Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences from the Tested Ones?

  • Raoul R. NigmatullinEmail author
  • Artem S. Vorobev
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 303)


Based on the new method (defined below as the discrete geometrical invariants—DGI(s)), one can show that it enables to find the statistical differences between random sequences that can be presented in the form of 2D curves. We generalized and considered the Weierstrass–Mandelbrot function and found the desired invariant of the fourth order that connects the WM-functions with different fractal dimensions. Besides, we consider an example based on real experimental data. A high correlation of the statistically significant parameters of the DGI obtained from the measured data (associated with reflection optical spectra of olive oil) with the sample temperature is shown. This new methodology opens wide practical applications in differentiation of the hidden interconnections between measured by the environment and external factors.


Weerstrass–Mandelbrot function Discrete geometrical invariants Equipment calibration Nano-noise “reading” 

2010 AMS Math. Subject Classification.

Primary 40A05 40A25 Secondary 45G05. 


  1. 1.
    Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman and Company, San Francisco (1983)CrossRefGoogle Scholar
  2. 2.
    Feder, J.: Fractals. Plenum Press, Ny and London (1988)CrossRefGoogle Scholar
  3. 3.
    Samko, S.B., Kilbas, A.A., Marichev, I.: Fractional Integrals and Derivatives—Theory and Applications. Gordon and Breach, New York (1993)zbMATHGoogle Scholar
  4. 4.
    Uchaikin, V.V. : Method of the Fractional Derivatives. Artishock Publishing House, Ulyanovsk (2008)Google Scholar
  5. 5.
    Babenko, Y.I.: Power Relations in a Circumference and a Sphere. Norell Press Inc., USA (1997)Google Scholar
  6. 6.
    Babenko, Yu.I.: The power law invariants od the point sets, Professional, S-Petersburg, ISBN 978-5-91259-095-5, www., (Russian Federation) (2014)Google Scholar
  7. 7.
    Nigmatullin, R.R., Budnikov, H.C., Sidelnikov, A.V., Maksyutova, E.I.: Application of the discrete geometrical invariants to the quantitative monitoring of the electrochemical background, Res. J. Math. Comput. Sci. (RJMCS). eSciPub LLC, Houston, TX USA. Website:
  8. 8.
    Berry, M.V., Lewis, Z.V.: On the weierstrass mandelbrot fractal function. Proc. R. Soc. London A 370, 459–484 (1980)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Butler, J.M., Johnson, J.E., Boone, W.R.: The heat is on: room temperature affects laboratory equipment–an observational study. J. Assis.t Reprod Genet., 1389–1393. Published online 2013 Aug 7. Scholar
  10. 10.
    Hoffman, G.R., Birtwistle, J.K.: Factors affecting the performance of a thin film magnetoresistive vector magnetometer. J. Appl. Phys. 53, 8266 (1982). Scholar
  11. 11.
    Mostafa Mohamed Abd El-Raheem, Hoda Hamid Al-Ofi, Abdullah Alhuthali, Ateyyah Moshrif AL-Baradi, Effect of preparation condition on the optical properties of transparent conducting oxide based on zinc oxide. Optics. 4(3), 17–24 (2015). Scholar
  12. 12.
    Saleem, M., Ahmad, N., Ali, H., Bilal, M., Khan, S., Ullah, R., Ahmed, M., Mahmood, S.: Investigating temperature effects on extra virgin olive oil using fluorescence spectroscopy. IOP Publishing: Laser Phys. 27 (2017), pp. 1–10. Scholar
  13. 13.
    Giuffrè, A.M., Zappia, C., Capocasale, M.: Effects of High Temperatures and Duration Of Heating on Olive Oil Properties for Food use and Biodiesel Production. Springer: Journal of the American Oil Chemists’ Society, June 2017, Volume 94, Issue 6, pp. 819–830 (2017)Google Scholar
  14. 14.
    Clodoveo, M.L., Delcuratolo, D., Gomes, T., Colelli, G.: Effect of different temperatures and storage atmospheres on Coratina olive oil quality, Elsevier. Food Chemistry 102(3), 571–576 (2007)CrossRefGoogle Scholar
  15. 15.
    Bodurov, I., Vlaeva, I., Marudova, M., Yovcheva, T., Nikolova, K., Eftimov, T., Plachkova, V.: Detection of adulteration in olive oils using optical and thermal methods. Bulg. Chem. Commun. 45(Special Issue B), 81–85 (2013)Google Scholar
  16. 16.
    Nigmatullin, R.R., Ceglie, C., Maione, G., Striccoli, D.: Reduced fractional modeling of 3D video streams: the FERMA approach. Nonlinear Dyn. (2014). Scholar
  17. 17.
    Nigmatullin, R.R.: New Noninvasive Methods for “Reading” of Random Sequences and Their Applications in Nanotechnology. Springer: New Trends in Nanotechnology and Fractional Calculus Applications, pp. 43–56Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Radioelectronics and Informative-Measurements Techniques DepartmentKazan National Research Technical University (KNRTU-KAI)KazanRussian Federation

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