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UDC 517.9
Dinamic measurement in spaces of "noise"
A.L. Shestakov, South Ural State University, Chelyabinsk, Russian Federation, admin@susu.ac.ru
G.A. Sviridyuk, South Ural State University, Chelyabinsk, Russian Federation, sviridyuk@74.ru
Yu.V. Hudyakov, South Ural State University, Chelyabinsk, Russian Federation, hudyakov74@gmail.com
Abstract
The new concept of the «white noise» was proposed by the authors, it is understood by the Nelson – Gliklikh’s derivative of the Wiener process. This approach extends to other «noise», which together make up the space of «noise». Precise dynamic measurement «noise» produced in these spaces through a mathematical model of the measuring device provided by the equations of Leontief type. As an example, the measured «noise» having the form a pulse, the amplitude of which is the Gauss random variable. The results of measurements are precise.
Keywords
the Wiener process, the Nelson – Gliklikh’s derivative, «white noise», dynamic measurement, space of «noise»
References
1. Arato M. Linear Stohastic Systems with Constant Coefficients. A Statistical Approach. Berlin;; Heidelberg; N.Y., Springer-Verlag, 1982.
2. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London; Dordrecht; Heidelberg; N.-Y., Springer, 2011.
3. Da Prato G., Zabczyk J. Stochastic equations in infinite dimensions. Cambridge, Cambridge University Press, 1992.
4. Kovacs M., Larsson S. Introduction to Stochastic Partial Differential Equations. Processing of «New Directions in the Mathematical and Computer Sciences», National Universities Commission. Abuja. Nigeria. October 8–12. 2007. Publications of the ICMCS. 2008, vol. 4, pp. 159–232.
5. Melnikova I.V, Filinkov A.I., Alshansky M.A. Abstract Stochastic Equations II. Solutions in Spaces of Abstract Stochastic Distributions. Journal of Mathematical Sciences. 2003, vol. 116, no 5, pp. 3620–3656.
6. Melnikova I.V., Filinkov A.I. Generalized Solutions to Abstract Stochastic Problems. J. Integ. Transf. and Special Funct. 2009, vol. 20, no. 3–4, pp. 199–206.
7. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2011, no. 17 (234), pp. 70–75.
8. Gantmacher F.R. The Theory of Matrices. AMS Chelsea Publishing, Reprinted by American Mathematical Society. 2000.
9. Shestakov A.L., Sviridyuk G.A. On Optimal Measurement of the «White Noise». Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2012, no. 27, pp. 99–108.
10. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht; Boston; Tokio, VSP, 2003.
11. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.
12. Gliklih Yu.E. The Study of Equations of Leontief type with White Noise Methods Derived an Average of Random Processes [Izuchenie uravnenij leont'evskogo tipa s belym shumom metodami proizvodnyh v srednem sluchajnyh processov]. Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2012, no. 27, pp. 24–34.
13. Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal measurement problem [Chislennoe reshenie zadachi optimal'nogo izmerenija]. Automation and Remote Control. 2012, no. 1, pp. 107–115.
14. Zamyshlyaeva А.А. Stochastic Partial Linear Equations of Sobolev type of High Order with Additive White Noise [Stohasticheskie nepolnye linejnye uravnenija sobolevskogo tipa vysokogo porjadka s additivnym belym shumom]. Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2012, no. 40, pp. 73–82.
15. Zagrebina S.A., Soldatova Е.А. Equation of Barenblatt-Zheltova-Kochina with White Noise [Uravnenie Barenblatta-Zheltova-Kochinoj s belym shumom]. Obozrenie priklad. i prom. matematiki. 2012, vol. 19, issue 2, pp. 252–254.
16. Sviridyuk G.A., Zagrebina S.А. The Showalter-Sidorov problem as a Phenomena of the Sobolev type Equations [Zadacha Shouoltera-Sidorova kak fenomen uravnenij sobolevskogo tipa]. Izvestiya Irkutskogo gosudarstvennogo universiteta, seriya Matematika. 2010, vol. 3, no. 1, pp. 104–125.
2. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London; Dordrecht; Heidelberg; N.-Y., Springer, 2011.
3. Da Prato G., Zabczyk J. Stochastic equations in infinite dimensions. Cambridge, Cambridge University Press, 1992.
4. Kovacs M., Larsson S. Introduction to Stochastic Partial Differential Equations. Processing of «New Directions in the Mathematical and Computer Sciences», National Universities Commission. Abuja. Nigeria. October 8–12. 2007. Publications of the ICMCS. 2008, vol. 4, pp. 159–232.
5. Melnikova I.V, Filinkov A.I., Alshansky M.A. Abstract Stochastic Equations II. Solutions in Spaces of Abstract Stochastic Distributions. Journal of Mathematical Sciences. 2003, vol. 116, no 5, pp. 3620–3656.
6. Melnikova I.V., Filinkov A.I. Generalized Solutions to Abstract Stochastic Problems. J. Integ. Transf. and Special Funct. 2009, vol. 20, no. 3–4, pp. 199–206.
7. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2011, no. 17 (234), pp. 70–75.
8. Gantmacher F.R. The Theory of Matrices. AMS Chelsea Publishing, Reprinted by American Mathematical Society. 2000.
9. Shestakov A.L., Sviridyuk G.A. On Optimal Measurement of the «White Noise». Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2012, no. 27, pp. 99–108.
10. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht; Boston; Tokio, VSP, 2003.
11. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.
12. Gliklih Yu.E. The Study of Equations of Leontief type with White Noise Methods Derived an Average of Random Processes [Izuchenie uravnenij leont'evskogo tipa s belym shumom metodami proizvodnyh v srednem sluchajnyh processov]. Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2012, no. 27, pp. 24–34.
13. Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal measurement problem [Chislennoe reshenie zadachi optimal'nogo izmerenija]. Automation and Remote Control. 2012, no. 1, pp. 107–115.
14. Zamyshlyaeva А.А. Stochastic Partial Linear Equations of Sobolev type of High Order with Additive White Noise [Stohasticheskie nepolnye linejnye uravnenija sobolevskogo tipa vysokogo porjadka s additivnym belym shumom]. Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2012, no. 40, pp. 73–82.
15. Zagrebina S.A., Soldatova Е.А. Equation of Barenblatt-Zheltova-Kochina with White Noise [Uravnenie Barenblatta-Zheltova-Kochinoj s belym shumom]. Obozrenie priklad. i prom. matematiki. 2012, vol. 19, issue 2, pp. 252–254.
16. Sviridyuk G.A., Zagrebina S.А. The Showalter-Sidorov problem as a Phenomena of the Sobolev type Equations [Zadacha Shouoltera-Sidorova kak fenomen uravnenij sobolevskogo tipa]. Izvestiya Irkutskogo gosudarstvennogo universiteta, seriya Matematika. 2010, vol. 3, no. 1, pp. 104–125.
Source
Bulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control, Radio Electronics, 2013, vol. 13, no. 2, pp. 4-11. (in Russ.) (The main)