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It was found that search engines provide a wide selection of detectors with technical characteristics. A significant range of characteristics ensures the use of detectors for solving problems in security systems. A search and sorting mode for security system alarms has been developed. It is based on the necessary and sufficient characteristics of detectors, which ensure their selection depending on the tasks set for security systems. Criteria for their selection are formed on the basis of user requirements and technical characteristics of detectors.

In this paper, a methodology for benchmarking study of the market infrastructure development in regions of Ukraine is developed. As indicators for comparison, the number of active enterprises in the sectors that make up the market infrastructure was chosen. The suitability of standardization of output data by the standard deviation of the mean (Z-index) is grounded. The aggregation of standardized data is carried out by a simple arithmetic mean. The aggregated data are evaluated by two methods: ranking and grouping with irregular intervals.

The article critically analyzes four methods of evaluating the rating of commercial content of the Internet service, namely: "Just plus"; "Plus / Minus"; "Star"; "Combined". The basis of the development of the information service model is the "Star" method, since it provides a detailed evaluation with a fixed scale of values. The mathematical support of the "Star" method for calculating the content rating has been constructed and optimized.

The article deals with the main quantification methods of the system of checks and balances. Taking into account the main historical stages of the development of ideas concerning the functioning of the checks and balances system, on the basis of the comparative method, the main features of understanding the functioning of different aspects of checks and balances system by using mathematical methods are emphasized.