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Ultrametricity in the theory of complex systems

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Abstract

We review applications of p-adic and ultrametric methods in the theory of complex systems. We consider the following examples: the p-adic parameterization of the Parisi matrix in the replica method; the method of hierarchical (interbasin) kinetics, which allows describing macromolecular dynamics by models of ultrametric diffusion; the two-dimensional 2-adic parameterization of the genetic code, which demonstrates that degenerations of the genetic code are described by local constancy domains of maps in the 2-adic metric. We discuss clustering methods for a family of metrics and demonstrate that the multiclustering (ensemble clustering) approach is related to the Bruhat–Tits building theory.

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References

  1. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, P-Adic Analysis and Mathematical Physics [in Russian], Nauka, Moscow (1994)

    Book  Google Scholar 

  2. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, English transl., World Scientific, Singapore (1994).

    Google Scholar 

  3. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, and I. V. Volovich, P-Adic Numbers Ultrametric Anal. Appl., 1, 1–17 (2009); arXiv:0904.4205v1 [math-ph] (2009).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. N. Kochubei, Pseudo-Differential Equations and Stochastics Over Non-Archimedean Fields, Marcel Dekker, New York (2001).

    Book  MATH  Google Scholar 

  5. V. Anashin and A. Khrennikov, Applied Algebraic Dynamics (de Gruyter Expo. Math., Vol. 49), de Gruyter, Berlin (2009).

    Book  MATH  Google Scholar 

  6. S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, Theory of p-Adic Distributions (London Math. Soc. Lect. Note Ser., Vol. 370), Cambridge Univ. Press, Cambridge (2010).

    MATH  Google Scholar 

  7. I. V. Volovich, Class. Q. Grav., 4, L83–87 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  8. M. Mezard, G. Parisi, and M. Virasoro, Spin-Glass Theory and Beyond (World Sci. Lect. Notes Phys., Vol. 9), World Scientific, Singapore (1987).

    MATH  Google Scholar 

  9. V. A. Avetisov, A. H. Bikulov, and S. V. Kozyrev, J. Phys. A: Math. Gen., 32, 8785–8791 (1999); arXiv:condmat/9904360v1 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. G. Parisi and N. Sourlas, Eur. Phys. J. B, 14, 535–542 (2000); arXiv:cond-mat/9906095v1 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  11. A. Yu. Khrennikov and S. V. Kozyrev, Phys. A, 359, 222–240 (2006); arXiv:cond-mat/0603685v1 (2006).

    Article  Google Scholar 

  12. A. Yu. Khrennikov and S. V. Kozyrev, Phys. A, 359, 241–266 (2006); arXiv:cond-mat/0603687v1 (2006).

    Article  Google Scholar 

  13. A. Yu. Khrennikov and S. V. Kozyrev, Phys. A, 378, 283–298 (2007); arXiv:cond-mat/0603694v1 (2006).

    Article  Google Scholar 

  14. D. M. Carlucci and C. De Dominicis, C. R. Acad Sci. Ser. IIB, 325, 527–530 (1997); arXiv:cond-mat/9709200v1 (1997).

    ADS  MATH  Google Scholar 

  15. C. De Dominicis, D. M. Carlucci, and T. Temesvári, Journal de Physique I (France), 7, 105–115 (1997); arXiv: cond-mat/9703132v1 (1997).

    Article  ADS  Google Scholar 

  16. M. Talagrand, Spin glasses, a Challenge for Mathematicians: Cavity and Mean Field Models (Ser. Mod. Surv. Math., Vol. 46), Springer, Berlin (2003).

    Google Scholar 

  17. O. M. Becker and M. Karplus, J. Chem. Phys., 106, 1495–1517 (1997).

    Article  ADS  Google Scholar 

  18. A. T. Ogielski and D. L. Stein, Phys. Rev. Lett., 55, 1634–1637 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  19. V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev, and V. A. Osipov, J. Phys. A: Math. Gen., 35, 177–189 (2002); arXiv:cond-mat/0106506v1 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. V. A. Avetisov, A. Kh. Bikulov, and A. P. Zubarev, Proc. Steklov Inst. Math., 285, 3–25 (2014).

    Article  MATH  Google Scholar 

  21. V. A. Avetisov, A. Kh. Bikulov, and V. Al. Osipov, J. Phys. A: Math. Gen., 36, 4239–4246 (2003).

    Article  ADS  MATH  Google Scholar 

  22. V. A. Avetisov, A. Kh. Bikulov, and A. P. Zubarev, J. Phys. A: Math. Theor., 42, 085003 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  23. S. V. Kozyrev, P-Adic Ultrametric Anal. Appl., 2, 122–132 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Albeverio and W. Karwowski, Stochastic Processes Appl., 53, 1–22 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  25. S. K. Nechaev and O. A. Vasiliev, Proc. Steklov Inst. Math., 245, 169–188 (2004); arXiv:cond-mat/0310079v1 (2003).

    Google Scholar 

  26. V. A. Avetisov, A. Kh. Bikulov, and A. P. Zubarev, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 1(22), 9–15 (2011).

    Article  Google Scholar 

  27. V. A. Avetisov, V. A. Ivanov, D. A. Meshkov, and S. K. Nechaev, JETP Lett., 98, 242–246 (2013).

    Article  ADS  Google Scholar 

  28. F. Murtagh, Multidimensional Clustering Algorithms (Compstat Lect., Vol. 4), Physica, Heidelberg (1985).

    MATH  Google Scholar 

  29. F. Murtagh and P. Contreras, P-Adic Numbers, Ultrametric Anal. Appl., 4, 46–56 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  30. F. Murtagh, P-Adic Numbers, Ultrametric Anal. Appl., 5, 326–337 (2013).

    Article  Google Scholar 

  31. V. A. Lemin, “Finite ultrametric spaces and computer science,” in: Categorical Perspectives (J. Koslowski and A. Melton, eds.), Birkhäuser, Boston, Mass. (2001), pp. 219–241.

    Chapter  Google Scholar 

  32. M. Greenfield, M. Marcolli, and K. Teh, P-Adic Numbers, Ultrametric Anal. Appl., 6, 81–104 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Marcolli and N. Tedeschi, P-Adic Numbers, Ultrametric Anal. Appl., 6, 135–154 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Yu. Lerner and M. D. Missarov, Lett. Math. Phys., 22, 123–129 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. A. Strehl, J. Ghosh, and C. Cardie, J. Machine Learning Research, 3, 583–617 (2002).

    Google Scholar 

  36. S. V. Kozyrev, Theor. Math. Phys., 164, 1163–1168 (2010).

    Article  MATH  Google Scholar 

  37. S. Albeverio and S. V. Kozyrev, P-Adic Numbers, Ultrametric Anal. Appl., 4, 167–178 (2012); arXiv:1204.5952v1 [cs.DS] (2012).

    Article  MathSciNet  MATH  Google Scholar 

  38. S. V. Kozyrev, Theor. Math. Phys., 180, 958–966 (2014); arXiv:1404.6960v1 [math.MG] (2014).

    Article  MathSciNet  MATH  Google Scholar 

  39. B. Dragovich and A. Dragovich, P-Adic Numbers, Ultrametric Anal. Appl., 1, 34–41 (2009); arXiv:q-bio/0607018v1 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  40. B. Dragovich, A. Dragovich, Computer J., 53, 432–442 (2010); arXiv:0707.3043v1 [q-bio.OT] (2007).

    Article  Google Scholar 

  41. A. Yu. Khrennikov and S. V. Kozyrev, Phys. A, 381, 265–272 (2007); arXiv:q-bio.QM/0701007v3 (2007).

    Article  Google Scholar 

  42. A. Yu. Khrennikov and S. V. Kozyrev, J. Theoret. Biology., 261, 396–406 (2009); arXiv:0903.0137v3 [q-bio.GN] (2009).

    Article  MathSciNet  Google Scholar 

  43. S. V. Kozyrev and A. Yu. Khrennikov, Dokl. Math., 81, 128–130 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Yu. Khrennikov and S. V. Kozyrev, P-Adic Numbers, Ultrametric Anal. Appl., 3, 165–168 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Yu. Khrennikov and S. V. Kozyrev, “2-Adic degeneration of the genetic code and energy of binding of codons,” in: Quantum Bio-Informatics III (L. Accardi, W. Freudenberg, and M. Ohya, eds.), World Scientific, Singapore (2010), pp. 193–204.

    Chapter  Google Scholar 

  46. S. V. Kozyrev, Proc. Steklov Inst. Math., 274, 1–84 (2011).

    Article  MathSciNet  Google Scholar 

  47. A. Monna, Analyse non-Archimedienne, Springer, New York (1970).

  48. Yu. I. Manin, “Reflections on arithmetical physics,” in: Conformal Invariance and String Theory (P. Dita and V. Georgescu, eds.), Acad. Press, Boston (1989), pp. 293–303.

    Google Scholar 

  49. I. Ya. Aref’eva, B. Dragovich, P. H. Frampton, and I. V. Volovich, Internat. J. Mod. Phys. A, 6, 4341–4358 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. L. Brekke and P. G. O. Freund, Phys. Rep., 233, 1–66 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  51. V. S. Vladimirov, Proc. Steklov Inst. Math., 245, 3–21 (2004).

    Google Scholar 

  52. V. S. Varadarajan, Proc. Steklov Inst. Math., 245, 258–265 (2004).

    MathSciNet  Google Scholar 

  53. A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems, and Biological Models, Kluwer, Dordrecht (1997).

    Book  Google Scholar 

Download references

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Correspondence to S. V. Kozyrev.

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This research was funded by a grant from the Russian Science Foundation (Project No. 14-50-00005).

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 185, No. 2, pp. 346–360, November, 2015.

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Kozyrev, S.V. Ultrametricity in the theory of complex systems. Theor Math Phys 185, 1665–1677 (2015). https://doi.org/10.1007/s11232-015-0371-2

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