Integral Representation and Explicit Formula at Rational Arguments for Apostol–Tangent Polynomials

The Fourier series expansion of Apostol–tangent polynomials is derived using the Cauchy residue theorem and a complex integral over a contour. This Fourier series and the Hurwitz–Lerch zeta function are utilized to obtain the explicit formula at rational arguments of these polynomials. Using the Lipschitz summation formula, an integral representation of Apostol–tangent polynomials is also obtained.

Assessing the Need of Using Artificial Intelligence within Legal Practice

The objective of the article is to analyze the content and particularities of the use of artificial intelligence in legal practice. Historical and legal, epistemological, and comparative scientific methods are used. It has been clarified that artificial intelligence is the ability of a computer (machine) to simulate human intelligence while solving certain tasks. This type of intelligence is also designed to solve complex integral tasks related to the collection, processing, storage, generalization, and other actions with information. It has been argued that the introduction of artificial intelligence technologies in all spheres of public life requires adequate legal regulation of all aspects of their use. The main guidelines for the development of artificial intelligence in legal practice have been identified, namely: development of innovative cybersecurity systems; determination of the list of administrative services, the decisions of which can be made by automated systems using special information processing algorithms; development of digital systems for the identification and verification of persons; use of artificial intelligence technologies to detect illegal activities in computer systems, registries, other socially dangerous phenomena; protection of personal data; development of technologies for e-government.

Development of Risk Assessment Tools for Enterprises

Th e purpose of this article is to develop tools for assessing the risks of enterprises, used both to assess the current activities of enterprises and during the implementation of investment proj-ects. It is supposed to be used both as the main risk assessment tool, for example, for small and medium-sized enterprises, and as an additional tool, mainly for large enterprises, for making a preliminary decision on the feasibility of the project. Th e developed tools include a comprehen-sive risk assessment system, which provides a fundamental basis for calculations, and a soft ware product that signifi cantly optimizes the calculations of a complex integral risk indicator. Th e risk assessment system represents individual risks, grouped by 51 risk groups. Making calculations using the developed risk assessment tools involves assigning to each risk the characteristics of assessing the level of infl uence by selecting the characteristic most appropriate to the current level of risk. A standard version of indicators is proposed for each risk and characteristics cor-responding to "high", "elevated", "medium", "moderate" and "low" risk levels. Based on the results of the analysis, a complex integral risk indicator is formed, which is a quantitative risk assessment, which allows you to classify an enterprise into one of three groups: "eff ective enterprise / project", "enterprise / project requiring improvements", "ineffi cient enterprise". To optimize the calcula-tions, the authors have developed a soft ware product that allows for analysis with a signifi cant reduction in the time spent on its implementation. In general, the use of the developed risk as-sessment system and tools makes it possible to signifi cantly optimize the risk assessment process and reduce the costs of analisys.

MODERN METHODS FOR ASSESSING THE LEVEL OF COMPETITIVENESS OF ENTERPRISES IN THE SERVICE SPHERE

The main trends in the development of service enterprises are the increase of foreign capital, the formation of new economic ties, the integration of the domestic market of services in European and world markets. At the same time, the positive dynamics of development of the domestic market of services is constantly accompanied by unpredictable changes, primarily related to the economic situation in Ukraine. The purpose of this study is to form a modern classification of methods for assessing the competitiveness of service enterprises. To achieve this goal, a range of research tasks was determined, including: 1) study of the essence of the efficiency and competitiveness of the activities of enterprises in the service sector, the order and main criteria for their assessment; 2) characteristics of existing methods for assessing the level of competitiveness of enterprises; 3) the formation of a modern classification of methods for assessing the level of competitiveness of enterprises in the service sector. Having studied different approaches to defining the essence of the efficiency and competitiveness of enterprises, the authors give their generalized definition, which defines the essence of the competitiveness of enterprises as a complex integral characteristic of its effective activity. Therefore, evaluating the efficiency of an enterprise, we also evaluate its competitiveness. Based on the above characteristics of existing methods for assessing the level of competitiveness of service enterprises, the authors propose a modern classification of these methods, which is based on the assertion that assessing the competitiveness of an enterprise is a sequential process that begins with assessing the market and the competitive environment, smoothly moving on to analyzing the position of the enterprise. among competitors, and ending with the evaluation of the product (service) itself.

Cadastral valuation of forest lands, taking into account the degree of development of their infrastructure

The article substantiates the need to improve methodology for cadastral valuation of forest lands and change the principles for calculating the rental rate of forest lands, on the basis of which the rent for the use of these lands will be determined in future. The methodology for determining the cadastral value of forest lands, taking into account the degree of development of their infrastructure, is presented. The infrastructure of lands of the forest fund is considered and an algorithm for assessing such infrastructure by means of geoinformation modeling is shown. A method for calculating a complex integral indicator (coefficient of infrastructure development) is presented, which is applicable to differentiate lands covered and not covered with tree vegetation, for their effective and rational management. The complex integral indicator determines the availability of forest plots, as it takes into account location of the forest fund infrastructure in relation to assessment plots, which makes it possible to obtain information about the time spent on covering the required distance. This methodology can be applied to assess the quality and condition of the existing infrastructure of the forestry sector in implementation of measures within the framework of the Strategy for the development of the forestry complex, providing for support and construction of infrastructure facilities on the forest fund lands. The method was tested in the Baltiysky district forestry of the Baltiysko-Belozersky taiga region. The results were as follows: the map of the forest fund infrastructure development within the boundaries of differentiation assessment according to the calculated integral indicator of infrastructure development was obtained; the regression model for calculating the rent rate taking into account the taxation value and integral indicator infrastructure development was received: the cadastral value of forest lands taking into account the developed model was calculated.

Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry

We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.

SCIENTIFIC AND METHODOLOGICAL WORK AS A FORM OF MANAGEMENT OF THE DEVELOPMENT OF PROFESSIONAL COMPETENCE OF A TEACHER

On the basis of consideration of theoretical, methodological and scientific literature, the essence of the basic concepts of the research problem have been analyzed. It has been established that control is inherent only in complex integral systems. It is determined that every teacher, head of an educational institution, at one’s position in cooperation with one’s colleagues, must perform one’s own specific functions, while simultaneously guiding one’s activities towards achieving the overall goal of the system. It is indicated that the management of the development of teacher's professional competence, as a rule, is carried out at several levels: hierarchical management (arises in connection with an order, instruction, requirement, the formation of regimes and restrictions, which are part of the main documentation of the educational institution); parity management (provides for democratic, interested, equal participation for all in the life and activities of an educational institution, in the process of which there are no authority and subordinates, but there is a useful joint activity focused on application of personal interests and qualities in common work; parity interaction makes it possible to discover hidden abilities of a teacher, there is mutual learning) and latent management (it aims at application of a large amount of information and opportunities from the outside to stimulate the teacher's creative search, push him to independently invent an idea or solution). It was found that scientific and methodological work in an educational institution is a holistic system based on the achievements of science and advanced pedagogical experience and on a specific analysis of the educational process, a system of interrelated actions and activities aimed at a comprehensive improvement of qualifications in the professional skills of each teacher and educator (including measures to manage professional self-education and self-education of teachers), to develop and increase the creative potential of the teaching staff, and ultimately to improve the educational process, to achieve the optimal level of education, upbringing and development of specific specialists.

System Approach in Complex Integral Design Methodology and Its Application in New Zealand

Many New Zealand houses are energy-inefficient, unhealthy, cold, mouldy, and damp. Therefore, a new approach to building design is imminent. This article proposes a framework for the transformation of housing that integrates construction planning and design, optimization, and control tools at strategic, tactical, and operational levels. The introduced Complex Integral Design New Zealand (CIDNZ) represents a comprehensive and balanced system-based design and delivery process that facilitates and accelerates cross-disciplinary and trans-disciplinary expertise and knowledge. CIDNZ delineates a new way of designing the process based on integral, complex, and systems thinking. The emerging novel understanding of sustainability, which guides the transformation process, might lead to a balance between individuals, groups, society, and existing ecosystems. CIDNZ comprises all stages in the life cycle of buildings and all significant factors in the architecture, engineering, and construction industry, particularly, people, processes, technology, and the environment. Therefore, the entire construction process that implements a system approach to buildings as a vital part of environmental systems, goes from the environment to humans and vice versa and offers unlimited possibilities. The consequent practical application of these principles might eliminate or reduce the design defects and lead accordingly to the reduction of costs involved in their rectification.

SCIENTIFIC CONTENT OF THE CATEGORY «SOCIAL-PROFESSIONAL MATURITY» ОF PERSONALITY

The concepts of «personal maturity» and «professional maturity» are often used in the literature, which are not identical since they describe different psychological realities. Personal maturity is defined by the authors as a key new formation of age, which radically changes the previous social situation of development, as a psychological new formation. That is, professional maturity may be a relatively autonomous entity that does not necessarily correlate with personal growth. Personal maturity is characterized by a process of internal personal transformation, the source of which is the individual's need for self-actualization and sense of identity. Autonomy is determined as the most important feature. Personal maturity within the functional approach framework is considered as a self-regulatory system, where the main function is the transformation (regulation) of the ontogenetic development process into a purposeful, systemic, personality-given process that acquires individual features in the means, conditions and purposes of this development. It is known that the effectiveness of socio-pedagogical activities largely depends on the level of training, theoretical knowledge and practical skills of the teacher, one’s skills and abilities to establish contact with the student. In our opinion, the best indicator of the quality of socio-pedagogical training, primarily, is the effective implementation of future professional activities. We believe that the criterion for ensuring the effective performance of professional duties is a high level of future teacher’s socio-professional maturity development. Social maturity consists of individual components like any complex, integral phenomenon. Certain states of personality that accompany one’s activities in various spheres of public life act as such components. There are many approaches in the scientific literature to building the structure of social maturity and to identifying the components that are of most importance and play the role of «supporting element» of the «core» of social maturity.