A method of solving symmetric polynomials of two variables is presented in the article. Several of such problems have been solved. There are very hard problems among them which were included in Mathematical Olympiads.

The mathematical education of a pripary schoolchild is one of the directions at primary school. It gives an opportunity to a schoolchild to master successfully the program of secondary and high school. The question of education actions formation for junior schoolchildren through geometric training is considered in this article.

An interdisciplinary connection in the school learning process is a necessary condition for successful teaching. Properly executed drawing of geometric shapes according to the rules of rectangular projection is one of the solutions of mathematical problems in geometry. Such ways of construction as the replacement of the projection planes, rotation and alignment demonstrate finding out of an actual size of a segment and a plane.

In this article the study of knowledge control patterns in math class with the help of computer-based testing. Displaying several models of construction of test tasks . Highlight the advantages of computer-based testing. The theoretical foundations of mathematical modeling.

The article presents a study of the effect of a resistor in a network neutral point from 6 to 35 kV on the conductive electromagnetic interference with the ground-fault current. A mathematical model is proposed.

This article reveals the possibility of solving life situations with combinatorial coefficients and shows the practicality of applying formulas on the theory of probability and mathematical statistics. The solution of tasks with combinatorial coefficients consists mainly in the decomposition of these coefficients by the basic formulas and their manipulations.

This article describes the ways in which students can develop logical thinking skills. Currently, any form of the exam takes precedence over logic. Accordingly, the article grouped logical tasks and provided their solutions.

The article provides a mathematical description of an induction motor with a device for indirect control of the electromagnetic moment and angular velocity of an induction motor in an electric drive, which allows substantiating on its basis a method for controlling the output variables of an asynchronous electric motor. A simulation model of an induction motor with an indirect control device for the output variables of an induction motor and the main results of the study are presented.

The article is devoted to the mathematical modulation of a complex of experimental data obtained in the process of experimental studies of processed cheese products, processed by mathematical methods. The purpose of this study is to establish the type and amount of the melting salt that promotes the formation of the plastic structure of the processed cheese product. The analysis of scientific research in the field of food technologies showed that mathematical modeling is used in the following areas: clarification of the modes of technological processes, design of recipes and assessment of the quality of finished products, as well as forecasting the shelf life of new products when they are put into production. The most relevant in describing the processes of food production are models of multivariate variance-regression analysis using methods of mathematical planning of the experiment. The authors carried out mathematical modeling and established graphical dependencies characterizing the degree of influence of the regulated factors X1 and X2 on the controlled ones that determine the quality and safety of processed cheese products. It is important that the mathematical analysis of the graphical dependences of the rheological parameters on the adjustable factors indicates the reliability of the data obtained. It is concluded that an increase in the melting salt dose leads to an increase in the – limit shear stress, which reflects the nature and state of the consistency of the processed cheese product. The process of normalization of the controlled factors by the maximum value was carried out. Graphical dependencies were built and regression analysis was performed, the results of which allow an objective assessment of the degree of influence of the type and dose of the melting salt on the chemical and organoleptic characteristics of the test products. The authors have proven that the optimal efficiency of transforming the structure of the constituent components of the recipe into a plastic structure of a processed cheese product that is stable during storage is provided by a combination of adjustable factors X1 (Solva 85) and X2 (Solva 120) taken in a ratio of 1: 1, with a total amount of 1,2 mas.%. At the same time, the quality indicators of the experimental products are characterized by the following values of the controlled factors: У1 – 1280 Pa; У2 – 9 points, У3 – 9,301 (2,0-2,2109 colony forming units /g).

ing the image of a dog. The purpose of this study is to consider the national and cultural specifics, reflected in the stable expressions of the English and Kazakh languages with the "dog" component. The article provides a brief overview of works related to the issues of phraseology in English and Kazakh. The work was based on general scientific methods: induction and deduction, analysis and synthesis. The work with the collected card index of stable expressions was carried out by the method of continuous sampling, using the descriptive-analytical method. An important place in the work with the material was given to the Вестник Инновационного Евразийского университета. 2020. № 4 ISSN 1729-536X 17 methods of definitional, contextual and typological analysis. Using the comparative-comparative method, national-specific and universal features of zoonyms in English and Kazakh languages were determined. The material of the research was a card index of phraseological units of the English and Kazakh languages (more than 160 units) which were selected using the method of continuous sampling from phraseological dictionaries of the English and Kazakh languages. When analyzing the authors of the article relied on materials from phraseological dictionaries in English and Kazakh languages. In the course of the analysis, groups of phraseological units were identified and their differences and similarities in the compared languages were identified, as well as their national and cultural specificity, based on the action of universal cultural codes; the national and cultural specificity of zoonyms is determined by a number of extralinguistic factors: the location of the country, the living conditions of the people, their customs and traditions, history, religion and socio-political structure. All this secures national-specific connotation to zoonyms and determines their productivity. The results of the study showed that through the prism of axiology, an analysis of English and Kazakh stable combinations with the "dog" component, reflecting the paradigm of values (general cultural and national-cultural) and anti-values of the corresponding ethnic communities, was presented, as well as universal values representing at the intercultural level. The results of the study showed that through the prism of axiology, the analysis reflects the paradigm of values (general cultural and national-cultural) and anti-values of the corresponding ethnic communities. The research results have both theoretical and practical significance; it will be additional material for elective courses on the subject of "Lexicology" taught at the faculties of philology and foreign languages in higher educational institutions.

The COVID-19 coronavirus pandemic has brought about a new economic reality, to which every economic entity must adapt, regardless of the scale: within a region, a country, or the whole world. The article examines the regional economic mechanism in the context of the pandemic in the context of the interaction of the market mechanism of self-development and the control influence of the regional authorities. The destabilization of the economic situation in the regions as a result of the pandemic and related restrictions poses the problem of revising the main directions of regional policy To achieve the set goals and objectives, the general methods of empirical and theoretical research were used: analysis, synthesis, induction, deduction, generalization, description. Three spheres play a key role in the economy of Pavlodar region: industry, trade and transport. Just how to remain positive dynamics in terms of industrial production and maintained the services will depend on the development of the region, the main indicators which are discussed in this article through the prism of the events in the context of the pandemic. The necessary health measures taken in response to the pandemic have led to a severe reduction in mobility and are accompanied by high economic costs. The quarantine measures have had a direct impact on businesses, especially microbusinesses. To support this sector, tax incentives were provided, lending was expanded, and measures were taken to further facilitate the business environment. The article reflects only a small list of issues that required changes in the economic mechanism in the context of the pandemic and the introduction of restrictive measures, as well as the strengthening of the influence of external control on the economy. At the same time, the state as whole and individual regions, taking into account their potential and the actual situation in the economy, need to find an optimal balance between the prohibitions and restrictions imposed to ensure the safety of human health and the preservation of the economy in a functional mode, work and income of employees. This problem of finding a balance between prohibitive and permissive measures requires further analysis. At the same time, the state needs to develop and apply tools of a systematic nature, focused on the long term.

Main problem: Currently, there are various socio-economic and environmental problems in the development of rural regions, among the key ones: access to basic services, lack of transport connectivity, inadequate social, housing infrastructure, and others. Achieving sustainable development of the territory is one of the important priorities, which is of great importance for any state. Sustainable development is understood as harmonious development from the perspective of the economy, social sphere and ecology. At the same time, the sustainable development of socio-economic and natural systems is not possible if many factors and conditions are not taken into account that can directly affect the current state and long-term sustainability of their functioning and interference is not created on these processes within the framework of the implementation of various policies by the state. Purpose: In this regard, the purpose of this study is to identify and study the key factors that determine and contribute to the balanced development of rural regions of the country in the context of sustainable development goals and objectives. Methods: Empirical methods of economic cognition, methods of induction and deduction, synthesis, as well as logical methods were used in the framework of scientific research. Results and their significance: When classifying significant factors that influence the formation of sustainable rural development, it is necessary to take into account an essential feature of sustainable development – a harmonious combination of economic, social and environmental components of social development. Taking into account the logic of this study, based on the variety of factors contributing to the sustainable development of rural a reas, their systematization has been carried out.

Singularly perturbed partial differential equations with small parameters with higher derivatives deserve special attention, which often arise in a variety of applied problems and are used in describing mathematical models of diffusion processes, absorption taking into account small diffusion, filtration of liquids in porous media, chemical kinetics, chromatography, heat and mass transfer, hydrodynamics and many other fields. It is necessary to consider the creation of an asymptotic classification of solutions of singularly perturbed equations using a well-known approach to solving the boundary value problem. In this case, the singular problem is understood as the problem of constructing the asymptotics of the solution of the Cauchy problem for a system of ordinary differential equations with a small parameter with a large derivative. The asymptotics of the solution in all cases is based on the last time interval or the construction of a boundary value problem for a system with a weak clot in an asymptotically large time interval. Purpose - to construct and substantiate the asymptotics of solving a singular initial problem for a system of two nonlinear ordinary differential equations with a small parameter; To date, a number of methods have been developed for constructing asymptotic expansions of solutions to various problems. This is the method of boundary functions developed in the works of A.B. Vasilyeva, M.I. Vishik, L.A. Lusternik, V.F. Butuzov; the regularization method of S. A. Lomov, methods of averaging, VKB, splicing of asymptotic decompositions of A.M. Ilyin and others. All the above methods allow us to obtain asymptotic expansions of solutions for wide classes of equations. At the same time, such singularly perturbed problems often arise, to which ready-made methods are not applicable or do not allow to obtain an effective result. Therefore, the development of methods for solving equations remains a very urgent problem. As a result of the study, an algorithm for constructing an asymptotic classification of the initial solution of the problem with a singular perturbation is given, and approaches to estimating the residual term are also shown.

The article considers international experience in automating the system of land relations, aimed at improving the quality and reducing the time for the provision of services by state bodies to citizens and organizations, providing access to information databases, which positively affects the development of effective government decisions for the effective use of land resources, and also improves social welfare. In Kazakhstan, the transfer of land cadastral information to electronic media continues, and the automated information system of the land cadastre is being improved. The task of creating a unified mechanism for state cadastral registration of land plots according to uniform rules and technologies for all, identification and individualization of land plots and objects of immovable property firmly associated with them, for subsequent state registration of rights to land plots and transfer of information to the territorial bodies of the State Revenue Committee, remains relevant. To study the experience of developed countries in the provision of public services in the system of land relations through the development of official land information systems. The article used general scientific methods from general to concrete one, synthesis, theoretical generalization, induction, deduction, and system analysis. Based on the studied experience of developed countries, relevant conclusions and proposals were made: information technology is a useful tool and can help overcome bureaucracy, backwardness, isolation, monopolistic practices, and inefficiency in the public and private sectors; the use of state bodies for effective regulation of geographic information (GIS) systems that allow solving a wide range of tasks, as well as the use of modern information technologies, contributes to the transition of the activities of state bodies to a qualitatively new level of state control, which allows for full transparency of information about the land fund and objects real estate of the country, as well as its availability to the population. The provision of public services in the field of land relations using information and communication technologies reduces entry barriers and transaction costs in business, increases competition, and improves mechanisms in this area.

The sustainable socio-economic development of the regions is determined by a number of factors, including the implementation of an effective budget policy, which plays a special role for the performance of social functions, regulating Inter-level organizational relations, strengthening the economic and financial independence of the regions.Effective management of budget policy makes it possible to purposefully use the mechanisms of budget regulation at all levels of management. At the same time, it involves the implementation of a number of functions, of which the control function is the most important. In the system of economic relations, it is assigned to a specialized financial body – the Treasury. Topical tasks are: regulation of budget flows in the region, strengthening control over the receipt and targeted use of budget funds; improving inter-budgetary relations; increase the revenue base of budgets by increasing their collection; etc. In the process of solving these tasks, a number of managerial, organizational and methodological problems arise, the solution of which contributes to improving the effectiveness of the budget policy of the region and, as a result, its economic development. Objective - to study the regional specifics of the implementation of budget policy in terms of the formation and execution of local budgets on the example of the Pavlodar budget, identify the existing problems in the region and determine the directions for improving the budget process. The methodological basis of the study is formed by general scientific and special methods of cognition: analysis and synthesis, deduction and induction, historical, logical, structural-functional and comparative analysis, as well as individual scientific methods: statistical-economic and computational-analytical. The formation of the local budget is carried out through the application of a single methodology, a single budget legislation and orientation to the Strategic Development Plan of the Republic of Kazakhstan until 2025, developed for the implementation of the main document of the state planning system for the medium – term period-the long-term development strategy of Kazakhstan until 2050. The implementation of all the tasks set out in the strategic development plans of the Republic will begin at the level of local budgets. The solution of State tasks is guaranteed, first of all, to meet the needs of the population at the local level. The role of regions in the entire process of reproduction cannot be overestimated, so the independence, flexibility and high degree of efficiency of using budget funds at the local level is the key to the successful development of society, production and business in any territory.

It is known that an increase in the volume of production in any sphere of production as a consequence increases the intensity of anthropogenic impact on the environment, including the atmosphere. This problem is especially typical for oil refineries. On the one hand, this is due to the complexity of oil refining processes and the release of various hydrocarbon compounds into the atmosphere during processing. Therefore, when implementing sustainable ecological and economic development in the Azerbaijani regions that are associated with oil refining, an increase in the volume of products should not create an increase in environmental impact. This problem is considered one of the factors directly affecting the economic situation of the enterprise, as well as creating certain environmental, social and economic problems for society as a whole. The purpose is to study the possibility of using the correlation-regression method in solving the problem of determining the presence of the decapling effect. The relationship between the volume of petroleum products produced and the amount of pollutants released into the atmosphere is random and is characterized by stochastic and statistical dependence. At the same time, on the basis of a mutual comparison of the methods of correlation and regression analysis and the method of production function, the effectiveness of correlation and regression analysis is determined. The methodological basis of the work was the scientific works of scientists-economists, mathematicians providing the definition of the decoupling effect at industrial enterprises. In the presented methodology, the obtained estimates of the performed in real time vary depending on the values of the set parameter. When developing the presented methodology, the STATISTIKA software package was used. This program was developed by StatSoft. Based on the obtained and experimentally verified data, the equation of linear regression dependence is determined. During the research, it was found that in some cases it is not possible to evaluate the statistical characteristics of a random variable or it is accompanied by serious errors. Therefore, when processing data, instead of mathematical expectations and variances, we used selective mathematical expectations and variances. The analysis of the ecological and economic situation of oil refineries based on correlation and regression analysis allows us to determine the presence of the decapling effect. The obtained correlation coefficient between the volume of production oil products and the volume of emissions into the atmosphere shows that there is no statistically significant relationship between these parameters. Thus, there is an achievement of the decoupling effect between the production of petroleum products and the volume of emissions of pollutants into the atmosphere.

The main problem: Modern research in the development of fermented milk products is focused on increasing the bioavailability of milk components, as well as the use of bacterial components of sourdough, which increase health properties. The use of the fermentation process of skimmed milk with a combined starter consisting of traditional for cottage cheese and starter cultures of probiotic cultures immobilized in a gel of biopolymers is very important. Purpose: The purpose of this study is to determine the optimal amount of starter, consisting of an association of probiotic cultures immobilized in a gel of biopolymers (membranes) to be added to fermented skimmed milk in order to enrich it with functional ingredients. Methods: A one-factor experiment was used. The culture association Propionibacterium freudenreichii subsp. was used as a regulatory factor. Shermanii, Bifidobacterium lactis and Streptococcus thermophilus, immobilized in a biopolymer gel, added to skimmed milk in the form of membranes (determined as a percentage of the mass of fermented milk). Controlled factors are the main indicators characterizing the efficiency of the skim milk fermentation process; these are active acidity, the logarithm of the number of viable cells of bifidobacteria, the logarithm of the number of viable cells of propionic acid bacteria, and organoleptic evaluation. Results and their significance: Based on the results of a mathematical analysis of the totality of values of controlled factors depending on the amount of starter cultures of probiotic cultures, mathematical models were built to determine the degree of influence of the starter on the quality indicators of the product, using the Table Curve 3D-v4 mathematical computer program.

Main problem: the article is devoted to the issue of designing basic recipes for dairy products using mathematical modeling methods. The author analyzed the problems in the field of food technology and concluded that mathematical modeling is used in the following areas: clarification of technological process modes, designing recipes and assessing the quality of finished products, as well as predicting the shelf life of new products when they are put into production. The use of dihydroquercetin as a drug that prevents the oxidation of milk fat in the design of dairy products is substantiated. Mathematical modeling was carried out on the basis of experimental and analytical material obtained in laboratory and production conditions. Based on the maximum value of the objective functions, the optimal normalized mixtures and the maximum allowable concentration of dihydroquercetin were selected. The analysis of the received mathematical dependences and models is carried out, the system of linear equations is made. Purpose: to study the effect of natural bioflavonoid antioxidant on the oxidative processes of milk fat and the viability of lactic acid cultures and their associations by mathematical modeling in order to use it in the technology of a new product; conduct an analysis of mathematical dependencies and models, compose a system of linear equations. Methods: the article uses the method of mathematical analysis and the matrix method. Results and their significance: a mathematical model was developed for the dependence of the viability of probiotic cultures on the mass fraction of dihydroquercetin when designing recipes for a creamy-protein curd product, a matrix of the chemical composition of dairy ingredients was presented, and a system of linear equations for basic recipes was compiled. In the course of the study, the spatial configurations of dihydroquercetin, as well as its effect on the oxidative processes of milk fat, were studied. Mathematical modeling of experimental data on the study of the effect of dihydroquercetin on the viability of microorganisms with probiotic properties was carried out. The normalization of the complex of obtained results on the study of the influence of the mass fraction of dihydroquercetin on the fermentation process was studied. Controlled factors characterizing the process of fermentation of model media with dihydroquercetin have been determined. A rationing of the dihydroquercetin complex of more than 0.50% was established; the target function decreases to its minimum value of 0.53 with a mass fraction of dihydroquercetin of 1.00%.

Main problem: achievement of socio-economic sustainability in each locality depends on coordinated actions of executive and representative authorities, which should be focused on creating appropriate conditions for ensuring social standards and improving quality of life of rural population, as well as the development of main economic basis – agricultural production. In this context, in today’s reality, the urgent problem of sustainable development of rural settlements is provision of appropriate industrial and social infrastructure, because systemic development of rural regions depends on this rural infrastructure, which acts as an element of territorial socio-economic subsystem. Purpose: identification and studying of key factors that determines and contributes to balanced development of industrial and social infrastructure of rural regions of country in context of sustainable development goals and objectives. According to the results of the study, the main distinctive features of sustainable development of rural areas are determined due to the provision of industrial and social infrastructure of rural regions, which has a correlation with the level and quality of life of the rural population, as well as the final results of agricultural production. Methods: general methodological principles, systematic approach and empirical methods of economic cognition act as methodological basis for implementation of this study: economic and statistical models, forecasting and modeling, methods of induction and deduction, synthesis, and also logical methods. Results and their significance: results of research can be used as a practical basis in activity of local executive bodies and local self-government bodies. Their use is possible for scientific substantiation of development and implementation of comprehensive plans for development of industrial and social infrastructure of rural areas of region within framework of Regional Development Plan, focused on qualitative improvement of economic basis of villages and the well-being of all segments of rural population.

The article contains the results of a research work devoted to an important problem - the development, production and delivery to the consumer of fermented dairy products of a healthy diet fortified (enriched) with milk protein, a vitamin-mineral complex, and probiotics and prebiotics. Purpose is based on mathematical modeling of experimental data, to determine an effective fermented food system for use as the basis of a fortified (enriched) curd product for a healthy specialized diet; to develop a recipe and technology for its production; to study the nutritional, biological and energy value of a new product. Modern technologies and equipment were used in experimental studies: ultrafiltration, fortification, etc. The studies were carried out by standard physicochemical, microbiological methods in three to five repetitions. Mathematical and statistical processing of experimental data was carried out using the program "Statistica-6.0". A recipe and technology for the production of a curd product for a healthy specialized diet has been developed. Its nutritional, biological and energy value has been studied. The developed recipe and biotechnological parameters for the production of the curd product were tested in the industrial conditions of the leading enterprise JSC "Lyubinskiy MKK". The quality and safety of the curd product was studied in the laboratory of techno-chemical control of FGANU "VNIMI" (Moscow).

Main problem: One of the priority directions for improving the management system at the enterprise is the organization of effective interaction between strategic and operational management through the formation of a mechanism for their interaction, orienting operational management to achieve strategic goals and objectives. In market conditions, issues of interaction between strategic and operational production management are particularly relevant, allowing solving complex economic problems of the enterprise related to the rational use of resources involved in production, reducing production costs, improving the quality of products, optimizing production potential in order to increase the efficiency of the entire enterprise as a whole and its structural divisions. The role of the mechanism of interaction between strategic and operational production management also lies in the possibility of coordinating those final strategic goals that the enterprise strives for with the actual internal production capabilities. The purpose of this article is to study the issues of interaction between operational and strategic management of an enterprise in a market environment. The theory and modern practice of production management in a competitive environment show that the implementation of operational production management is a natural stage in the implementation of strategic management, therefore, there is a close relationship between them, which manifests itself in their interaction. Methods: The methods of analysis and generalization, the method of systematic and comparative analysis, structural and functional method, economic and mathematical methods, graphical and tabular method were used in the work. Results and their significance: the authors consider the theoretical aspects of the relationship between strategic and operational production management at an enterprise in a competitive environment. There is a low level of compliance of operational tasks of production with the strategic goals of development of domestic enterprises and the special role of information systems designed to link operational and strategic management tasks together. Accordingly, the key aspects of information technology implementation are investigated, including the need for employee training, strategic communications planning, information flow management and the use of data analytics. Examples of successful application of information technologies in various industries and organizations are given.