Vologda State Technical University

Introduction. As a result of the digital revolution, the entire system of scientific knowledge has undergone a metasystemic transition to a fundamentally new level of research. This level of research is based on the triad: model, algorithm, and programme. Aim. The present research aims to develop the theoretical and methodological foundations for integrating fundamental and applied knowledge through the modernisation of mathematical and computer training for university students, thereby enhancing their general cultural development and professional success. Methodology and research methods. In this research, the systemic and synergetic approaches, alongside the cultural studies approach, play leading roles. Results. The methodological aspects of the synergy between fundamental and applied knowledge in higher education, as well as the role of mathematics and computer science within this context, have been examined. The levels of synergy between fundamental and applied knowledge in higher education have been identified and characterised. It has been established which specific content within specialised teaching of the mathematical foundations of computer science plays a crucial role in the proficient and effective use of computer algebra systems and computer technologies. The significant role of incorporating a discrete mathematics component in teaching the mathematical foundations of computer science has been substantiated. Scientific novelty. The theoretical and methodological foundations for the synergy between fundamental and applied knowledge have been established, based on the modernisation of mathematical and computing education for students in higher education institutions. Drawing on a new "mathematical-computational" research culture, the levels of synergy between fundamental and applied knowledge in the modernisation of higher education have been identified and characterised. The importance of fostering non-mathematics students' understanding of the role of Big Ideas, and the methods of modern mathematics and computer science derived from them, within their chosen professional fields has been substantiated, with the aim of achieving a high level of IT proficiency. Practical significance. The features of modernising core educational programmes and curricula for student training have been described, based on the characterised levels of synergy between fundamental and applied knowledge in higher education. The methodological specifics of selecting content for the mathematical foundations of computer science have also been outlined.

A (discrete) group is called amenable whenever there exists a finitely additive right invariant probablity measure on it. For Thompson's group F the problem whether it is amenable is a long-standing open question. We consider presentation of F in terms of non-spherical semigroup diagrams. There is a natural partition of F into 7 parts in terms of these diagrams. We show that for any measure with the above properties on F, all but one of these sets have zero measure. This helps to clarify the structure of Folner sets in F provided the group is amenable.

The Russian economy faces an imperative to enhance national competitiveness as a prerequisite for maintaining socio-economic security. The Strategy for Scientific and Technological Development of the Russian Federation (STR Strategy) is conceived as a long-term framework aimed at ensuring the country’s sustainable, dynamic, and balanced advancement toward global competitiveness. Although the strategic directions articulated within the STR formally reflect the conceptual principles of the knowledge economy, they lack concrete mechanisms for stakeholder interaction, the realization of which is essential for achieving the strategic objectives of socio-economic development across all levels of governance. Since the inception of the STR Strategy, the conceptual model for managing the economic development of the Russian Federation within the paradigm of a knowledge-based economy has remained insufficiently defined. In this regard, the primary objective of the present study is to substantiate a methodological framework for constructing an evaluation system that measures the effectiveness of managing the economic development of socio-economic systems in the context of a knowledge economy. The research methodology is grounded in the dialectical, systemic, deductive, indicative, and synergetic approaches, which together provide a comprehensive analytical basis for examining the dynamic interrelations among innovation, competitiveness, and socioeconomic stability. The scientific novelty of the study lies in the development of theoretical propositions, methodological guidelines, and practical instruments for the organization of monitoring systems aimed at evaluating the innovative potential of socio-economic systems at multiple hierarchical levels. The study synthesizes interdisciplinary theoretical foundations concerning the nature and mechanisms of ensuring national competitiveness within a knowledge-based economy. The research design employs a deductive approach encompassing four analytical levels: cross-country comparison, national scale, regional level, and organizational level. Using R. Solow’s model of economic growth as an analytical framework, the paper examines key determinants of global economic development and establishes that the total factor productivity (TFP) exerts a decisive influence on Russia’s economic performance. The analysis identifies regions of the Russian Federation exhibiting structural weaknesses in socio-economic development and empirically confirms the dependence of regional socio-economic security on the degree of innovative activity within organizations. The proposed analytical toolkit provides a foundation for information and analytical support in assessing the contribution of organizational innovation to the effectiveness of socio-economic systems at various levels of aggregation. Future research should focus on elaborating a comprehensive mechanism for managing national socio-economic security within the institutional framework of the knowledge economy, thereby enhancing the strategic resilience and global competitiveness of the Russian Federation.

The purpose of the study is to form an integrated approach to the innovative development of tourist destinations in the Russian North. The main methods: a systematic approach to research, an author’s methodological approach to assessing tourist potential, economic and statistical methods, data visualization and other methods of analysis and synthesis. The study was carried out on the materials of the tourist destination «Beloozero» of the Vologda Oblast as part of the tourist macroregion «Russian North and Arctic». Results: through the author’s methodology, an assessment of the potential for the development of nature-oriented tourism in the territory of the Beloozero destination at the initial and modern stages of its development was carried out, based on the results of the assessment, a significant potential for the development of nature-oriented types of tourism was identified; prospects for the development of nature-oriented types of tourist activity were determined, an innovative approach to the spatial organization and development of the Beloozero destination was proposed by forming a tourist intermunicipal cluster specializing in nature-oriented tourism. Scientific novelty: the originality of the developed integrated approach is determined by the specifics of the process of forming a tourist space, which contributes to the innovative development of intermunicipal destinations, based on the results of using the author’s methodology for assessing the potential of nature-oriented tourism. Practical significance: the developed approach can be applied by regional and municipal authorities and management in order to develop northern tourist destinations in modern conditions. Conclusion: the study makes it possible to assert that in the modern realities of the intensification of tourist activities in the northern tourist ter-rhetoric of the Russian Federation, the application of an innovative approach to the spatial organization and development of the destinations of the Russian North will contribute.

In this paper, we consider a priori estimates and examine the existence of bounded solutions for a class of systems of ordinary differential equations whose right-hand sides have an arbitrary order of growth with respect to the independent and dependent variables. For the right-hand sides of the equations, we find one-sided estimates that provide a priori estimates for bounded solutions; using the methods for calculating the rotation of vector fields and the method of periodic truncations, we prove theorems on the existence of periodic and bounded solutions.