The purpose of the article is to determine the role of the middle class in political democracy and economic democracy in Ukraine. The role of the Ukrainian middle class in the revolutionary events of 2004 and 2013-2014 – the Ukrainian Maidans, the influence of the middle class on the results of the parliamentary elections in Ukraine and its presentation in the Verkhovna Rada of Ukraine are noted. The pendulum-like oscillation of self-identification of the middle class between "service to the oligarchs" and "resistance to democracy," the basis of civil society, is noted. The emphasis is placed on the fact that democracy is the rule of the middle class, and the principle of equal freedom is at the core of economic democracy. The compromise between capitalism and democracy is the need to distribute the public good under the supervision of political democracy in the interests of the middle class. Methodology. The theory of economic democracy was used to study the essence and characteristics of the middle class in Ukraine. By means of sociological methods of empirical research middle class identification by indicators of income, education, labor (employment), consumer behavior is given. Differences of "European" middle class from Ukrainian middle class are marked. In connection with the fact that the level of income of the Ukrainian middle class does not correspond to the indicators of EU countries, the necessity of application of subjective indicators to determine self-identification of middle class representatives in Ukraine is substantiated. The definition of the middle class from the opposite, i.e., what the middle class is not (not the oligarchs, not the authorities, not the working class, etc.) is also applied. Results. The trajectories of the middle class in Ukraine have been defined – from active participation in the Ukrainian revolutions to the role of a servant of the oligarchic regime. The middle class is well positioned to establish a regime of political democracy and to oppose the oligarchic-lumpen alliance. What hinders this is the insufficient activity of the middle class during election campaigns and especially after them. The oligarchs' established monopoly in the economy and media space allows for effective election campaigns in the mass media and the financing of political parties. This nullifies attempts of the middle class to establish political democracy in Ukraine. The growing role of the new middle class – intellectuals and representatives of the creative industries – has been noted. The self-identification factors of the middle class include a sense of justice, responsibility for the future and the practice of democratic values. According to sociological research, representatives of the Ukrainian middle class strive for democratic changes, overcoming corruption, deconstruction of the oligarchic regime, and implementation of market reforms. They have higher education, are interested in art, go to restaurants, and can sometimes afford vacations abroad. The political parties that declared support for the middle class and defined it as a target group in the elections included the "Team of Winter Generation," "Veche," "Self Reliance", and "Voice". The first two were perceived as oligarchic projects, while the last two parties made it to parliament, respectively, in the eighth and ninth convocations. Separately, the 2019 elections, in which the voice of the middle class became truly powerful, are examined separately. The "old faces," representatives of the post-Soviet oligarchy, and bureaucrats lost in the majoritarian districts. The middle class refused to support those politicians and parties that had become "servants of the oligarchs" in parliament, taking an important step in building not only a political democracy, but also an economic democracy. Practical implications. The results of the study can be used in the process of European integration of Ukraine, the implementation of reforms to develop political democracy as the rule of the middle class and economic democracy as a democratic system of redistribution of public resources. Value/originality. This study of the middle class and economic democracy allows us to understand the economic factors that influence politics, the role of the middle class and its characteristics in the process of establishing political democracy and economic democracy in Ukraine.

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Context. Effective solution of a number of application problems related to transportation, as a rule, depends on the solution of two problems: the correct formation of the composition of the column of technique and the successful choice of the route of its movement. Each of the problems is optimization, the methods of solving which are currently being worked out. Theoretical studies of each of the individual problems and their practical applications indicate their interdependence, which has not yet been fully studied. Practical applications necessitate the development of a suitable scientific and methodological apparatus. Objective. The purpose of this work is development of a method for solving a complex optimization problem of forming a column of vehicles and choosing the route of its movement on a non-stationary road network. Method. The mathematical model of solving the optimization problem of complex formation of the composition of the column of machinery and the choice of its route of motion is proposed. A heterogeneous set was used to describe the array from which the vehicles were selected. A graph was used to describe the road network. As a criterion for the optimality of the complex problem is the minimization of time spent on moving. The peculiarity of the model is to take into account the possibility of dynamically changing the time weights of edges of the graph when implementing the movement of a column of machinery along the chosen route. Based on the use of this model, a method is proposed, which provides a comprehensive choice of the composition of the column of equipment and optimal routes of its movement on a non-stationary road network. Results. The article proposes an algorithm that provides the solution of the optimization problem of complex formation of the composition of machinery column and the choice of its route of motion in terms of time-fixed edges that describe the network of roads. The features of application of the proposed algorithm are given. Using the developed software, the choice of technique from an existing inhomogeneous array and the choice of a route on a graph with a non-stationary time weight of edges was investigated. The example shows the imperfection of decisions regarding the complex formation of the column composition and the choice of its optimal route of travel on a non-stationary network of roads obtained using classical methods. Conclusions. Not taking into account the impact of a possible change in traffic conditions, as evidenced by a change in the time weights of the edges of the graph describing the road network, on the composition of the column of machinery can lead to suboptimality of the obtained solutions using classical methods of forming the composition of the column and finding the shortest route in the graph. The method proposed in this study can be used to obtain the optimum composition of the column and the route, taking into account the change in road conditions during the movement of the column. The obtained results extend the possibilities of the...

д-р техн. наук, доцент, начальник кафедри загальнонаукових та інженерних дисциплін, Національна академія Державної прикордонної служби України імені Богдана Хмельницького, Хмельницький, Україна. Боровик О. В.-д-р техн. наук, професор, начальник навчального відділу, Національна академія Державної прикордонної служби України імені Богдана Хмельницького, Хмельницький, Україна. Боровик Л. В.-канд. психол. наук, доцент, доцент кафедри загальнонаукових та інженерних дисциплін, Національна академія Державної прикордонної служби України імені Богдана Хмельницького, Хмельницький, Україна. АНОТАЦІЯ Актуальність. Важливою передумовою надійної охорони державного кордону України є ефективне виявлення порушників кордону. З цією метою у прикордонному відомстві використовуються різні сучасні системи та комплекси спостереження, окремі прикордонні наряди оснащуються новітніми технічними засобами охорони кордону. Різноманітність можливостей окремих засобів спостереження при їх комплексному використанні робить важливим визначення ефективного розміщення точок спостереження з урахуванням особливостей місцевості. Мета. Метою роботи є розробка методики ефективного розміщення елементів складних систем спостереження прикордонного відомства. Метод. У роботі запропонований евристичний підхід до вирішення задачі визначення ефективного розміщення різних за можливостями технічних засобів спостереження. Розроблена методика ґрунтується на циклічному використанні відомого методу комбінаторної оптимізації розміщення точок спостереження для всіх складових складної системи спостереження прикордонного відомства з коригуванням після кожної ітерації смуги перекриття. При цьому з застосуванням геообробки враховується загороджуючий негативний вплив рельєфу місцевості та можливості різних технічних засобів спостереження в різних умовах. Для оцінки ефективності функціонування складових системи спостереження використовується показник, що відображає частку площі смуги перекриття, на якій забезпечується виявлення цілей. Результати. В результаті використання запропонованої методики отримується раціональна структурна побудова складної системи спостереження органу охорони кордону і множина неконтрольованих точок місцевості. Ця множина може використовуватись при плануванні несення служби підрозділами охорони кордону. Наведений опис алгоритмічної і програмної реалізації запропонованої методики. З використанням розробленого програмного забезпечення проведена оптимізація розміщення трьох типів елементів складної системи спостереження (веж системи оптико-електронного спостереження, мобільних тепловізійних комплексів, постів спостереження). Висновки. Сучасна побудова складних систем спостереження прикордонного відомства у багатьох випадках не є оптимальною. Для підвищення ефективності їх функціонування можливо використати запропоновану методику розстановки елементів цих систем на місцевості. КЛЮЧОВІ СЛОВА: система спостереження, показник ефективності системи спостереження, математичні методи, алгоритм.

The article is devoted to the analysis of the possibilities of using modern applications in the teaching of higher mathematics for future border guard officers of the State Border Guard Service of Ukraine. The article describes the possibilities of the software stand developed on the basis of the mathematical package of the MatLab system, which allows cadets to perform spectral analysis of signals, which is important for solving professional problems of future professional activity. In addition, the article describes the process of the stand and presents its program code. The interface of the developed software stand allows to choose a window for the analysis of a kind of pulses, to receive their images, corresponding radio pulses, and also their spectra. The obtained amplitude spectra allows to compare and state the similarities and differences in the radio signal and signal spectra. In addition, if it is necessary to demonstrate the form of an arbitrary (periodic or non-periodic) signal or a signal with noise, the presented software product allows the cadet to generate a signal depending on its characteristics. The article notes that conducting classes using this software product develops the motivation of cadets to study fundamental disciplines, promotes interest and increases their knowledge and skills in higher mathematics and related disciplines, as the classes have a purposeful informational and emotional impact on future officers, their analytical-synthetic skills and critical thinking develop. In addition, the article considers the features of experimental verification of the classes effectiveness using a program stand to improve the knowledge and skills of cadets in professionally-oriented disciplines, as well as the reliability of the results using mathematical statistics.