S.Ye. Saukh

ABSTRACT

The need of using the method of incomplete column-row (ICR) is substantiated. ICR method is the method of factorization of matrices in the composition with quasi-Newton methods for solving nonsmooth algebraic systems of equations. The factorization method enables direct solution of approximated systems of Newton’s equations and eliminates the use of iterative methods for approximate solving initial systems of Newton’s equations.

KEYWORDS

variational inequality problems, non-smooth equations, sparce matrix, column-row factorization.

REFERENCES

1. Facchinei, F. and Pang, J.-S. (2003), Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I., Springer, New York, USA.
2. Facchinei, F. and Pang, J.-S. (2003), Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. II., Springer, New York, USA.
3. Ruggiero, V. and Tinti, F. (2006), “A preconditioner for solving large scale variational inequality problems by a semismooth inexact approach”, Intern. Journal of Computer Mathematics, no. 10, pp. 723-739.
4. Petra, S. (2008), “Semismooth Least Squares Methods for Complementarity Problems”, Ph.D. Thesis, Wurzburg, available at: http:// www.opus-bayern.de/uni-uerzburg/volltexte/2006/1866/pdf/dissertation_petra.pdf.
5. Kanzow, C. (2003), “Inexact Semismooth Newton Methods for Large-Scale Complementarity Problems”, Wurzburg, Institute of Applied Mathematics and Statistics, available at: http://www.mathematik.uni-wuerzburg.de/ ~kanzow/paper/InSemiP.pdf
6. Fischer, A. (1992), “A special Newton-type optimization method”, Optimization, Vol. 24, no. 3-4, pp. 269-284.
7. Saad, Y. (2003), Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, USA.
8. Saukh, S.Ye. (2015), “Method of shearing matrix elements of the Clarke’s generalized Jacobian for providing numerical stability of the quasi-Newton methods of solving of the variational inequalities”, Elektronnoe modelirovanie, Vol. 37, no. 4, pp. 13-18.
9. Saukh, S.Ye. (2007), “CR-factorization method for large-scale matrices”, Elektronnoe modelirovanie, Vol. 29, no. 6, pp. 3-22. 
10. Saukh, S.Ye. (2010), “Incomplete column-row factorization of matrices for solving of large-scale system of equations”, Elektronnoe modelirovanie, Vol. 32, no. 6, pp. 3-14.
11. Tinti, F. (2003), “VIPLIB: A matlab collection of variational inequality problems”, available at: http://dm.unife.it/tinti/VIPs/test-problems.html (accessed July 31, 2015).

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S.V. Listrovoy, A.V. Sidorenko

ABSTRACT

An algorithm for solving the k-SAT-problem for the average polynomial time and 3-SAT-problem for the polynomial time. The proposed method can significantly reduce the time to solve SAT-problems.

KEYWORDS

SAT-problem, polynomial reducibility.

REFERENCES

1. Hirsch, E.A. (2000), “New Worst-Case Upper Bounds for SAT”, Journal of Automated Reasoning, Vol. 24, no. 4, pp. 397-420.
2. Listrovoy, S.V. and Minuchin, S.V. (2012), “The method of solving the problems of the minimum vertex cover in an arbitrary graph and the problem of the lowest coverage”, Elektronnoe modelirovanie, Vol. 34, no. 1, pp. 29-43.
3. Papadimitriou, H. and Stayglits, K. (1985), Kombinatornaya optimizatsiya. Algoritmy i slozhnost [Combinatorial optimization. Algorithms and complexity], Mir, Moscow, Russia.
4. SAT Live, available at: http: // www.satlive.org.

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Yu.D. Polissky

ABSTRACT

A method is considered for increasing the speed of response of the operation of determining a number belonging to the given half of range in the system of residual classes. The approach is based on reordering of modules with the preliminary estimation of variants of ordering and choice of the most preferable variant of ordering on this iteration. Thus the estimation of the variant of ordering consists in the subtraction from every remainder of a certain constant and in the count of quantity of the obtained zero residuals. The variant with the greatest quantity of modules, which residuals are equal to zero, are most preferable.

KEYWORDS

residual classes, complicated operations, modules, range.

REFERENCES

1. Akushskiy, I.Ya. and Yuditskiy, D.I. (1968), Mashinnaya arifmetika v ostatochnykh klassov [Machine arithmetic in the residual classes], Sov. radio, Moscow, Russia.
2. Polissky, Yu.D. (2008), “Forming of position descriptions during tabular realization of algorithms of the system of residual classes”, Sbornik trudov konferentsii “Modelirovanie-2008” [Conference proceedings «Simulation-2008»], Vol. 2, Kiev, IPME, May, 14-16, 2008, pp. 489-495.
3. Polissky, Yu.D. (2014), “About intercommunication of unmodule operations in the system of residual classes”, Matematychne modelyuvannya, no. 2 (31), pp. 3-6.
4. Polissky, Yu.D. (2014), “Algorithm of implementation of complex operations in the system of residual classes with the help of numbers presentation in reverse codes”, Elektronnoe modelirovanie, Vol. 36, no. 4, pp. 117-122.
5. Polissky, Yu.D. (2015), “Choice of criterion of belonging of number to this half of range in the system of residual classes”, Problemy matematychnoho modelyuvannya. Materialy naukovo-metodychnoi konferentsii [Problems of mathematical modeling. Proc. of scientificmethodical conference], Dnepropetrovsk, May, 27-29, 2015, pp. 91-92.

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A.A. Popov

ABSTRACT

Modern national and foreign mathematical models of technogenic risk assessment have been considered in the paper. The models classification based on quantification of the risk of exposure to hazardous chemical facilities is proposed. They were classified by the source of origin, the impact of the object and purpose. The mathematical formulas that determine the risk according to the proposed classification are given. The choice of the models is substantiated as well as the possibility of using the mathematical models in solving practical problems is described. The analysis of natural and technogenic hazards which create the greatest threat to people has also been done.

KEYWORDS

mathematical model, technogenic risk, danger, assessment, chemically dangerous objects.

REFERENCES

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