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Fractal Atomicity, a Fundamental Concept in the Dynamics of Complex Systems

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13th Chaotic Modeling and Simulation International Conference (CHAOS 2020)

Part of the book series: Springer Proceedings in Complexity ((SPCOM))

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Abstract

Applying a fractal method of analyzing the dynamics of the structural units of any complex system, a mathematical concept is built, namely that of fractal atomicity. The construction of such a concept involves defining dynamic variables in the form of fractal functions, defining scale resolutions, defining a principle of scale covariance as a fundamental principle of motion, equations of evolution, etc. Finally, some specific mathematical properties of the fractal atom are also established.

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References

  1. M. Agop, A. Gavriluţ, G. Ştefan, B. Doroftei, Implications of non-differentiable entropy on a space-time manifold. Entropy 17, 2184–2197 (2015)

    Article  Google Scholar 

  2. K.M.R. Audenaert, Subadditivity of q-entropies for \(q>1\). J. Math. Phys. 48 (2007)

    Google Scholar 

  3. P. Cavaliere, F. Ventriglia, On nonatomicity for non-additive functions. J. Math. Anal. Appl. 415(1), 358–372 (2014)

    Article  MathSciNet  Google Scholar 

  4. I. Chiţescu, Finitely purely atomic measures and \(\cal{L}^{p}\)-spaces. An. Univ. Bucureşti Şt. Natur. 24, 23–29 (1975)

    Google Scholar 

  5. I. Chiţescu, Finitely purely atomic measures: coincidence and rigidity properties. Rend. Circ. Mat. Palermo 50(3), 455–476 (2001)

    Google Scholar 

  6. L. Drewnowski, Topological rings of sets, continuous set functions, integration, I, II, III. Bull. Acad. Polon. Sci. 20, 269–276, 277–286, 439–445 (1972)

    Google Scholar 

  7. A. Gavriluţ, Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel, Baire multivalued set functions. Fuzzy Sets Syst. 160, 1308–1317 (2009). Erratum in Fuzzy Sets Syst. 161, 2612–2613 (2010)

    Google Scholar 

  8. A. Gavriluţ, Fuzzy Gould integrability on atoms. Iran. J. Fuzzy Syst. 8(3), 113–124 (2011)

    Google Scholar 

  9. A. Gavriluţ, Regular Set Multifunctions (Pim Publishing House, Iaşi, 2012)

    MATH  Google Scholar 

  10. A. Gavriluţ, A. Croitoru, On the Darboux property in the multivalued case. Annals of the University of Craiova. Math. Comput. Sci. Ser. 35, 130–138 (2008)

    Google Scholar 

  11. A. Gavriluţ, A. Croitoru, Non-atomicity for fuzzy and non-fuzzy multivalued set functions. Fuzzy Sets Syst. 160, 2106–2116 (2009)

    Article  MathSciNet  Google Scholar 

  12. A. Gavriluţ, A. Croitoru, Pseudo-atoms and Darboux property for set multifunctions. Fuzzy Sets Syst. 161(22), 2897–2908 (2010)

    Article  MathSciNet  Google Scholar 

  13. A. Gavriluţ, M. Agop, An Introduction to the Mathematical World of Atomicity Through a Physical Approach (ArsLonga Publishing House, Iaşi, 2016)

    Google Scholar 

  14. A. Gavriluţ, A. Iosif, A. Croitoru, The Gould integral in Banach lattices. Positivity 19(1), 65–82 (2015)

    Article  MathSciNet  Google Scholar 

  15. S. Gudder, Quantum measure and integration theory. J. Math. Phys. 50 (2009)

    Google Scholar 

  16. S. Gudder, Quantum integrals and anhomomorphic logics (2009), arXiv:quant-ph (0911.1572)

  17. S. Gudder, Quantum measure theory. Math. Slovaca 60, 681–700 (2010)

    Article  MathSciNet  Google Scholar 

  18. S. Gudder, Quantum measures and the coevent interpretation. Rep. Math. Phys. 67, 137–156 (2011)

    Article  MathSciNet  Google Scholar 

  19. S. Gudder, Quantum measures and integrals

    Google Scholar 

  20. J.B. Hartle, The Quantum Mechanics of Cosmology. Lectures at Winter School on Quantum Cosmology and Baby Universes, Jerusalem, Israel, Dec 27, 1989–Jan 4, 1990 (1989)

    Google Scholar 

  21. J.B. Hartle, Spacetime quantum mechanics and the quantum mechanics of spacetime, in Proceedings of the Les Houches Summer School on Gravitation and Quantizations, ed. by J. Zinn-Justin, B. Julia, Les Houches, France, 6 Jul–1 Aug 1992 (North-Holland, 1995), arXiv:gr-qc/9304006

  22. P.M. Iannaccone, M. Khokha, Fractal Geometry in Biological Systems: An Analitical Approach (1995)

    Google Scholar 

  23. M. Khare, A.K. Singh, Atoms and Dobrakov submeasures in effect algebras. Fuzzy Sets Syst. 159(9), 1123–1128 (2008)

    Article  MathSciNet  Google Scholar 

  24. J. Li, R. Mesiar, E. Pap, Atoms of weakly null-additive monotone measures and integrals. Inf. Sci. 134–139 (2014)

    Google Scholar 

  25. J. Li, R. Mesiar, E. Pap, E.P. Klement, Convergence theorems for monotone measures. Fuzzy Sets Syst. 281, 103–127 (2015)

    Google Scholar 

  26. B.B. Mandelbrot, The Fractal Geometry of Nature, Updated and augm. edn. (W.H. Freeman, New York, 1983)

    Google Scholar 

  27. I. Mercheş, M. Agop, Differentiability and Fractality in Dynamics of Physical Systems (World Scientific, 2015)

    Google Scholar 

  28. E. Pap, The range of null-additive fuzzy and non-fuzzy measures. Fuzzy Sets Syst. 65(1), 105–115 (1994)

    Article  MathSciNet  Google Scholar 

  29. E. Pap, Null-Additive Set Functions. Mathematics and Its Applications, vol. 337 (Springer, 1995)

    Google Scholar 

  30. E. Pap, Handbook of measure theory, in Some Elements of the Classical Measure Theory (2002), pp. 27–82

    Google Scholar 

  31. E. Pap, A. Gavriluţ, M. Agop, Atomicity via regularity for non-additive set multifunctions. Soft Comput. (Found.) 1–6 (2016). https://doi.org/10.1007/s00500-015-2021-x

  32. K.P.S.B. Rao, M.B. Rao, Theory of Charges (Academic Press Inc., New York, 1983)

    MATH  Google Scholar 

  33. R. Salgado, Some identities for the q-measure and its generalizations. Mod. Phys. Lett. A 17, 711–728 (2002)

    Article  MathSciNet  Google Scholar 

  34. B. Schweizer, A. Sklar, Probabilistic Metric Spaces (Elsevier Science Publishing Co., Inc., 1983). Republished in 2005 by Dover Publications, Inc., with a new preface, errata, notes, and supplementary references

    Google Scholar 

  35. R.D. Sorkin, Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9, 3119–3128 (1994)

    Article  MathSciNet  Google Scholar 

  36. R.D. Sorkin, Quantum measure theory and its interpretation, in Quantum Classical Correspondence: Proceedings of the 4\(^{\text{th}}\) Drexel Symposium on Quantum Non-integrability, ed. by D.H. Feng, B.-L. Hu (International Press, Cambridge Mass, 1997), pp. 229–251

    Google Scholar 

  37. R. Sorkin, Quantum dynamics without the wave function. J. Phys. A: Math. Theory 40, 3207–3231 (2007)

    Article  Google Scholar 

  38. R. Sorkin, Quantum mechanics as quantum measure theory, arXiv:gr-qc/9401003

  39. S. Surya, P. Waldlden, Quantum covers in q-measure theory (2008), ArXiv: quant-ph 0809.1951

  40. H. Suzuki, Atoms of fuzzy measures and fuzzy integrals. Fuzzy Sets Syst. 41, 329–342 (1991)

    Article  MathSciNet  Google Scholar 

  41. C. Wu, S. Bo, Pseudo-atoms of fuzzy and non-fuzzy measures. Fuzzy Sets Syst. 158, 1258–1272 (2007)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Physics, “Gheorghe Asachi” Technical University from Iaşi, Iaşi, Romania

    Maricel Agop

  2. Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Iaşi, Romania

    Alina Gavriluţ

  3. N. Oblu Clinical Emergency Hospital Iaşi, Iaşi, Romania

    Lucian Eva

  4. Faculty of Psychology and Education Sciences, “Alexandru Ioan Cuza” of Iaşi, Iaşi, Romania

    Gabriel Crumpei

Authors
  1. Maricel Agop
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  2. Alina Gavriluţ
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  3. Lucian Eva
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  4. Gabriel Crumpei
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Corresponding author

Correspondence to Alina Gavriluţ .

Editor information

Editors and Affiliations

  1. ManLab, Technical University of Crete, Chania, Greece

    Christos H. Skiadas

  2. Department of Management Science and Technology, Hellenic Mediterranean University, Agios Nikolaos, Crete, Lasithi, Greece

    Yiannis Dimotikalis

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Agop, M., Gavriluţ, A., Eva, L., Crumpei, G. (2021). Fractal Atomicity, a Fundamental Concept in the Dynamics of Complex Systems. In: Skiadas, C.H., Dimotikalis, Y. (eds) 13th Chaotic Modeling and Simulation International Conference. CHAOS 2020. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-70795-8_3

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