigate the vibrational dynamics of the system and conditions of resonance phenomena in it, as well as to describe the method for determining the resonance curves to increase the margin of safety of industrial equipment. Methodology. Methods for studying resonance amplitudes and frequencies, determining the strength characteristics of equipment are based on the use of asymptotic methods of nonlinear mechanics, wave motion theory and theory of special Ateb-functions. Findings. In this work the conditions of resonance amplitude and frequency depending on the system parameters were obtained analytically for these nonlinear vibrational systems of elastic shaft and the overall method for determining the resonance curves was described. Originality. For the first time a complete analysis of the impact of physical, mechanical and geometrical factors of the dynamic pricess on the resonant frequency and amplitude in systems such as elastic shaft was conducted on the UDC 534.1

Abstract. We establish the necessary and sufficient conditions of existence of nontrivial quasi-polynomial solutions of the problem in a layer for homogeneous partial differential equation with s + 1 variables of second order in time variable and generally infinite order in other s (spatial) variables with Dirichlet boundary conditions in time. We apply the differential-symbol method for constructing such quasi-polynomial solutions. We also give examples of problems for which we construct other solutions besides of quasi-polynomial ones.

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Abstract. The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable.

517.95We propose an operational method of solving the Cauchy problem for partial differential equations and systems of partial differential equations. We demonstrate its superiority to the known methods. We give a number of illustrative examples of applications of the method.The efforts of many scholars have amassed a large number of methods of solving both partial differential equations and initial-value problems connected with them. Among such methods are the well-known operator method [ 1,7,16], the method of integral transforms [22], the method of initial functions [6] and others. The application of each such method to a specific Cauchy problem, wherever this can be done, makes it possible to obtain different presentations of its solution. To the methods of solving the Cauchy problem mentioned above one may add the operational method [9-11, 13, 19], which is based on generalized separation of variables [9,12,14]. This method is in a certain sense "dual" to the operator method whose starting point is the classical operational calculus.The classical operational calculus is a method of solving ordinary differential equations in which the symbol p = d/dx is manipulated like an ordinary variable. For the required solution one obtains expressions of the formwhere f(x) is a known function. Such a calculus was first proposed in [4], and then greatly developed in [25]. The basic problem in this calculus is obviously to find an effective interpretation of the symbols g(p). We note that a complete mathematical justification of the symbolic calculus was not given in these papers. The justification was given much later, when a connection was established between this calculus and the Laplace transform. What is important in this topic was the algebraic approach to justifying the operational calculus proposed in [17]. The technique of infinite-order differential operators which is inherent in the classical symbolic calculus, has recently been successfully applied to study Cauchy problems for partial differential equations (see, for example, [7,8] and the bibliography they contain).The formalism of infinite-order differential operators has frequently been applied in recent years to study specific problems of mechanics [3,6,15,20]. In these papers the solutions of initial-and boundary-value problems are given using the expressions (1) where x e ~", p = (Pl , P2 ..... P,), Pj = 6~/~ , J = 1, n.The operational method of generalized separation of variables considered in the present paper makes it possible to solve the Cauchy problem using expressions of the form where fj are known functions (initial functions or right-hand sides of the equations), and # e C'.The proposed operational method of generalized separation of variables has certain advantages over those mentioned above, and the present paper is devoted to a description of those advantages. The characteristics of the method can be illustrated using the examples of Cauchy problems for the equations of the direct and inverse heatconduction problems, wave equations for the ca...

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