The aspects of mathematical modeling of the dynamics of complex processes in complex onboard systems are considered. The structural connections between the elements of modeled systems and their dynamics are formalized using a system of differential equations. Qualitative analysis of the synthesized model allowed establishing new non-local conditions for stable functioning of modeled combined onboard systems. The results obtained make it possible to expand the working areas of technical devices and the devices included in onboard systems.

We consider the problem of stability of nonzero integral manifolds of a nonlinear finitedimensional system of ordinary differential equations whose right-hand side is a periodic vector-valued function of the independent variable containing a parameter. We assume that the system has a trivial integral manifold for all values of the parameter and the corresponding linear subsystem does not possess the property of exponential dichotomy. The aim of this work is to find sufficient conditions for stability, instability, and asymptotic stability of a local nonzero integral manifold. For this purpose, we use the method of Lyapunov functions modified to the problem considered and singularities of the right-hand sides of the system.

The relevance of the article is caused by the rapidly growing importance of qualitative higher education. Higher levels of scientific and technological processes lead to the ever-growing demand for highly-qualified specialists where low-skilled workers can lose their jobs without any stop in the trend to change the usual style of life. Specialists should be able to possess a very important skill, i.e. the skill to adapt to high-tech changes in the working environment. Changing requirements to workers’ professional competencies cause changing the direction of educational systems development forcing them to be modified. One of the new fundamental directions in this modification is the emergence and improvement of educational smart technologies. However, due to the specific character of smart education, the task to assess its quality becomes more complicated creating difficulties primarily for employers. The article offers a new approach to assess the quality of modern education based on the mathematical models proposed by the authors. Methods to build and analyze differential equations as well as methods of the mathematical theory of fuzzy sets are used allowing us to transfer qualitative educational criteria into quantitative ones and further perform their analysis. The model combines key advantages of existing methods to assess the quality of education. The model offered is flexible as it can be adapted to any number of various criteria and factors and can be used as the basis for the system of education quality indicators applicable both for job seekers and employers.

High rates of scientific and technological progress determine the constant increase in demand in the labor market for highly qualified specialists, primarily those engaged in professional activities using IT. In modern conditions, the most relevant skill for graduates of higher educational institutions is the ability to adapt to dynamic changes and improve the information and communication technologies themselves and the means of their implementation. Accordingly, this leads to a significant modernization of the sphere of higher education and the totality of educational processes. In particular, approaches to quality assessment and ways to adequately determine its level are changing. The authors in this article propose to effectively assess the quality of modern education, using mathematical models developed by them. These models are synthesized by the authors based on the theory of fuzzy sets. Which, in turn, allows us to generalize a number of qualitative and quantitative criteria. And, consequently, to ensure the adaptation of models for the implementation of educational processes depending on a variety of identifiable factors. The results of the research carried out by the authors of this article can be very useful for a wide range of representatives of various communities of participants in socio-economic relations.

In this paper, we consider particular cases of hyperbolic parallelograms obtained by transferring characteristic properties of rectangles and squares on the Euclidean plane associated with their diagonals to the Lobachevsky plane. The existence of these quadrangles in the Cayley–Klein model in a circle of the Euclidean plane is proved.

In this paper, we consider the stability problem for nonzero integral manifolds of a nonlinear, finite-dimensional system of ordinary differential equations whose right-hand side is a vector-valued function containing a parameter and periodic in an independent variable. We assume that the system possesses a trivial integral manifold for all values of the parameter and the corresponding linear subsystem does not possess the exponential dichotomy property.We find sufficient conditions for the existence of a nonzero integral manifold in a neighborhood of the equilibrium of the system and conditions for its stability or instability. For this purpose, based of the ideas of the Lyapunov method and the method of transform matrices, we construct operators that allow one to reduce the solution of this problem to the search for fixed points.

In the context of the functioning and evolution of the modern information society, the processes that are actually implemented in the economy, public life, and even politics, in a certain way significantly correlate with the totality of information processes in numerous social networks. Individuals and legal entities, business structures and non-profit organizations, social movements and political parties create multimedia content and post it on social networks, contributing to its active promotion in the virtual world. Various principles, methods and tools are used by the owners of such content to generate active interest in information resources posted on social networks and attract new users. The purpose of such actions is to activate and optimize their real activities, regardless of the sphere of interests. In the article, the authors analyzed a number of aspects that indicate the synchronization of real and virtual processes. Such conclusions were made based on the analysis and processing of a significant amount of statistical information about the functioning of several specific groups of accounts in social networks, in particular in the VKONTAKTE network. A number of models are proposed that allow predicting the implementation of information processes in social networks in the short and medium term and their synchronization with real processes.

The empirical distributions of various statistical indicators that influence the implementation of algorithms for information exchange between users of the social network •Vkontakte• are analyzed. The authors of the article synthesized regression models that can be used to predict the dynamics of the number of users of the social network •Vkontakte•, their activity, as well as other parameters that characterize the functioning and evolution of various communities of computer social networks.

In order to model information dissemination in social networks, a special methodology of sampling statistical data formation has been implemented. The probability distribution laws of various characteristics of personal and group accounts in four social networks are investigated. Stochastic aspects of interrelations between these indicators were analyzed. The classification of groups of social network users is proposed, and their characteristic features and main empirical regularities of mutual transitions are marked. Regression models of forecasting changes in the number of users of the selected groups have been obtained.

To create analytical models for disseminating information in social networks, a special technique has been implemented for selected statistics in the network VKontakte (VK). The statistical features of the change in the number of subscribers to social groups in terms of the pageviews are investigated. The classification of the groups of social network users is proposed; their differences and common patterns of mutual transitions are determined.

Is considered a number of aspects of evolution of Internet business, in particular, assessment of efficiency of this innovation technology in terms of satisfaction of requests and needs of end users is implemented. The mathematical model is synthesized and modeling of set of processes is carried out. The analysis of results of such modeling allowed authors to formulate an entity of the concept “trust level” to information telecommunication technologies and to offer a way of its assessment, by definition of the so-called “index of trust of Internet users”. This index also to some extent reflects priorities of end users and can quite serve as a peculiar indicator of drivers of growth of the Internet in general and the Internet of things in particular.

In the context of the Fourth Industrial Revolution, information processing and information and communication technologies (ICT) are actively used in all areas of socio-economic relations. In fact, they are the driver of the formation and dynamic development of the global information society. Information systems, hardware and software systems, mobile devices and applications today allow you to successfully solve many important tasks of practical activity in the economy, social sphere, medicine, education, management infrastructure of megacities and even in politics. The consequence of the effective use of ICT is the formation of completely new social, economic and political activities of mankind, striving for progress and ensuring its existence in harmony with nature. However, the above-mentioned technological innovations and trends, despite the many positive aspects of creating the conditions for the evolution of society as a whole and each individual, can have a very negative impact on the immediate ICT user, on his well-being, on health and performance. A number of studies of such destructive effects have now been carried out. But, many nuances have not yet been fully explored. The authors believe that it is necessary to conduct a systematic analysis of the relevant processes and develop tools for the rapid identification of negative factors, as well as on a scientific basis to form methodical recommendations for the implementation of the complex measures to protect people from the destructive effects of ICT in the formation of modern information and environmental systems. The authors attempt to organize and classify elements of negative impact of ICT on people and society, to assess risks, to determine the likely consequences.

Для целей изучения динамики информационных обменов в социальных сетях и прогнозирования на этой основе реальных общественных акций предложена классификации групп пользователей сетей на четыре категории: маркетинговые, политические, блогерские, индивидуальные. Изучены эмпирические распределения различных статистических показателей активности названных категорий в сети «ВКонтакте» применительно к коротким (до 100 дней) и длинным (более 100 дней) промежуткам времени. Показано существование устойчивых нормальных, равномерных и экспоненциальных распределений исследованных статистических показателей и статистически устойчивых связей между ними, пригодных для обоснования динамики развития информационных обменов в социальных сетях и дающих возможность строить регрессионные модели для оценки количества участников различных общественных акций, включая протестные движения. Делается вывод, что следующий шаг в исследованиях данного направления должен быть связан с более глубоким изучением факторов, влияющих на распространение информации в социальных сетях. К ним относятся: дальнейшая дифференциация статистических исследований применительно к маркетинговым, политическим и блогерским сообществам; более детальное изучение устойчивости статистических зависимостей на длинных и коротких временных отрезках; сочетание статистического анализа структуры выделенных сообществ с имитационным моделированием динамики переходов между их группами с учетом латентных (ненаблюдаемых) состояний. For the purpose of studying the dynamics of information exchanges in social networks and forecasting real public actions on this basis, the classification of groups of network users into four categories is proposed: marketing, political, blogging and individual. We studied the empirical distributions of various statistical indicators of the activity of these categories in the Vkontakte network in relation to short (up to 100 days) and long (more than 100 days) time intervals. It shows the existence of stable normal, uniform and exponential distributions of the studied statistical indicators and statistically stable relationships between them, which are suitable for substantiating the dynamics of information exchanges in social networks and make it possible to build regression models for estimating the number of participants in various public actions, including protest movements. It is concluded that the next step in research in this area should be associated with a deeper study of the factors that affect the dissemination of information in social networks. These include: further differentiation of statistical research in relation to marketing, political and blogging communities; a more detailed study of the stability of statistical dependencies over long and short time periods; a combination of statistical analysis of the structure of selected communities with simulation of the dynamics of transitions between their groups, taking into account latent (unobservable) states.

Рассматривается задача устойчивости ненулевых интегральных многообразий нелинейной конечномерной системы обыкновенных дифференциальных уравнений, правая часть которой является периодической вектор-функцией по независимой переменной и содержит параметр. Предполагается, что у изучаемой системы имеется тривиальное интегральное многообразие при всех значениях параметра, а соответствующая линейная подсистема не обладает свойством экспоненциальной дихотомии. Целью работы является нахождение достаточных условий устойчивости, неустойчивости и асимптотической устойчивости локального ненулевого интегрального многообразия. Для этой цели применяется метод функций Ляпунова, модифицированный к рассматриваемой задаче и особенностям правых частей изучаемой системы дифференциальных уравнений.

Рассматривается задача устойчивости ненулевых интегральных многообразий нелинейной конечномерной системы обыкновенных дифференциальных уравнений, правая часть которой является периодической вектор-функцией по независимой переменной и содержит параметр. Предполагается, что у изучаемой системы имеется тривиальное интегральное многообразие при всех значениях параметра, а соответствующая линейная подсистема не обладает свойством экспоненциальной дихотомии. Целью работы является нахождение достаточных условий существования в окрестности состояния равновесия системы ненулевого интегрального многообразия, а также условий его устойчивости и неустойчивости. Для этой цели на основе идей метода функций Ляпунова и метода преобразующей матрицы строятся операторы, позволяющие свести решение указанной задачи к поиску их неподвижных точек.

The synchronization of the processes of functioning of all components in socio-economic sphere determines the pace of its development. The education is an integral part of this sphere. Its quality depends on many different factors, primarily on the consistency of the results of educational processes and current demands of the labor market. Currently, there is no scientific interpretation of the notion of “quality of education”, methodology for determining its individual indicators and their integration into the generalized criterion. The purpose of this publication is to substantiate an innovative approach to the interpretation of the above-mentioned concept and to determine its adequate indicators, based on the principle of synchronization between the set of requirements of employers and real competencies of University graduates. Usually, the society evaluates the quality of education by determining the level of demand for specialists in the labor market, as well as their real ability to adapt in changing conditions. To a greater degree, the level of demand depends on the criteria, which are developed by employers for the selection of candidates for vacant positions. The adaptability closely correlates with intellectual potential, innovative behavior, leadership qualities and the ability to think creatively. The authors consider that the above criteria should be synchronized. The authors made an experiment and, using the theory of possibilities, synthetically produced a number of mathematical models, based on the obtained results. These models allowed to formulate and justify an integral quality indicator of the quality of the implementation of educational processes.

The article explores several aspects of the possible existence of informational actions and counteractions in the field of economy. The analysis of similar processes has been made utilizing structure-functional and structure-dynamic modelling. The basic models that allow revealing the essence of the phenomena under analysis are the systems of differential equations that are congruent with the constructions, which have used in the traditional dynamic system. The combination of contextual and subsidiary DFD-diagrams, developed following the technology SADT, allows understanding the functional logic of performed actions. The experimental part of the research had based on the use of imitation platform Analogic and the real statistic data that testified to the dynamic marketing behaviour of a range of business entities. These entities promoted their products in the market using information-communication technologies. The scholars of the study integrated structurefunctional and structure-dynamic patterns of informational actions and counteractions in the field of the economy that have successfully tested. The results of modelling possess a high level of compatibility with the empirical data. It provides adequate prediction and assessment of the possibility of implementing various scenarios of events, and, consequently, in economy management.

Рассмотрены частные случаи гиперболического параллелограмма, которые получены при перенесении на плоскость Лобачевского характеристических свойств прямоугольника и квадрата евклидовой плоскости, связанных с их диагоналями. Доказано существование этих четырехугольников на модели Кэли - Клейна в круге евклидовой плоскости.

The article presents and discusses the results of modeling a wide-band horn antenna with a ridge transition. Based on the model and experimental measurements, an analysis of the set of antenna characteristics is carried out and its modification is proposed. In the modification of the antenna, a radar absorbing material with a thickness of 10 mm is used, which significantly reduces the uneven distribution of the field in the near zone of the antenna. The distribution of the field amplitude for the modernized antenna (with the application of radar absorbing material) has an enlarged region of the “quasiplane” wave. The phase distribution of the field becomes more compact.