The mathematical apparatus of synergy and economic applications

The subject of the research is the mathematical apparatus of synergy and its possible applications in the data economy of the Russian Federation aimed at increasing the efficiency, sustainability, and security of the national economic system under the sanctions pressure from Western countries. The authors focus on the issues of effective interaction among economic agents in theoretical-stochastic conditions, in finance, and during firm mergers. Key mathematical aspects of synergy in relation to the utility function, which contribute to the necessary structural changes in the Russian data economy on the path to a knowledge-based economy, are considered. In the literature review phase, the authors examine the adaptation of foreign experience, which allows for the identification of the advantages and limitations of the proposed approaches. The study aims to develop a systematic and data-oriented approach to assessing the effectiveness of interactions among economic agents and decision-makers at the state governance level. The research utilizes methods of economic-mathematical modeling, mathematical tools from game theory, the Cobb-Douglas two-factor production function, Cournot economic model, the statements of Debreu's theorem, and principles of the new economic paradigm of sufficiency, formulated by the authors in previous works. The scientific novelty of the study lies in the transition from verbal descriptions of synergy and its properties to mathematical ones, including in the context of asymptotic methods, and finding a quantitative expression for these concepts in some specific cases. The main results of the conducted research are: mathematical descriptions of the properties of synergy in theoretical-stochastic conditions, in finance, and in firm mergers; synergy of family utility functions, synergy of financial operations; formulation of a general remark about the synergistic effect and the possibility of transitioning from competition principles to synergy. The practical significance of the research results lies in their potential application by state authorities for improving economic policy and creating an effective system for assessing the performance of economic actors in order to ensure national security. The research gains particular relevance under the conditions of sanctions pressure on the Russian economy, where the task of enhancing efficiency becomes a top priority. Additionally, synergy can be used as a new indicator of socio-economic development, determining the quality of life and effectiveness of state governance in the context of sustainable development in Russia and mitigating potential threats.