S.E. Saukh

ABSTRACT

Infinitesimal elements of matrix components of generalized Fischer-Burmeister have been analyzed, conditions of singularity initiation of such matrix components have been determined. An analogous analysis has been made in conditions of finite-digit calculations of small elements of the same matrix components. The conditions of initiation of badly determined and singular matrix components of generalized Clark Jacobians for non-smooth systems of nonlinear algebraic
equation, which are formed using special methods of problems transformation into variation inequalities, have been studied. A method of displacement of small elements of matrix components of generalized Clark Jacobian ensuring numerical stability of quasi-Newton methods for iteration solution of variation equations has been developed.

KEYWORDS

generalize Clark Jacobian, quasi-Newton methods, numerical stability, displacement method, small quantities.

REFERENCES

1. Facchinei, F. and Pang, J.-S. (2003), Finite-dimensional variational inequalities and complementarity problems, Vol. I, New York, Springer, USA.
2. Facchinei, F. and Pang, J.-S. (2003), Finite-dimensional variational inequalities and complementarity problems, Vol. II, New York, Springer, 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-wuerzburg/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. Billups, S.C., Dirkse, S.P. and Ferris, M.C. (1997), “A comparison of large scale mixed complementarity problem solvers”, Computational Optimization and Applications, no. 7, pp. 3-25.
7. Fischer, A. (1992), “A special Newton-type optimization method”, Optimization, Vol. 24, no. 3-4, pp. 269-284.
8. Dirkse, S.P. and Ferris,M.C. (1995), “The PATH solver: A non-monotone stabilization scheme formixed complementarity problems”, Optimization Methods and Software, no. 5, pp. 125-156.
9. Saad, Y. (2003), Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics, Philadelphia, USA.
10. “Energy Research Centre of the Netherlands” (2015),COMPETES input data, available at: http: // www. ecn.nl/fileadmin/ecn/units/bs / COMPETES/ flowgate-information.xls (accessed June 17, 2015), cost-functions.xls (accessed June 17, 2015).
11. Saukh, S.E. (2013), “Methods of computer modeling of competitive equilibrium in electric power markets”, Electronnoe modelirovanie, Vol. 35, no. 5, pp. 11-26.
12. Saukh, S.E. (2007), “Method of CR-factorization of great dimension matrices”, Electronnoe modelirovanie, Vol 29, no. 6, pp. 3-22.

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L.I. Timchenko, Y.F. Kutayev, S.V. Cheporniuk, N.I. Kokriatskaya, A.A. Yarovyy

ABSTRACT

A method of S-preparation has been developed, which, allowing for the preliminary conveyor formation of correlated image convolution sums, is characterized by high noise immunity and adaptivity to uncertainty and variability of the signal clutter situation. This method allows one to determine coordinates of the true shift of the image background with the accuracy of up to one resolution step. Correlation algorithms have been classified. Based on the mentioned processing methods, a schematic diagram of the correlation analysis unit has been developed and realized.

KEYWORDS

correlation, method of S-preparation, loop preparation, images, gradient.

REFERENCES

1. Ima, J., Jensen, J.R., and Tullis, J.A. (2008), “Object-based change detection using correlation image analysis and image segmentation”, International Journal of Remote Sensing, Vol. 29, no. 2, pp. 399-423.
2. Kozhemyako, V.P., Kutaev, Y.F, Timchenko, L.I., Chepornyuk, S.V., Hamdi, R.R., Gertsiy, A.A. and Ivasyuk I.D. (1998), “The Q-transformation method applying to the facial images Normalization”, Proceedings of International ICSC IFAC Symposium on NEURAL COMPUTATION, NC’98, Vienna, September 23-25, 1998, pp. 287-291.
3. Perveen, S. and James, L.A. (2012), “Changes in correlation coefficients with spatial scale and implications for water resources and vulnarability data”, The Professional Geographer, Vol. 64, (X), pp. 1-12. 4. Sp uler, M., Rosenstiel, W. and Bogdan, M. (2012), “One class SVM and canonical correlation analysis increase performance in a c-VEP based brain-computer interface (BCI)”, Proceedings of 20th European Symposium on Artificial Neural Networks, Bruges, Belgium, April, 2012, pp. 103-108.
5. Yarovyy, A., Timchenko, L., and Kokriatskaia, N. (2012), “Theoretical aspects of parallel-hierarchical multi-level transformation of digital signals”, Proceedings of the 11th International Conference on Development and Application Systems, Suceava, Universitatea Stefan cel Mare Suceava, Romania, May, 2012, pp. 1-9.
6. Sharin, A., Khan, M.R., Imtiaz, H., Sarwar, M.S.U. and Fattah, S.A. (2010), “An efficient face recognition algorithm based on frequency domain cross-correlation function”, Electrical and Computer Engineering (ICECE), International Conference, Dhaka, Bangladesh, December, 2010, pp. 183-186.
7. Zhao, Q., Rutkowski, T.M., Zhang, L. and Cichocki, A. (2010), “Generalized optimal spatial filtering using a kernel approach with application to EEG classification”, Cognitive Neurodynamics, Vol. 4, no. 4, pp. 355-358.
8. Pannekoucke, O., Berre, L. and Desroziers, G. (2008), “Background error correlation length-scale estimates and their sampling statistics”, Quarterly Journal of the Royal Meteorological Society, Vol. 134, pp. 497-511.
9. Donev, A., Torquato, S., and Stillinger, F.H. (2005), “Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings”, Physical Review, E 71, 011105, pp. 1-14.
10. Zhou, Z. and Tang, X. (2008), “New families of binary low correlation zone sequences based on interleaved quadratic form sequences”, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E91-A, (11), pp. 3406-3409.
11. Zou, K.H. and Hall, W.J. (2002), “On estimating a transformation correlation coefficient”, Journal of Applied Statistics, Vol. 29, no. 5, pp. 745-760.
12. Awwal, A.A.S., Rice, K.L., and Taha, T.M. (2009), “Fast implementation of matched-filter-based automatic alignment image processing”, Optics & Laser Technology, Vol. 41, no. 2, pp. 193-197.
13. Cherkasov, A., Sprous, D.G. and Chen, R. (2003), “Three-dimensional correlation analysis. A novel approach to the quantification of substituent effects”, The Journal of Physical Chemistry A, Vol. 107, no. 45, pp. 9695-9704.
14. Peña-Ortega, C. and Velez-Reyes, M. (2010), “Evaluation of different structural models for target detection in hyperspectral imagery”, Proceedings of SPIE 2010, Orlando, Florida, pp. 76952H-76952H-11.
15. Shawakfen, O.Q., Gertsiy, A.A., Timchenko, L.I., Kutaev, Y.F., Zlepko, S.M. and Shveyki, N. (1999), “Method of recursive-contour preparing for image normalization”, Proceedings of the IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing, Antalya, Turkey, pp. 414-418.
16. Thirumalai, V. and Frossard, P. (2012), “Distributed representation of geometrically correlated images with compressed linear measurements”, IEEE Transactions on Image Processing, Vol. 21, no. 7, pp. 3206-3219.
17. Kou, G., Lu, Y., Yi, Peng and Shi, Y. (2012), “Evaluation of classification algorithms using MCDM and rank correlation”, International Journal of Information Technology & Decision Making (IJITDM), Vol. 11, no. 01, pp. 197-225.
18. Zhao, J., Zhang, J., and Yin, J. (2009), “A parallel differentialcorrelation acquisitionalgorithm in time domain”, Wireless Communications, Networking and Mobile Computing. WiCom’09. 5th International Conference, pp. 1-4.
19. Kozhemyako, V., Timchenko, L. and Yarovyy, A. (2008), “Methodological principles of pyramidal and parallel-hierarchical image processing on the base of neural-like network systems”, Advances in Electrical and Computer Engineering, Vol. 8, no. 2, pp. 54-60.
20. Timchenko, L.I., Kutaev, Y.F., Chepornyuk, S.V., Grudin, M.A., Harvey, D.M., and Gertsiy, A.A. (1997), “A brain-like approach to multistage hierarchial image”, Proceedings Image Analysis and Processing, Springer-Verlag, Italy, pp. 246-253.

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A.Ya. Bomba, O.V. Prysiazhniuk

ABSTRACT

Mathematical model of singularly perturbed process of convection-diffusion-adsorption mass transfer of two types of pollutant in a medium consisting of particles of microporous structure is developed. The corresponding boundary value problem is solved in a curvilinear quadrangular parallelepiped. An asymptotic expansion of its decision is constructed, which makes it possible to complement autonomously the convective component of solution with mass transfer and diffusion components, and take into account the amendments on the output of the filter flow and the impact of lateral sources. The results of numerical calculations are given.

KEYWORDS

convection, diffusion, mass transfer, nanoporous medium.

REFERENCES

1. Ruthven, D.M. (1984), Principles of adsorption and adsorption processes, Wiley-Interscience, New York, USA.
2. Verigin, N.N. and Sherzhukov, B.S. (1969), “Diffusion and mass transfer at liquid filtration in porous media”, Razvitie issledovaniy po teorii filtratsii v SSSR (1917-1967), pp. 237-313.
3. Bomba, À.Ya., Bulavatskyy, V.M. and Skopetskyy, V.V. (2007), Neliniyni matematychni modeli protsesiv heohidrodynamiky [Nonlinear mathematical models of the hydrodynamics], Naukova Dumka, Kyiv, Ukraine.
4. Bomba, À.Ya. and Klymyuk, Yu. E. (2014), Matematychne modelyuvannya prostorovykh syngulyarno-zburenykh protsesiv typu filtratsiya-konvektsiya-difuziya [Mathematical modeling of spatial singularly perturbed process type filtration-convection-diffusion], Asol, Rivne, Ukraine.
5. Rolando, M.A. Roque-Malherbe (2012), Adsorption and Diffusion in Nanoporous Materials, CRC Press, Taylor & Francis, Boca Raton, USA.
6. Quirke, N. (2006), Adsorption and Transport at the Nanoscale, CRC Press, Taylor & Francis, Boca Raton, USA.
7. Petryk, M. (2007), “Mathematical modeling of mass transfer in symmetric heterogeneous and nanoporous media with a system of n-interface interactions”, Cybernetics and System Analysis, Vol. 43, no. 1, pp. 94-111.
8. Petryk, Ì.R., Fressard, G. and Mihalyk, D.M. (2009), “Modeling and analysis of concentration fields of nonlinear competitive binary diffusion among particles”, Problemy upravleniya i informatiki, no. 4, pp. 73-83.
9. Deineka, V.S., Petryk, M.R. and Fressard, G. (2011), “Identification of kinetic parameters of components of multicomponent mass transfer in inhomogeneous media nanoporous system kompetetive diffusion”, Kibernetika i systemnyi analiz, no. 5, pp. 45-64.
10. Bomba, A.Ya., Prysiazhniuk, I.M. and Prysiazhniuk, O.V. (2013), “An asymptotic method for solving a class of singularly perturbed model problems of mass transfer process in different porous environments”, Dopovidi NAN Ukrayiny, no. 3, pp. 28-34.
11. Samarskiy, A.A. and Gulin, A.V. (2000), Chislennye metody matematicheskoy fiziki [Numerical methods of mathematical physics], Nauchnyi mir, Moscow, Russia.
12. Voevodin, A.F. and Goncharova, O.N. (2001) “Method of splitting into physical processes for numerical investigation of convection problems”, Matematicheskoe modelirovanie, Vol. 13, no. 5, pp. 90-96.

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E.I. Vladimirskiy, B.I. Ismailov

ABSTRACT

Statements of the synchronization and control problems for nonlinear physical systems of a fractional order have been considered. The criterion of the problem realization is the concept of the average time of Poincare recurrence. A generalized compact metrical space with Poincare dimensionality <τ> was proposed. The algorithm is provided for determination the memory losses when training system.

KEYWORDS

systems with memory, Poincare recurrences, fractality, fractionality, visualization.

REFERENCES

1. Tarasov, V.E. (2011), “Models of theoretical physics with integro-differentiation of fractional order”, Abstract of Dr. Sci. (Phys.-Math.) dissertation, Moscow, Russia.
2. Tarasov, V.E. (2012), “The fractional oscillator as an open system”, Cent. Eur. Phys., Vol. 10, no. 2, pp. 382-389. DOI 10.2478/s11534-012-0008-0.
3. Tarasov, V.E. (2010), “Fractional Dynamics of Open Quantum Systems”, Fractional Dynamics Nonlinear Physical Science, Vol. 0, pp. 467-490. DOI 10.1007/978-3-642-14003-7_20.
4. Nigmatullin, R.R. (1992), “Fractional integral”, Teoreticheskaya i matematicheskaya fizika, Vol. 90, no. 3, pp. 354-367.
5. Uchaykin, V.V., “Fractional-differential model of dynamic system”, available at www.rfbr.ru/rffi/ru/books/0_14704.
6. Vladimirskiy, E.I. (2012), “Times of Poincare recurrence under interaction of chaotic and stochastic systems”, Vostochno-Evropeyskiy zhurnal peredovyh tehnologiy, no. 6/4 (60), pp. 4-8.
7. Vladimirskiy, E.I. and Ismaylov, B.I. (2014), “Fractional structure “mixing-transport” as the open system”, Vostochno-Evropeyskiy zhurnal peredovyh tehnologiy, no. 4/4 (70), pp. 4-9.
8. Vladimirskiy, E.I. and Ismaylov, B.I. (2014), “Visualization of the Poincare recurrence times in the open systems of fractional order”, Novye zadachi tehnicheskikh nauk i puti ikh resheniya. Sb. statey Mezhdun. NPK [New tasks of technical sciences and the ways of its solutions], Septemb.1, 2014, Aeterna, Ufa, Russia, pp. 4-8.
9. Altman, Eduardo G., Elton, C. da Silva and Ibere, L. Caldas (2003), “Memory Effect on the Poincare Recurrence Time Series”, available at: arXiv: nlin/0304027v2 [nlin. CD] 2 Sep. 2003.

10. Santhanam, M.S. and Holger Kantr (2008), “Return interval distribution of extreme events and long term memory”, available at: arXiv: 0803.1706v1 [g-fin ST] 12 Mar. 2008.
11. Afraymovich, V., Ugalde, E. and Urias, H. (2011), Fraktalnye razmernosti dlya vremen vozvrascheniya Puankare [Fractal dimensions for Poincare recurrence times], NITs Regulyarnaya i khaoticheskaya dinamika, Izhevskiy institut kompyuternyh issledovaniy, Moscow-Izhevsk, Russia.
12. Anischenko, V.S. and Astakhov, S.V. (2013), “Theory of Poincare recurrences and its application to the problems of nonlinear physics”, UFN, Vol. 183, no. 10, pp. 1009-1028. DOI: 10.3367/UFNr.0183.201310a.1009.
13. Vladimirskiy, E.I. and Ismaylov, B.I. (2011), “Nonlinear recurrent analysis as a mathematicalmodel of chaotic processes control”, Informacionnye tehnologii,Vol. 177, no. 5, pp. 42-45.
14. Olemskoy, A.I. and Flat, A.Ya. (1993), “Use of fractal conception in physics of condensed medium”, UFN, Vol. 163, no. 12, pp.1-49.
15. Ibedou, Ismail and Takahisa, Miyata (2008), “The theorem of Pontrjagin-Schnirelmann and approximate sequences”, New Zeeland Journal of Mathematics, Vol. 38, pp. 121-128.
16. Vladimirskiy, E.I. and Ismaylov, B.I. (2011), Sinergeticheskie metody upravleniya haoticheskimi sistemami [Synergetic methods of control of chaotic systems], ELM, Baku, Azerbaijan.
17. Butkovskiy, A.G., Postnov, S.S. and Postnova E.A. (2013), “Fractional integro-differential calculus and its application in the control theory. I. Mathematical bases and interpretation problem”, Avtomatika i Telemekhanika, no. 4, pp. 3-42.
18. Grigorenko, I. and Grigorenko, E. (2003), “Chaotic dynamics of the fractional Lorenz system”, Phys.Rev.Let., Vol. 91, pp. 1-4.
19. Matouk, A.E. (2009), “Chaos synchronization between two different fractional systems of Lorenz family”, Mathematical Problems Engineering . DOI : 10.1155/ 2009/ 572 724.
20. Ke-hui Sun, Jian Ren and Shui-sheng Qiu (2009), “Co-coupled synchronization of fractional-order unified chaotic systems”, available at: arXiv: 0909.2410[nlin.CD].
21. Petras, Ivo (2011), Fractional-order nonlinear systems. Modeling, analysis and simulation, Springer, USA.
22. Kronover, R. (2008), Fraktaly i haos v dinamicheskikh sistemakh [Fractals and chaos in dynamic systems], Tekhnosfera, Moscow Russia.

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