A typical DPLL algorithm for the Boolean satisfiability problem splits the input problem into two by assigning the two possible values to a variable; then it simplifies the two resulting formulas. In this paper we consider an extension of the DPLL paradigm. Our algorithms can split by an arbitrary linear combination of variables modulo two. These algorithms quickly solve formulas that explicitly encode linear systems modulo two, which were used for proving exponential lower bounds for conventional DPLL algorithms.We prove exponential lower bounds on the running time of DPLL with splitting by linear combinations on 2-fold Tseitin formulas and on formulas that encode the pigeonhole principle.Raz and Tzameret introduced a system R(lin) which operates with disjunctions of linear equalities with integer coefficients. We consider an extension of the resolution proof system that operates with disjunctions of linear equalities over F 2 ; we call this system Res-Lin. Res-Lin can be p-simulated in R(lin) but currently we do not know any superpolynomial lower bounds in R(lin). Tree-like proofs in Res-Lin are equivalent to the behavior of our algorithms on unsatisfiable instances. We prove that Res-Lin is implication complete and also prove that Res-Lin is polynomially equivalent to its semantic version. We prove a space-size tradeoff for Res-Lin proofs of 2-fold Tseitin formulas.

Background. According to the World Health Organization, 92% of the world's population lives in places where air quality levels exceed recommended limits. Recently, Ukraine had the most deaths per every 100,000 people (out of 120 countries) attributed to atmospheric air pollution. High levels of atmospheric air pollution have been observed not only in typically industrial regions, but in Ukraine's capital, Kyiv, as well. Objectives. The aim of the present study was to establish the state of air pollution in Kyiv and perform a risk assessment of associated human health effects. Methods. Using official statistics and state monitoring data, the study aimed to identify and analyze risks to the health of Kyiv's population associated with air pollution. The following methods were used: systematic, functional and comparative analysis, risk theory, mathematical modeling, probability theory and mathematical statistics, as well as geographic information system technologies for digital map design and objective-oriented methodology for software design systems. Results. The risk values across different areas of the city varied significantly, indicating that atmospheric air quality remains unstable. Areas with the highest and lowest risk values were identified. Conclusions. The environmental state of atmospheric air in Kyiv requires greater attention and additional research to identify the causes of air pollution, along with implementation of measures to improve air quality. Competing Interests. The authors declare no competing financial interests.

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The authors carried out a thorough study of the features of the spread of hazardous chemicals in the surface layer of the atmosphere in the event of an emergency at the site of a nuclear power plant. In order to ensure the continuous operation of the stations in their territories, various ancillary technogenic facilities are located and operate, which release emissions of non-radiation pollutants into the atmosphere. Under various negative circumstances of a technical and natural nature, emergencies may occur due to significant chemical pollution of the atmospheric air in and outside the sanitary protection zone. The prevention of such emergencies is based on environmental monitoring in the locations of man-made objects and their preventive forecast. Implementation of these measures is not possible without the use of effective methods based on mathematical models of environmental pollution by anthropogenic objects, and the hardware and software that implement these methods. The main stages of the development of information and technical methods of prevention of such emergencies are given and described. Different scenarios of emergencies are described as a result of the release of chemicals into the atmosphere at these sites. A conceptual scheme for the distribution of impurities in the atmosphere due to man-made emissions has been developed. The peculiarities of atmospheric air propagation under stationary and non-stationary emission conditions are described in detail. It is established that the most determinants of influence on the concentration distribution of impurities are: mode and conditions of emission, type of source, direction, and velocity of the wind, state of the atmosphere, chemical interaction with other substances in the atmospheric air, gravitational deposition, leaching of sediments, absorption of the underlying surface. surface, terrain. The results obtained will be used in the process of developing mathematical models for the propagation of pollutants in the atmospheric air from the emissions of nuclear power plants during relevant emergencies.

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