ON MATHEMATICAL MODELING OF (NANO) TECHNOLOGIES RELATED TO NAVIGATION PROBLEMS ON THE BASE OF GENERALIZATIONS OF THE PICARD METHOD

N. M. Glazunov

Abstract


The aim of the paper is the mathematical modeling of nanotechnology problems of navigation based on generalizations of the Picard method. The Picard method for solving systems of ordinary differential equations, and its extensions on the basis of hyperlogarithms  and iterated path integrals, are presented. The derivation of the Picard-Fuchs differential equations for connections in bundles on schemes is given. The results can be used to study the corresponding differential equations and to calculate the Taylor coefficients of (dimensionally regularized) Feynman amplitudes with rational parameters.


Keywords


Picard method; ordinary differential equation; iterated integral; hyperlogarithm; nanotechnology; Picard–Fuchs differential equation; multiple zeta value

References


A. Tsourdos, B. White, and M. Shanmugavel, Cooperative Path Planning of Unmanned Aerial Vehicles, Chichester, John Wiley& Sons Ltd, 2011.

A. A. Chikrii, Conflict controlled processes, Boston, London, Dordrecht, Kluwer Acad. Publ., 1997.

J Weiner and F. Nunes, Light-Matter Interaction: Physics and Engineering at the Nanoscale, 1st edition, Oxford University Press, 2013.

Quantum chemistry, in. Annual reports in computational chemistry, vol. 8, edited by R. Wheeler, Elsevier, Boston, 1–70, 2012.

S. Maier, Plasmonics: Fundamentals and Applications, Springer, 2007.

Yu. G. Kryvonos, V. P. Kharchenko, and N. M. Glazunov, “Differential-algebraic equations and dynamical systems on manifolds, Springer Link,” Cybernetics and system analysis, vol. 52, issue 3, pp. 408–418, 2016.

I. A. Lappo-Danilevsii, Applications of functions of matrices to the theory of linear systems of ordinary differential equations, Gostekhizdat, Moscow, 1957. (in Russian)

L. S. Pontryagin, Ordinary differential equations, Nauka, Moscow, 1965. (in Russian)

V. Arnol`d, Ordinary differential equations, Berlin, Heidelberg, Springer-Verlag, 1992.

A. Parshin, A generalization of Jacobian variety, Izv. Akad. Nauk SSSR Ser Mat., 30, 175–182, 1966.

K. Chen. Iterated path integrals, Bull. Amer. Math. Soc. 83, 831–879, 1977.

J. Blumlein, D. J. Broadhurst, and J. A. M. Vermaseren, “The Multiple Zeta Value Data Mine,” Comput. Phys. Commun. 181, 582–625, 2010.

Y. André. Une introduction aux motifs (motifs purs, motifs mixtes, periodes). Publie par la Societe mathematique de France, AMS dans Paris, Providence, RI, 2004.

D. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys. Lett. B, 393, 403–412, 1997.

H. Poincare, “Sur les groupes d’equations lineaires,” Acta Mathematica, no. 4, 1884.

A. Grothendieck, P. Deligne, and N. Katz, with M. Raynaud, and D. S. Rim, Groupes de monodromie en geometrie algebriques, Lect. Notes Math. 288, 340, 1972-73.

N. Glazunov, “Class Fields, Riemann Surfaces and (Multiple) Zeta Values,” Proceedings of the 3-d Int. Conf. Computer Algebra & Information Technologies, Mechnikov National University, Odessa, 2018, pp.152–154.

N. M. Glazunov, “Merging control of heterogeneous unmanned aerial vehicular platoon and dualities,” 2017 IEEE 4th International Conference “Actual problems of unmanned aerial vehicles development (APUAVD)”, Proceedings, IEEE Ukraine Section SP/AES, NAU, Kyiv, 2017, pp. 41–43.


Full Text: PDF

Refbacks

  • There are currently no refbacks.


Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.