p-Adic mathematical physics: the first 30 years

Review Articles

Abstract

p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

Key words

p-adic numbers adeles ultrametrics p-adic mathematical physics p-adic wavelets complex systems hierarchy p-adic string theory quantum theory gravity cosmology stochastic processes biological systems proteins genetic code cognitive science 

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References

  1. 1.
    S. Albeverio, A. Yu. Khrennikov and V. M. Shelkovich, Theory of p-Adic Distributions: Linear and Nonlinear Models, London Mathematical Society Lecture Note Series 370 (Cambridge Univ. Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi, Dubai, Tokyo, 2010).MATHCrossRefGoogle Scholar
  2. 2.
    S. Albeverio and S. V. Kozyrev, “Pseudodifferential p-adic vector fields and pseudodifferentiation of a composite p-adic function,” p-Adic Numbers Ultrametric Anal. Appl. 2 (1), 21–34 (2010) [arXiv:1105.1506].MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    S. Albeverio and W. Karwowski, “A random walk on p-adic numbers–generator and its spectrum,” Stoch. Proc. Theory Appl. 53, 1–22 (1994).MATHCrossRefGoogle Scholar
  4. 4.
    S. Albeverio and Ya. Belopolskaya, “Stochastic processes in Qp associated with systems of nonlinear PDEs,” p-Adic Numbers Ultrametric Anal. Appl. 1 (2), 105–117 (2009).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. Albeverio and S. V. Kozyrev, “Frames of p-adic wavelets and orbits of the affine group,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 18–33 (2009) [arxiv:0801.4713].MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    S. Albeverio, S. Evdokimov and M. Skopina, “p-Adic multiresolution analysis and wavelet frames,” J. Fourier Anal. Appl. 16 (5), 693–714 (2010) [arXiv:0802.1079v1].MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    S. Albeverio, S. Evdokimov and M. Skopina, “p-Adic multiresolution analyses,” (2008) [arXiv:0810.1147].MATHGoogle Scholar
  8. 8.
    S. Albeverio, S. Evdokimov and M. Skopina, “p-Adic non-orthogonal wavelet bases,” Proc. Steklov Inst. Math. 265, 1–12 (2009).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    S. Albeverio, A. Yu. Khrennikov and V. M. Shelkovich, “Harmonic analysis in the p-adic Lizorkin spaces: fractional operators, pseudo-differential equations, p-adic wavelets, Tauberian theorems,” J. Fourier Anal. Appl. 12 (4), 393–425 (2006).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    S. Albeverio and S. V. Kozyrev, “Multidimensional basis of p-adic wavelets and representation theory,” p-Adic Numbers Ultrametric Anal. Appl. 1 (3), 181–189 (2009) [arXiv:0903.0461].MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    S. Albeverio and S. V. Kozyrev, “Multidimensional p-adic wavelets for the deformed metric,” p-Adic Numbers Ultrametric Anal. Appl. 2 (4), 265–277 (2010) [arXiv:1105.1524].MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    S. Albeverio, A. Yu. Khrennikov and V. M. Shelkovich, “Pseudo-differential operators in the p-adic Lizorkin space,” in p-AdicMathematical Physics, AIP Conference Proceedings 826, 195–205 (2006).Google Scholar
  13. 13.
    S. Albeverio and S. V. Kozyrev, “Multidimensional ultrametric pseudodifferential equations,” Proc. Steklov Math. Inst. 265, 19–35 (2009) [arXiv:0708.2074].MathSciNetMATHGoogle Scholar
  14. 14.
    S. Albeverio, A. Yu. Khrennikov and V. M. Shelkovich, “p-Adic semilinear evolutionary pseudodifferential equations in Lizorkin spaces,” Dokl. Akad. Nauk 415 (3), 295–299 (2007) [Dokl. Math. 76 (1), 539–543 (2007)].MATHGoogle Scholar
  15. 15.
    S. Albeverio, S. Kuzhel, and S. Torba, “p-Adic Schrödinger-type operator with point interactions,” J. Math. Anal. Appl. 338, 1267–1281 (2008).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    S. Albeverio and S. V. Kozyrev, “Clustering by hypergraphs and dimensionality of cluster systems,” p-Adic Numbers Ultrametric Anal. Appl. 4 (3), 167–178 (2012) [arXiv:1204.5952v1].MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    S. Albeverio, R. Cianci and A. Yu. Khrennikov, “p-Adic valued quantization,” p-Adic Number Ultrametric Anal. Appl. 1 (2), 91–104 (2009).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    V. Anashin and A. Khrennikov, Applied Algebraic Dynamics, de Gruyter Expositions in Mathematics (de Gruyter, 2009).MATHCrossRefGoogle Scholar
  19. 19.
    A. Ansari, J. Berendzen, S. F. Bowne, H. Frauenfelder, I. E. T. Iben, T. B. Sauke, E. Shyamsunder and R. D. Young, “Protein states and proteinquakes,” Proc. Natl. Acad. Sci. USA 82, 5000–5004 (1985).CrossRefGoogle Scholar
  20. 20.
    I. Ya. Arefeva, B. Dragovich and I. V. Volovich, “On the p-adic summability of the anharmonic oscillator,” Phys. Lett. B 200, 512–514 (1988).MathSciNetCrossRefGoogle Scholar
  21. 21.
    I. Ya. Aref’eva, B. Dragovich, P. H. Frampton and I. V. Volovich, “The wave function of the Universe and p-adic gravity,” Int. J. Mod. Phys. A 6, 4341–4358 (1991).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    I. Ya. Aref’eva, “Nonlocal string tachyon as a model for cosmological dark energy,” AIP Conf. Proc. 826, 301–311 (2006) [astro-ph/0410443].MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    I. Ya. Aref’eva, A. S. Koshelev and S. Yu. Vernov, “Crossing of the w = −1 barrier by D3-brane dark energy model,” Phys. Rev. D 72, 064017 (2005) [astro-ph/0507067].CrossRefGoogle Scholar
  24. 24.
    I. Ya. Aref’eva, L. V. Joukovskaya and S. Yu. Vernov, “Bouncing and accelerating solutions in nonlocal stringy models,” (2007) [hep-th/0701189].Google Scholar
  25. 25.
    I. Ya. Aref’evaand I. V. Volovich, “Quantization of the Riemann zeta-function and cosmology,” Int. J. Geom. Meth. Mod. Phys. 4, 881–895 (2007).MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    I. Ya. Arefeva and I. V. Volovich, “Cosmological daemon,” JHEP, 2011 8 (2011) 102, 32 pp. [arXiv:1103.0273].MATHCrossRefGoogle Scholar
  27. 27.
    I. Ya. Arefeva and I. V. Volovich, “The master field for QCD and q-deformed quantum field theory,” Nucl. Phys. B 462, 600–612 (1996) [hep-th/9510210].MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    V. A. Avetisov, A. H. Bikulov and S. V. Kozyrev, “Application of p-adic analysis to models of spontaneous breaking of replica symmetry,” J. Phys. A: Math. Gen. 32 (50), 8785–8791 (1999) [arXiv:condmat/9904360].MATHCrossRefGoogle Scholar
  29. 29.
    V. Avetisov, P. L. Krapivsky and S. Nechaev, “Native ultrametricity of sparse random ensembles,” J. Phys. A: Math. Theor. 49 (3), (2016).Google Scholar
  30. 30.
    V. A. Avetisov, A. Kh. Bikulov, S. V. Kozyrev and V. A. Osipov, “p-Adic models of ultrametric diffusion constrained by hierarchical energy landscapes,” J. Phys. A: Math. Gen. 35 (2), 177–189 (2002) [arXiv:condmat/0106506].MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    V. A. Avetisov, A. Kh. Bikulov and V. A. Osipov, “p-Adic models for ultrametric diffusion in conformational dynamics of macromolecules,” Proc. Steklov Inst. Math. 245, 48–57 (2004).MathSciNetMATHGoogle Scholar
  32. 32.
    V. A. Avetisov and A. Kh. Bikulov, “Protein ultrametricity and spectral diffusion in deeply frozen proteins,” Biophys. Rev. Lett. 3 (3), 387 (2008) [arXiv:0804.4551].CrossRefGoogle Scholar
  33. 33.
    V. A. Avetisov, A. Kh. Bikulov and A. P. Zubarev, “First passage time distribution and number of returns for ultrametric random walk,” J. Phys. A:Math. Theor. 42, 085003–085020 (2009) [arXiv:0808.3066].MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    V. A. Avetisov and Yu. N. Zhuravlev, “An evolutionary interpretation of the p-adic ultrametric diffusion equation,” Dokl. Math. 75 (3), 435–455 (2007) [arXiv:0808.3066].MATHCrossRefGoogle Scholar
  35. 35.
    V. A. Avetisov and A. Kh. Bikulov, “Ultrametricity of fluctuation dynamic mobility of protein molecules,” Proc. Steklov Inst. Math. 265 (1), 75–81 (2009).MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    V. A. Avetisov, A. Kh. Bikulov and A. P. Zubarev, “Ultrametric random walk and dynamics of protein molecules,” Proc. Steklov Inst. Math. 285, 3–25 (2014).MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    V. A. Avetisov, A. Kh. Bikulov and V. A. Osipov, “p-Adic description of characteristic relaxation in complex systems,” J. Phys. A:Math. Gen. 36 (15), 4239–4246 (2003).MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    V. A. Avetisov, A. Kh. Bikulov and A. P. Zubarev, “Mathematical modeling of molecular nanomachines,” Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki 1 (22), 9–15 (2011) [in Russian].CrossRefGoogle Scholar
  39. 39.
    V. A. Avetisov, V. A. Ivanov, D. A. Meshkov and S. K. Nechaev, “Fractal globule as a molecular machine, JETP Lett. 98 (4), 242–246 (2013).CrossRefGoogle Scholar
  40. 40.
    N. Barnaby, T. Biswas and J. M. Cline, “p-Adic inflation,” (2006) [hep-th/0612230].Google Scholar
  41. 41.
    J. J. Benedetto and R. L. Benedetto, “A wavelet theory for local fields and related groups,” J. Geom. Anal. 3, 423–456 (2004).MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    R. L. Benedetto, “Examples of wavelets for local fields,” in Wavelets, Frames, and Operator Theory (College Park, MD, 2003) pp. 27–47 (Am. Math. Soc., Providence, RI, 2004).CrossRefGoogle Scholar
  43. 43.
    A. Kh. Bikulov and I. V. Volovich, “p-Adic Brownian motion,” Izv. Math. 61 (3), 537–552 (1997).MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    A. Kh. Bikulov, “Stochastic p-adic equations of mathematical physics,” Theor. Math. Phys. 119 (2), 594–604 (1999).MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    A. Kh. Bikulov, A. P. Zubarev and L. V. Kaidalova, “Hierarchical dynamical model of financial market near the crash point and p-adic analysis,” Vest. Samarsk. Gosud. Tekhn. Univer. Seriya Fiziko-Matem. Nauki 42, 135–141 (2006) [in Russian].CrossRefGoogle Scholar
  46. 46.
    K. Binder and A. P. Young, “Spin glasses: Experimental facts, theoretical concepts, and open questions,” Rev. Mod. Phys. 58, 801–976 (1986).CrossRefGoogle Scholar
  47. 47.
    O. M. Becker and M. Karplus, “The topology of multidimensional protein energy surfaces: Theory and application to peptide structure and kinetics,” J. Chem. Phys. 106, 1495–1517 (1997).CrossRefGoogle Scholar
  48. 48.
    A. Blumen, J. Klafter and G. Zumofen, “Random walks on ultrametric spaces: low temperature patterns,” J. Phys. A:Math. Gen. 19, L861 (1986).CrossRefGoogle Scholar
  49. 49.
    P. E. Bradley, “On p-adic classification,” p-Adic Numbers Ultrametric Anal. Appl. 1 (4), 271–283 (2009).MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    P. E. Bradley, “A p-adic RANSAC algorithm for stereo vision using Hensel lifting,” p-Adic Numbers Ultrametric Anal. Appl. 2 (1), 55–67 (2010).MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    P. E. Bradley, “From image processing to topological modelling with p-adic numbers,” p-Adic Numbers Ultrametric Anal. Appl. 2 (4), 293–304 (2010).MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    P. E. Bradley, “Ultrametricity indices for the Euclidean and Boolean hypercubes,” p-Adic Numbers Ultrametric Anal. Appl. 8 (4), 298–311 (2016).MathSciNetCrossRefGoogle Scholar
  53. 53.
    L. Brekke and P. G. O. Freund, “p-Adic numbers in physics,” Phys. Rep. 233 (1), 1–66 (1993).MathSciNetCrossRefGoogle Scholar
  54. 54.
    G. Calcagni, “Cosmological tachyon from cubic string field theory,” JHEP 05 012 (2006) [hep-th/0512259].Google Scholar
  55. 55.
    L. F. Chacon-Cortes and W. A. Zuniga-Galindo, “Nonlocal operators, parabolic-type equations, and ultrametric random walks,” J. Math. Phys. 54 (11), 113503 (2013).MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    D. M. Carlucci and C. De Dominicis, “On the replica Fourier transform,” Compt. Rendus Ac. Sci. Ser. IIB Mech. Phys. Chem. Astr. 325, 527 (1997) [arXiv:cond-mat/9709200].MATHGoogle Scholar
  57. 57.
    O. Casas-Sanchez and W. A. Zuniga-Galindo, “Riesz kernels and pseudodifferential operators attached to quadratic forms over p-adic fields,” p-Adic Numbers Ultrametric Anal. Appl. 5 (3), 177–193 (2013).MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    I. Daubechies, Ten Lectures on Wavelets, CBMS-NSR Series in Appl. Math. (SIAM, 1992).MATHCrossRefGoogle Scholar
  59. 59.
    C. De Dominicis, D. M. Carlucci and T. Temesvari, “Replica Fourier tansforms on ultrametric trees, and block-diagonalizing multi-replica matrices,” J. Physique I (France) 7, 105–115 (1997) [arXiv:condmat/9703132].CrossRefGoogle Scholar
  60. 60.
    I. Dimitrijevic, B. Dragovich, J. Stankovic, A. S. Koshelev and Z. Rakic, “On nonlocal modified gravity and its cosmological solutions,” Springer Proc. Math. & Stat. 191, 35–51 (2016) [arXiv:1701.02090[hep-th]].CrossRefGoogle Scholar
  61. 61.
    I. Dimitrijevic, B. Dragovich, J. Grujic and Z. Rakic, “Some Cosmological Solutions of a Nonlocal Modified Gravity,” Filomat 29 (3), 619–628 (2015) [arXiv:1508.05583[hep-th]].MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    I. Dimitrijevic, B. Dragovich, J. Grujic and Z. Rakic, “New cosmological solutions in nonlocal modified gravity,” Romanian J. Phys. 58 (5-6), 550–559 (2013) [arXiv:1302.2794[gr-qc]].MathSciNetMATHGoogle Scholar
  63. 63.
    I. Dimitrijevic, B. Dragovich, J. Grujic, A. S. Koshelev and Z. Rakic, “Cosmology of modified gravity with a non-local f(R),” (2015) [arXiv:1509.04254[hep-th]].Google Scholar
  64. 64.
    D. D. Dimitrijevic, G. S. Djordjevic and Lj. Nesic, “Quantum cosmology and tachyons,” Fortsch. Physik (Progr. Phys.) 56 (4-5), 412–417 (2008).MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    G. S. Djordjević, B. Dragovich, Lj. D. Nešićand I. V. Volovich, “p-Adic and adelicminisuperspace quantum cosmology,” Int. J. Mod. Phys. A 17 (10), 1413–1433 (2002) [arXiv:gr-qc/0105050].MATHCrossRefGoogle Scholar
  66. 66.
    G. S. Djordjevic, L. Nesic and D. Radovancevic, Signature Change in p-Adic and Noncommutative FRW Cosmology, International Journal ofModern Physics A, 29, 1450155 (2014)MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    D. D. Dimitrijevic, G. S. Djordjevic and M. Milosevic, “Classicalization and quantization of tachyon-like matter in (non)Archimedean spaces,” Roman. Rep. Phys. 68 (1), 5–18 (2016).Google Scholar
  68. 68.
    G. S. Djordjevic, Lj. Nesic and D. Radovancevic, “Two-oscillator KantowskiЦSachs model of the Schwarzschild black hole interior,” Gen. Relativ. Gravit. 48, 106 (2016).MathSciNetCrossRefGoogle Scholar
  69. 69.
    B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On p-adic mathematical physics,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 1–17 (2009) [arXiv:0904.4205].MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    B. Dragovich, “On measurements, numbers and p-adicmathematical physics,” p-Adic Numbers Ultrametric Anal. Appl. 4 (2), 102–108 (2012).MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    B. Dragovich, “p-Adic perturbation series and adelic summability,” Phys. Lett. B 256 (3, 4), 392–39 (1991).MathSciNetCrossRefGoogle Scholar
  72. 72.
    B. G. Dragovich, “Power series everywhere convergent on R and Q p,” J. Math. Phys. 34 (3), 1143–1148 (1992) [arXiv:math-ph/0402037].MathSciNetCrossRefGoogle Scholar
  73. 73.
    B. G. Dragovich, “On p-adic aspects of some perturbation series,” Theor. Math. Phys. 93 (2), 1225–1231 (1993).MathSciNetCrossRefGoogle Scholar
  74. 74.
    B. G. Dragovich, “Rational summation of p-adic series,” Theor. Math. Phys. 100 (3), 1055–1064 (1994).MathSciNetCrossRefGoogle Scholar
  75. 75.
    B. Dragovich, “On p-adic series inmathematical physics,” Proc. Steklov Inst. Math. 203, 255–270 (1994).Google Scholar
  76. 76.
    B. Dragovich, “On p-adic series with rational sums,” Scientific Review 19–20, 97–104 (1996).MathSciNetGoogle Scholar
  77. 77.
    B. Dragovich, “On some p-adic series with factorials,” in p-Adic Functional Analysis, Lect. Notes Pure Appl. Math. 192, 95–105 (Marcel Dekker, 1997) [arXiv:math-ph/0402050].MathSciNetMATHGoogle Scholar
  78. 78.
    B. Dragovich, “On p-adic power series,” in p-Adic Functional Analysis, Lect. Notes Pure Appl. Math. 207, 65–75 (Marcel Dekker, 1999) [arXiv:math-ph/0402051].MathSciNetMATHGoogle Scholar
  79. 79.
    B. Dragovich, “On some finite sums with factorials,” Facta Universitatis: Ser. Math. Inform. 14, 1–10 (1999) [arXiv:math/0404487 [math.NT]].MathSciNetGoogle Scholar
  80. 80.
    M. de Gosson, B. Dragovich and A. Khrennikov, “Some p-adic differential equations,” in p-Adic Functional Analysis, Lect. Notes Pure Appl. Math. 222, 91–112 (Marcel Dekker, 2001) [arXiv:math-ph/0010023].MathSciNetMATHGoogle Scholar
  81. 81.
    B. Dragovich and N. Z. Misic, “p-Adic invariant summation of some p-adic functional series,” p-Adic Numbers Ultrametric Anal. Appl. 6 (4), 275–283 (2014) [arXiv:1411.4195v1 [math. NT]].MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    B. Dragovich, A. Yu. Khrennikov and N. Ž. Mišić, “Summation of p-adic functional series in integer points,” Filomat 31 (5), 1339–1347 (2017), [arXiv:1508.05079 [math.NT]].Google Scholar
  83. 83.
    B. Dragovich, “On summation of p-adic series,” accepted for publication in Contem. Math., AMS, [arXiv:1702.02569 [math. NT]] (2017).Google Scholar
  84. 84.
    N. J. A. Sloane, The on-line encyclopedia of integer sequences, https://oeis.org/.Google Scholar
  85. 85.
    B. Dragovich, “Zeta strings,” [arXiv:hep-th/0703008] (2007).Google Scholar
  86. 86.
    B. Dragovich, “Zeta-nonlocal scalar fields,” Theor. Math. Phys. 157 (3), 1669–1675 (2008) [arXiv:0804.4114[hep-th]].MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    B. Dragovich, “Some Lagrangians with zeta function nonlocality,” [arXiv:0805.0403 [hep-th]] (2008).MATHGoogle Scholar
  88. 88.
    B. Dragovich, “Lagrangians with Riemann zeta function,” Rom. J. Phys. 53 (9-10), 1105–1110 (2008) [arXiv:0809.1601[hep-th]].MathSciNetMATHGoogle Scholar
  89. 89.
    B. Dragovich, “Towards effective Lagrangians for adelic strings,” Fortschr. Phys. 57 (5-7), 546–551 (2009) [arXiv:0902.0295[hep-th]].MathSciNetMATHCrossRefGoogle Scholar
  90. 90.
    B. Dragovich, “The p-adic sector of the adelic string,” Theor. Math. Phys. 163 (3), 768–773 (2010) [arXiv:0911.3625[hep-th]].MathSciNetMATHCrossRefGoogle Scholar
  91. 91.
    B. Dragovich, “Nonlocal dynamics of p-adic strings,” Theor. Math. Phys. 164 (3), 1151–1155 (2010) [arXiv:1011.0912[hep-th]].MathSciNetMATHCrossRefGoogle Scholar
  92. 92.
    B. Dragovich, “Adeles in mathematical physics,” (2007) [arXiv:0707.3876[math-ph]].Google Scholar
  93. 93.
    B. Dragovich, “Adelic wave function of the Universe,” in Proc. Third A. Friedmann Int. Seminar on Grav. and Cosmology, Eds. Yu. N. Gnedin, A. A. Grib and V. M. Mostepanenko, pp. 311–321 (Friedmann Lab. Publishing, St. Petersburg, 1995).Google Scholar
  94. 94.
    B. Dragovich and Lj. Nešić, “p-Adic and adelic generalization of quantum cosmology,” Gravitat. Cosmol. 5, 222–228 (1999) [arXiv:gr-qc/0005103].MathSciNetMATHGoogle Scholar
  95. 95.
    B. Dragovich, “p-Adic and adelic cosmology: p-adic origin of dark energy and dark matter,” AIP Conf. Proc. 826, 25–42 (2006) [arXiv:hep-th/0602044].MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    B. Dragovich, “Towards p-adic matter in the universe,” Springer Proc. Math & Stat. 36, 13–24 (2013) [arXiv:1205.4409[hep-th]].MathSciNetMATHGoogle Scholar
  97. 97.
    B. Dragovich, “Adelic model of harmonic oscillator,” Theor. Math. Phys. 101, 1404–1415 (1994) [arXiv:hep-th/0402193].MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    B. Dragovich, “Adelic harmonic oscillator,” Int. J. Mod. Phys. A 10, 2349–2365 (1995) [arXiv:hepth/0404160].MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    B. Dragovich, “p-Adic and adelic quantum mechanics,” Proc. Steklov Inst. Math. 245, 72–85 (2004) [arXiv:hep-th/0312046].MathSciNetMATHGoogle Scholar
  100. 100.
    G. Djordjevićand B. Dragovich, “p-Adic path integrals for quadratic actions,” Mod. Phys. Lett. A 12 (20), 1455–1463 (1997) [arXiv:math-ph/0005026].MathSciNetMATHCrossRefGoogle Scholar
  101. 101.
    G. Djordjević, B. Dragovich and Lj. Nešić, “Adelic path intergals for quadratic Lagrangians,” Infin. Dimens. Anal. Quan. Prob. Relat. Topics 6, 179–195 (2003) [arXiv:hep-th/0105030].MATHCrossRefGoogle Scholar
  102. 102.
    B. Dragovich and Z. Rakic, “Path integrals for quadratic Lagrangians on p-adic and adelic spaces,” p-Adic Numbers Ultrametric Anal Appl. 2 (4), 322–340 (2010) [arXiv:1011.6589[math-ph]].MathSciNetMATHCrossRefGoogle Scholar
  103. 103.
    B. Dragovich, A. Khrennikov and D. Mihajlovic, “Linear fractional p-adic and adelic dynamical systems,” Rep. Math. Phys. 60, 55–68 (2007) [arXiv:math-ph/0612058].MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    B. Dragovich and A. Yu. Dragovich, “A p-adic model of DNA sequence and genetic code,” (2006) [arXiv:qbio.GN/0607018].MATHGoogle Scholar
  105. 105.
    B. Dragovich and A. Yu. Dragovich, “A p-adic model of DNA sequence and genetic code,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 34–41 (2009) [arXiv:q-bio.GN/0607018].MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    B. Dragovich and A. Yu. Dragovich, “p-Adic modelling of the genome and the genetic code,” Comp. J. 53, 432–442 (2010) [arXiv:0707.3043, doi:10. 1093/comjnl/bxm083].MATHCrossRefGoogle Scholar
  107. 107.
    B. Dragovich, “Genetic code and number theory,” FactaUniversitatis: Phys. Chem. Techn. 14 (3), 225–241 (2016) [arXiv:0911.4014[q-bio. OT]].Google Scholar
  108. 108.
    B. Dragovich, “p-Adic structure of the genetic code,” (2012) [arXiv:1202.2353[q-bio. OT]].MATHGoogle Scholar
  109. 109.
    B. Dragovich, “On ultrametricity in bioinformation systems,” in Conference Proceedings Theoretical Approaches to Bioinformation Systems (ABIS. 2013 Conference, Belgrade, 17-22. 09. 2013; (published by Institute of Physics, Belgrade, 2014).Google Scholar
  110. 110.
    B. Dragovich, A. Yu. Khrennikov and N. Ž. Mišić, “Ultrametrics in the genetic code and the genome,” to be published in Appl. Math. Comput. (2017).Google Scholar
  111. 111.
    B. Dragovich and D. Joksimović, “On possible uses of p-adic analysis in econometrics,” Megatrend Revija 4 (2), 5–16 (2007).Google Scholar
  112. 112.
    S. N. Evans, “Local field Brownian motion,” J. Theor. Probab. 6, 817–850 (1993).MathSciNetMATHCrossRefGoogle Scholar
  113. 113.
    Yu. A. Farkov, “Orthogonal wavelets with compact support on locally compact Abelian groups,” Izv. Ross. Akad. Nauk, Ser. Mat. 69 (3), 193–220 (2005) [Izv. Math. 69, 623–650 (2005)].MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    S. Fischenko and E. I. Zelenov, “p–Adic models of turbulence,” in p-Adic Mathematical Physics, AIP Conference Proceedings 286, 174–191 (Melville, New York, 2006).MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    K. H. Fischer and J. A. Hertz, Spin Glasses (Cambridge Univ. Press, 1993).Google Scholar
  116. 116.
    H. Frauenfelder, S. G. Sligar and P. G. Wolynes, “The energy landscapes and motions of proteins,” Science 254 5038, 1598–1603 (1991).CrossRefGoogle Scholar
  117. 117.
    H. Frauenfelder, B. H. McMahon and P. W. Fenimore, “Myoglobin: the hydrogen atom of biology and paradigm of complexity, PNAS 100 (15), 8615–8617 (2003).CrossRefGoogle Scholar
  118. 118.
    Y. V. Fyodorov, A. Ossipov and A. Rodriguez, “The Anderson localization transition and eigenfunction multifractality in an ensemble of ultrametric random matrices,” J. Stat. Mech.: Theory Exp. 12, L12001 (2009).CrossRefGoogle Scholar
  119. 119.
    N. Ganikhodjaev, F. Mukhamedov and C. H. Pah, “Phase diagram of the three states Potts model with next nearest neighbour interactions on the Bethe lattice,” Phys. Lett. A 373 (1), 33–38 (2008).MathSciNetMATHCrossRefGoogle Scholar
  120. 120.
    I. M. Gelfand, M. I. Graev and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions (Saunders, Philadelphia, 1969).Google Scholar
  121. 121.
    C. Hara, R. Iijima, H. Kaneko and H. Matsumoto, “Orlicz norm and Sobolev-Orlicz capacity on ends of tree based on probabilistic Bessel kernels,” p-Adic Numbers Ultrametric Anal. Appl. 7 (1), 24–38 (2015).MathSciNetMATHCrossRefGoogle Scholar
  122. 122.
    S. Haran, “Analytic potential theory over p-adics,” Ann. Inst. Fourier 43, 905–944 (1993).MathSciNetMATHCrossRefGoogle Scholar
  123. 123.
    K. H. Hoffmann and P. Sibani, “Diffusion in hierarchies,” Phys. Rev. A 38, 4261–4270 (1988).MathSciNetCrossRefGoogle Scholar
  124. 124.
    A. IlićStepić, Z. Ognjanović, N. Ikodinovićand A. Perović, “p-Adic probability logics,” p-Adic Numbers Ultrametric Anal. Appl. 8 (3), 177–203 (2016).MathSciNetCrossRefGoogle Scholar
  125. 125.
    A. Imai, H. Kaneko and H. Matsumoto, “A Dirichlet space associated with consistent networks on the ring of p-adic integers,” p-Adic Numbers Ultrametric Anal. Appl. 3 (4), 309–325 (2011).MathSciNetMATHCrossRefGoogle Scholar
  126. 126.
    R. S. Ismagilov, “On the spectrum of the self-adjoint operator in L2(K) where K is a local field: an analog of the Feynman-Kac formula,” Theor. Math. Phys. 89, 1024–1028 (1991).CrossRefGoogle Scholar
  127. 127.
    W. C. Lang, “Wavelet analysis on the Cantor dyadic group,” Houston J. Math. 24, 533–544 (1998).MathSciNetMATHGoogle Scholar
  128. 128.
    W. C. Lang, “Orthogonal wavelets on the Cantor dyadic group,” SIAM J. Math. Anal. 27, 305–312 (1996).MathSciNetMATHCrossRefGoogle Scholar
  129. 129.
    M. L. Lapidus and Lu Hung, “Self-similar p-adic fractal strings and their complex dimensions,” p-Adic Numbers Ultrametric Anal. Appl. 1 (2), 167–180 (2009).MathSciNetMATHCrossRefGoogle Scholar
  130. 130.
    J. W. de Jong, Graphs, spectral triples and Dirac zeta functions p-Adic Numbers Ultrametric Anal. Appl. 1 (3) 286–296 (2009).MATHGoogle Scholar
  131. 131.
    H. Kaneko and K. Yasuda, “Capacities associated with Dirichlet space on an infinite extension of a local field,” ForumMath. 17, 1011–1032 (2005).MathSciNetMATHGoogle Scholar
  132. 132.
    K. Kamizono, “p-Adic Brownian motion over Qp,” Proc. Steklov Inst. Math. 265, 115–130 (2009).MathSciNetMATHCrossRefGoogle Scholar
  133. 133.
    H. Kaneko and A. N. Kochubei, “Weak solutions of stochastic differential equations over the field of p-adic numbers,” Tohoku Math. J. 59, 547–564 (2007).MathSciNetMATHCrossRefGoogle Scholar
  134. 134.
    H. Kaneko, “Fractal theoretic aspects of local field,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 51–57 (2009).MathSciNetMATHCrossRefGoogle Scholar
  135. 135.
    W. Karwowski and K. Yasuda, “Dirichlet forms for diffusion in R2 and jumps on fractals: The regularity problem,” p-Adic Numbers Ultrametric Anal. Appl. 2 (4), 341–359 (2010).MathSciNetMATHCrossRefGoogle Scholar
  136. 136.
    E. King and M. A. Skopina, “Quincunx multiresolution analysis for L 2(Q2 2),” p-Adic Numbers Ultrametric Anal. Appl. 2 (3), 222–231 (2010).MathSciNetMATHCrossRefGoogle Scholar
  137. 137.
    A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer, Dordrecht, 1994).MATHCrossRefGoogle Scholar
  138. 138.
    A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer Acad. Publ., Dordrecht, 1997).MATHCrossRefGoogle Scholar
  139. 139.
    A. Yu. Khrennikov, Non-Archimedean Analysis and Its Applications (Fizmatlit, Moscow, 2003) [in Russian].MATHGoogle Scholar
  140. 140.
    A. Yu. Khrennikov and S. V. Kozyrev, “Wavelets on ultrametric spaces,” Appl. Comput. Harm. Anal. 19, 61–76 (2005).MathSciNetMATHCrossRefGoogle Scholar
  141. 141.
    A. Yu. Khrennikov and S. V. Kozyrev, “Ultrametric random field,” Infin. Dimens. Anal. Quan. Prob. Related Topics 9 (2), 199–213 (2006) [arXiv:math/0603584].MathSciNetMATHCrossRefGoogle Scholar
  142. 142.
    A. Yu. Khrennikov, S. V. Kozyrev, K. Oleschko, A. G. Jaramillo and M. de Jesús Correa López, “Application of p-adic analysis to time series,” Infin. Dimens. Anal. Quant. Probab. Relat. Topics, 16 (4), 1350030, 15 pp. (2013) [arXiv:1312.3878].MathSciNetMATHCrossRefGoogle Scholar
  143. 143.
    A. Yu. Khrennikov and V. M. Shelkovich, “Non-Haar p-adic wavelets and their application to pseudodifferential operators and equations,” Appl. Comput. Harm. Anal. 28 (1), 1–23 (2010) [arXiv:0808.3338v1].MATHCrossRefGoogle Scholar
  144. 144.
    A. Yu. Khrennikov, V. M. Shelkovich and M. Skopina, “p-Adic refinable functions and MRA-based wavelets,” J. Approx. Theory 161, 226–238 (2009) [arXiv:0711.2820].MathSciNetMATHCrossRefGoogle Scholar
  145. 145.
    A. Yu. Khrennikov and V. M. Shelkovich, “Distributional asymptotics and p-adic Tauberian and Shannon- Kotelnikov theorems,” Asympt. Anal. 46 (2), 163–187 (2006).MathSciNetMATHGoogle Scholar
  146. 146.
    A. Yu. Khrennikov and V. M. Shelkovich, “p-Adic multidimensional wavelets and their application to p-adic pseudo-differential operators,” (2006) [arXiv:math-ph/0612049].MATHGoogle Scholar
  147. 147.
    A. Yu. Khrennikov, V. M. Shelkovich and M. Skopina, “p-Adic orthogonal wavelet bases,” p-Adic Numbers Ultrametric Anal. Appl. 1 (2), 145–156 (2009).MathSciNetMATHCrossRefGoogle Scholar
  148. 148.
    A. Yu. Khrennikov and V. M. Shelkovich, “Non-Haar p-adic wavelets and pseudodifferential operators,” Dokl. Akad. Nauk 418 (2), 167–170 (2008) [Dokl. Math. 77 (1), 42–45 (2008)].MathSciNetMATHGoogle Scholar
  149. 149.
    A. Yu. Khrennikov and V. M. Shelkovich, “An infinite family of p-adic non-Haar wavelet bases and pseudodifferential operators,” p-Adic Numbers Ultrametric Anal. Appl. 1 (3), 204–216 (2009).MathSciNetMATHCrossRefGoogle Scholar
  150. 150.
    A. Yu. Khrennikov and S. V. Kozyrev, “Wavelets and the Cauchy problem for the Schrödinger equation on analytic ultrametric space,” in Proceedings of the 2nd Conference on Mathematical Modelling ofWave Phenomena 2005 (14–19 August 2005, Växjö, Sweden), eds. B. Nilsson, L. Fishman, AIP Conference Proceedings 834, 344–350 (Melville, New York, 2006).Google Scholar
  151. 151.
    A. Khrennikov, V. M. Shelkovich and J. H. Van Der Walt, “Adelic multiresolution analysis, construction of wavelet bases and pseudo-differential operators,” J. Fourier Anal. Appl. 19, 1323–1358 (2013).MathSciNetMATHCrossRefGoogle Scholar
  152. 152.
    A. Yu. Khrennikov and S. V. Kozyrev, “Replica symmetry breaking related to a general ultrametric space I: Replica matrices and functionals,” Physica A: Stat. Mech. Appl. 359, 222–240 (2006) [arXiv:condmat/0603685].CrossRefGoogle Scholar
  153. 153.
    A. Yu. Khrennikov and S. V. Kozyrev, “Replica symmetry breaking related to a general ultrametric space II: RSB solutions and the n → 0 limit,” Physica A: Stat. Mech. Appl. 359, 241–266 (2006) [arXiv:condmat/0603687].CrossRefGoogle Scholar
  154. 154.
    A. Yu. Khrennikov and S. V. Kozyrev, “Replica symmetry breaking related to a general ultrametric space III: The case of general measure,” Physica A: Stat. Mech. Appl. 378 (2), 283–298 (2007) [arXiv:condmat/0603694].CrossRefGoogle Scholar
  155. 155.
    A. Yu. Khrennikov, F. M. Mukhamedov and J. F. Mendes, “On p-adic Gibbsmeasures of the countable state Potts model on the Cayley tree,” Nonlinearity 20, 2923–2937 (2007).MathSciNetMATHCrossRefGoogle Scholar
  156. 156.
    A. Khrennikov, S. Kozyrev and A. Mansson, “Hierarchical model of the actomyosin molecular motor based on ultrametric diffusion with drift,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 (2), 1550013, 16 pp. (2015) [arXiv:1312.7528].MathSciNetMATHCrossRefGoogle Scholar
  157. 157.
    A. Yu. Khrennikov and S. V. Kozyrev, “Replica procedure for probabilistic algorithms as a model of gene duplication,” Dokl. Math. 84 (2), 726–729 (2011) [arXiv:1105.2893].MathSciNetMATHCrossRefGoogle Scholar
  158. 158.
    A. Yu. Khrennikov, “p-Adic information space and gene expression,” in Integrative Approaches to Brain Complexity, Eds. S. Grant, N. Heintz and J. Noebels, p. 14 (Wellcome Trust Publ., 2006).Google Scholar
  159. 159.
    A. Yu. Khrennikov and S. V. Kozyrev, “Genetic code on the dyadic plane,” Physica A: Stat. Mech. Appl. 381, 265–272 (2007) [arXiv:q-bio.QM/0701007].CrossRefGoogle Scholar
  160. 160.
    A. Yu. Khrennikov and S. V. Kozyrev, “2-Adic clustering of the PAMmatrix,” J. Theor. Biol. 261, 396–406 (2009) [arXiv:0903.0137].CrossRefGoogle Scholar
  161. 161.
    A. Yu. Khrennikov and S. V. Kozyrev, “Genetic code and deformation of the 2-dimensional 2-adic metric,” p-Adic Numbers Ultrametric Anal. Appl. 3 (2), 165–168 (2011).MathSciNetMATHCrossRefGoogle Scholar
  162. 162.
    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 pp. 193–204, eds. L. Accardi, W. Freudenberg and M. Ohya (World Scientific, 2010).CrossRefGoogle Scholar
  163. 163.
    A. N. Kochubei, Pseudo-Differential Equations and Stochastics overNon-Archimedean Fields (Marcel Dekker, New York, USA, 2001).MATHCrossRefGoogle Scholar
  164. 164.
    A. N. Kochubei, “A non-Archimedean wave equation,” Pacif. J. Math. 235, 245–261 (2008).MathSciNetMATHCrossRefGoogle Scholar
  165. 165.
    A. N. Kochubei, “Analysis and probability over infinite extensions of a local field,” Potential Anal. 10, 305–325 (1999).MathSciNetMATHCrossRefGoogle Scholar
  166. 166.
    A. N. Kochubei, “Hausdorff measure for a stable-like process over an infinite extension of a local field,” J. Theor. Probab. 15, 951–972 (2002).MathSciNetMATHCrossRefGoogle Scholar
  167. 167.
    A. N. Kochubei, Analysis in Positive Characteristic (Cambridge Univ. Press, Cambridge, 2009).MATHCrossRefGoogle Scholar
  168. 168.
    A. N. Kochubei, “Parabolic equations over the field of p-adic numbers,” Math. USSR Izv. 39, 1263–1280 (1992).MathSciNetCrossRefGoogle Scholar
  169. 169.
    A. N. Kochubei, “Radial solutions of non-Archimedean pseudodifferential equations,” Pacific J. Math. 269 (2), 355–369 (2014).MathSciNetMATHCrossRefGoogle Scholar
  170. 170.
    S. V. Kozyrev, “Ultrametricity in the theory of complex systems,” Theor. Math. Phys. 185 (2), 46–360 (2015).MathSciNetMATHCrossRefGoogle Scholar
  171. 171.
    S. V. Kozyrev, “Methods and applications of ultrametric and p-adic analysis: From wavelet theory to biophysics,” Proc. Steklov Inst. Math. 274 suppl. (1), 1–84 (2011).MathSciNetMATHCrossRefGoogle Scholar
  172. 172.
    S. V. Kozyrev, A. Yu. Khrennikov and V. M. Shelkovich, “p-Adic wavelets and their applications,” Proc. Steklov Inst. Math. 285, 157–196 (2014).MathSciNetMATHCrossRefGoogle Scholar
  173. 173.
    S. V. Kozyrev and A. Yu. Khrennikov, “Pseudodifferential operators on ultrametric spaces and ultrametric wavelets,” Izv. Ross. Akad. Nauk., Ser. Mat. 69 (5), 133–148 (2005) [Izv. Math. 69, 989–1003 (2005)] [arXiv:math-ph/0412062].MathSciNetMATHCrossRefGoogle Scholar
  174. 174.
    S. V. Kozyrev, “Wavelets and spectral analysis of ultrametric pseudodifferential operators,” Mat. Sbornik 198 (1), 103–126 (2007) [Sb. Math. 198, 97–116 (2007)] [arXiv:math-ph/0412082].MathSciNetMATHCrossRefGoogle Scholar
  175. 175.
    S. V. Kozyrev, “Wavelet theory as p-adic spectral analysis,” Izvest. Math. 66 (2), 367–376 (2002) [arXiv:math-ph/0012019].MathSciNetMATHCrossRefGoogle Scholar
  176. 176.
    S. V. Kozyrev, “p-Adic pseudodifferential operators and p-adic wavelets,” Theor. Math. Phys. 138 (3), 322–332 (2004) [arXiv:math-ph/0303045].MathSciNetMATHCrossRefGoogle Scholar
  177. 177.
    S. V. Kozyrev, “Toward an ultrametric theory of turbulence,” Theor. Math. Phys. 157 (3), 1711–1720 (2008) [arXiv:0803.2719].MATHCrossRefGoogle Scholar
  178. 178.
    S. V. Kozyrev and A. Yu. Khrennikov, “Localization in space for a free particle in ultrametric quantum mechanics,” Dokl. Akad. Nauk 411 (3), 319–322 (2006) [Dokl. Math. 74 (3), 906–909 (2006)].MathSciNetMATHGoogle Scholar
  179. 179.
    S. V. Kozyrev, “p-Adic pseudodifferential operators: Methods and applications,” Proc. Steklov Inst. Math. 245, 143–153 (2004).MATHGoogle Scholar
  180. 180.
    S. V. Kozyrev, V. Al. Osipov and V. A. Avetisov, “Nondegenerate ultrametric diffusion,” J. Math. Phys. 46, 063302–063317 (2005).MathSciNetMATHCrossRefGoogle Scholar
  181. 181.
    S. V. Kozyrev and A. Yu. Khrennikov, “p-Adic integral operators in wavelet bases,” Dokl. Akad. Nauk 437 (4), 457–461 (2011) [Dokl. Math. 83 (2), 209–212 (2011)].MathSciNetMATHGoogle Scholar
  182. 182.
    S. V. Kozyrev, “Dynamics on rugged landscapes of energy and ultrametric diffusion,” p-Adic Numbers Ultrametric Anal. Appl. 2 (2), 122–132 (2010).MathSciNetMATHCrossRefGoogle Scholar
  183. 183.
    S. V. Kozyrev, “Model of protein fragments and statistical potentials,” p-Adic Numbers Ultrametric Anal. Appl. 8 (4), 325–337 (2016) [arXiv:1504.03940].MathSciNetCrossRefGoogle Scholar
  184. 184.
    S. V. Kozyrev and A. Yu. Khrennikov, “2-Adic numbers in genetics and Rumer’s symmetry,” Dokl. Math. 81 (1), 128–130 (2010).MathSciNetMATHCrossRefGoogle Scholar
  185. 185.
    S. V. Kozyrev, “Multidimensional clustering and hypergraphs,” Theor. Math. Phys. 164 (3), 1163–1168 (2010).MATHCrossRefGoogle Scholar
  186. 186.
    S. V. Kozyrev, “Cluster networks and Bruhat-Tits buildings,” Theor. Math. Phys. 180 (2), 958–966 (2014) [arXiv:1404.6960].MathSciNetMATHCrossRefGoogle Scholar
  187. 187.
    A. V. Kosyak, A. Khrennikov and V. M. Shelkovich, “Wavelet bases on adele rings,” Dokl. Math. 85, 75–79 (2012).MathSciNetMATHCrossRefGoogle Scholar
  188. 188.
    A. V. Kosyak, A. Khrennikov and V. M. Shelkovich, “Pseudodifferential operators on adele rings and wavelet bases,” Dokl. Math. 85, 358–362 (2012).MathSciNetMATHCrossRefGoogle Scholar
  189. 189.
    Yu. I. Manin, “Numbers as functions,” p-Adic Numbers Ultrametric Anal. Appl. 5 (4), 313–325 (2013).MathSciNetMATHCrossRefGoogle Scholar
  190. 190.
    Yu. I. Manin, “Painleve VI equations in p-adic time,” p-Adic Numbers Ultrametric Anal. Appl. 8 (3), 217–224 (2016).MathSciNetMATHCrossRefGoogle Scholar
  191. 191.
    M. Greenfield, M. Marcolli and K. Teh, “Twisted spectral triples and quantum statistical mechanical systems,” p-Adic Numbers Ultrametric Anal. Appl. 6 (2), 81–104 (2014).MathSciNetMATHCrossRefGoogle Scholar
  192. 192.
    M. Marcolli and N. Tedeschi, “Multifractals, Mumford curves and eternal inflation,” p-Adic Numbers Ultrametric Anal. Appl. 6 (2), 135–154 (2014).MathSciNetMATHCrossRefGoogle Scholar
  193. 193.
    J. Marcinek and M. Marcolli, “KMS weights on higher rank buildings,” p-Adic Numbers Ultrametric Anal. Appl. 8 (1), 45–67 (2016).MathSciNetMATHCrossRefGoogle Scholar
  194. 194.
    M. Marcolli, “Cyclotomy and endomotives,” p-Adic Numbers Ultrametric Anal. Appl. 1 (3), 217–263 (2009).MathSciNetMATHCrossRefGoogle Scholar
  195. 195.
    Y. Meyer, Wavelets and Operators (Cambridge Univ. Press, Cambridge, 1992).MATHGoogle Scholar
  196. 196.
    M. Mezard, G. Parisi and M. Virasoro, Spin-Glass Theory and Beyond (World Scientific, Singapore, 1987).MATHGoogle Scholar
  197. 197.
    M. D. Missarov and R. G. Stepanov, “Asymptotic properties of combinatorial optimization problems in padic space,” p-Adic Numbers Ultrametric Anal. Appl. 3 (2), 114–128 (2011).MathSciNetMATHCrossRefGoogle Scholar
  198. 198.
    M. D. Missarov, p-Adic renormalization group solutions and the euclidean renormalization group conjectures p-Adic Numbers Ultrametric Anal. Appl. 4 (2), 109–114 (2012).MathSciNetMATHGoogle Scholar
  199. 199.
    A. Monna, Analyse non-Archimedienne (Springer-Verlag, New York, 1970).MATHCrossRefGoogle Scholar
  200. 200.
    A. Morozov, “Are there p-adic knot invariants?,” Theor. Math. Phys. 187 (1), 447–454 (2016) [arXiv:1509.04928].MathSciNetMATHCrossRefGoogle Scholar
  201. 201.
    F. Mukhamedov and U. Rozikov, “On inhomogeneous p-adic Potts model on a Cayley tree,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2), 277–290 (2005).MathSciNetMATHCrossRefGoogle Scholar
  202. 202.
    F. Mukhamedov and H. Akin, “The p-adic Potts model on the Cayley tree of order three,” Theor. Math. Phys. 176 (3), 1267–1279 (2013).MathSciNetMATHCrossRefGoogle Scholar
  203. 203.
    F. Mukhamedov, “A dynamical system approach to phase transitions for p-adic Potts model on the Cayley tree of order two,” Rep. Math. Phys. 70 (3), 385–406 (2012).MathSciNetMATHCrossRefGoogle Scholar
  204. 204.
    F. Mukhamedov, “On p-adic quasi Gibbs measures for q + 1-state Potts model on the Cayley tree,” p-Adic Numbers Ultrametric Anal. Appl. 2 (3), 241–251 (2010).MathSciNetMATHCrossRefGoogle Scholar
  205. 205.
    F. Mukhamedov, “On the existence of generalizedGibbs measures for the one-dimensional p-adic countable state Potts model,” Proc. Steklov Inst. Math. 265 (1), 165–176 (2009).MathSciNetMATHCrossRefGoogle Scholar
  206. 206.
    F. Mukhamedov, M. Saburov and O. Khakimov, “On p-adic Ising-Vannimenus model on an arbitrary order Cayley tree,” J. Stat. Mech.: Theory Exper. 5, P05032 (2015).MathSciNetCrossRefGoogle Scholar
  207. 207.
    F. Mukhamedov and O. Khakimov, “On periodic Gibbs measures of p-adic Potts model on a Cayley tree,” p-Adic Numbers Ultrametric Anal. Appl. 8 (3), 225–235 (2016).MathSciNetMATHCrossRefGoogle Scholar
  208. 208.
    F. Murtagh and A. Heck, Multivariate Data Analysis (Springer Science & BusinessMedia, 2012).MATHGoogle Scholar
  209. 209.
    F. Murtagh and P. Contreras, “Algorithms for hierarchical clustering: an overview,” Wiley Interdisci. Reviews: Data Mining and Knowledge Discovery 2 (1), 86–97 (2012).Google Scholar
  210. 210.
    F. Murtagh and P. Legendre, “Ward’s hierarchical agglomerative clustering method: which algorithms implementWard’s criterion?,” J. Classif. 31 (3), 274–295 (2014).MATHCrossRefGoogle Scholar
  211. 211.
    F. Murtagh, “Sparse p-adic data coding for computationally efficient and effective big data analytics,” p-Adic Numbers Ultrametric Anal. Appl. 8 (3), 236–247 (2016).MathSciNetMATHCrossRefGoogle Scholar
  212. 212.
    F. Murtagh, Multidimensional Clustering Algorithms (Physica-Verlag, Heidelberg, 1985).MATHGoogle Scholar
  213. 213.
    F. Murtagh, “From data to the p-adic or ultrametric model,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 58–68 (2009).MathSciNetMATHCrossRefGoogle Scholar
  214. 214.
    S. K. Nechaev and O. A. Vasiliev, “On the metric structure of ultrametric spaces,” Proc. Steklov Inst. Math. 245, 169–188 (2004) [arXiv:cond-mat/0310079].MathSciNetMATHGoogle Scholar
  215. 215.
    A. N. Nekrasov, “Analysis of the information structure of protein sequences: a new method for analyzing the domain organization of proteins,” J. Biomol. Struct. Dyn. 21 (5), 615–624 (2004).CrossRefGoogle Scholar
  216. 216.
    A. N. Nekrasov, A. A. Anashkina and A. I. Zinchenko, “A new paradigm of protein structural organization,” in Proceedings of the 2-nd International Conference “Theoretical Approaches to Bioinformatic Systems” (TABIS. 2013), pp. 1–23 (Belgrade, Serbia, Sept. 17–22, 2013).Google Scholar
  217. 217.
    I. Ya. Novikov and M. A. Skopina, “Why are Haar bases in various structures the same?,”Mat. Zametki 91 (6), 950–953 (2012).MathSciNetMATHCrossRefGoogle Scholar
  218. 218.
    A. T. Ogielski and D. L. Stein, “Dynamics on ultrametric spaces,” Phys. Rev. Lett. 55, 1634–1637 (1985).MathSciNetCrossRefGoogle Scholar
  219. 219.
    G. Parisi and N. Sourlas, “p-Adic numbers and replica symmetry breaking,” Europ. Phys. J. B 14, 535–542 (2000) [arXiv:cond-mat/9906095].MathSciNetCrossRefGoogle Scholar
  220. 220.
    G. Rammal, M. A. Toulouse and M. A. Virasoro, “Ultrametricity for physicists,” Rev. Mod. Phys. 58, 765–788 (1986).MathSciNetCrossRefGoogle Scholar
  221. 221.
    J. J. Rodriguez-Vega and W. A. Zuniga-Galindo, “Taibleson operators, p-adic parabolic equations and ultrametric diffusion,” Pacif. J. Math. 237, 327–347 (2008).MathSciNetMATHCrossRefGoogle Scholar
  222. 222.
    V. M. Shelkovich and M. Skopina, “p-Adic Haar multiresolution analysis and pseudo-differential operators,” J. Fourier Anal. Appl. 15 (3), 366–393 (2009).MathSciNetMATHCrossRefGoogle Scholar
  223. 223.
    S. M. Torba and W. A. Zuniga-Galindo, “Parabolic type equations and Markov stochastic processes on adeles,” J. Fourier Anal. Appl. 19 (4), 792–835 (2013).MathSciNetMATHCrossRefGoogle Scholar
  224. 224.
    M. Talagrand, Spin glasses, a Challenge forMathematicians (Springer-Verlag, 2003).Google Scholar
  225. 225.
    C. Micheletti, F. Seno and A. Maritan, “Recurrent oligomers in proteins: An optimal scheme reconciling accurate and concise backbone representations in automated folding and design studies,” Proteins: Struct. Funct. Genet. 40, 662–674 (2000).CrossRefGoogle Scholar
  226. 226.
    A. G. de Brevern, C. Etchebest and S. Hazout, “Bayesian probabilistic approach for predicting backbone structures in terms of protein blocks,” Proteins: Struct. Funct. Genet. 41, 271–287 (2000).CrossRefGoogle Scholar
  227. 227.
    A. Y. Grosberg, S. K. Nechaev and E. I. Shakhnovich, “The role of topological constraints in the kinetics of collapse of macromolecules,” J. Physique 49, 2095–2100 (1988).CrossRefGoogle Scholar
  228. 228.
    J. D. Halverson, J. Smrek, K. Kremer, A. Y. Grosberg, “From a melt of rings to chromosome territories: the role of topological constraints in genome folding,” Rep. Prog. Phys. 77, 022601, 24 pp. (2014).Google Scholar
  229. 229.
    M. Imakaev, K. Tchourine, S. Nechaev and L. Mirny, “Effects of topological constraints on globular polymers,” Soft Matter 11, 665–671 (2015).CrossRefGoogle Scholar
  230. 230.
    G. Fudenberg, G. Getz, M. Meyerson and L. A. Mirny, “High order chromatin architecture shapes the landscape of chromosomal alterations in cancer,” Nature Biotech. 29, 1109–1113 (2011).CrossRefGoogle Scholar
  231. 231.
    B. Bonev and G. Cavalli, “Organization and function of the 3D genome,” Nature Rev. Genet. 17, 661–678 (2016).CrossRefGoogle Scholar
  232. 232.
    G. E. Hinton and R. R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” Science 313, 504–507 (2006).MathSciNetMATHCrossRefGoogle Scholar
  233. 233.
    Y. Bengio, “Learning deep architectures for AI,” Found. TrendsMach. Learn. 2 (1), (2009).Google Scholar
  234. 234.
    C. Hennig, M. Meila, F. Murtagh and R. Rocci, Handbook of Cluster Analysis (CRC Press, 2015).MATHGoogle Scholar
  235. 235.
    C. Linnaeus, Systema naturae (Leiden, 1735).Google Scholar
  236. 236.
    V. A. Lemin, “Finite ultrametric spaces and computer science,” pp. 219–241, in Categorical Perspectives, Trends in Mathematics (Springer, 2001).MATHGoogle Scholar
  237. 237.
    F. Bruhat and J. Tits, “Groupes reductifs sur un corps local, I. Donnees radicielles valuees,” Publ. Math. IHES 41, 5–251 (1972).MATHCrossRefGoogle Scholar
  238. 238.
    P. B. Garrett, Buildings and Classical Groups (Chapman and Hall, London, 1997).MATHCrossRefGoogle Scholar
  239. 239.
    A. Strehl, J. Ghosh and C. Cardie, “Cluster ensembles–a knowledge reuse framework for combining multiple partitions,” J. Mach. Learn. Res. 3, 583–617 (2002).MathSciNetMATHGoogle Scholar
  240. 240.
    E. Bauer and R. Kohavi, “An empirical comparison of voting classification algorithms: bagging, boosting, and variants,” Mach. Learn. 36, 105–139 (1999).CrossRefGoogle Scholar
  241. 241.
    D. H. Huson, R. Rupp and C. Scornavacca, Phylogenetic Networks (Cambridge Univ. Press, Cambridge, 2010).CrossRefGoogle Scholar
  242. 242.
    A. Dress, K. T. Huber, J. Koolen, V. Moulton and A. Spillner, Basic Phylogenetic Combinatorics (Cambridge Univ. Press, Cambridge, 2012).MATHGoogle Scholar
  243. 243.
    E. V. Koonin, The Logic of Chance: The Nature and Origin of Biological Evolution (FT Press Science, 2012).Google Scholar
  244. 244.
    J.-L. Starck, F. Murtagh and J. Fadili, Sparse Image and Signal Processing: Wavelets and Related Geometric Multiscale Analysis (Cambridge Univ. Press, 2015).MATHCrossRefGoogle Scholar
  245. 245.
    M. N. Khokhlova and I. V. Volovich, “Modeling theory and hypergraph of classes,” Proc. Steklov Inst. Math. 245, 266–272 (2004).MATHGoogle Scholar
  246. 246.
    J. Q. Trelewicz and I. V. Volovich, “Analysis of business connections utilizing theory of topology of random graphs,” in p-AdicMathematical Physics AIP Conf. Proc. 826, 330–344 (Melville, New York, 2006).Google Scholar
  247. 247.
    S. Albeverio and W. Karwowski, Diffusion on p-Adic Numbers (Bielefeld-Bochum Stochastik, 1990).MATHGoogle Scholar
  248. 248.
    A. Khrennikov and M. Nilsen. p-Adic Deterministic and Random Dynamics, Math. Appl. 574 (Kluwer Acad. Publishers, Dordrecht, 2004).Google Scholar
  249. 249.
    S. Evans, “Local fields, Gaussian measures, and Brownian motions,” Topics in Lie Groups and Probability: Boundary Theory (J. Taylor ed.) CRM Proceedings and Lecture Notes 28, American Math. Society.Google Scholar
  250. 250.
    G. Parisi, “Infinite number of order parameters for spin-glasses,” Phys. Rev. Lett. 43, 1754 (1979).CrossRefGoogle Scholar
  251. 251.
    M. Mezard, G. Parisi, N. Sourlas, G. Toulouse and M. Virasoro, “Nature of the spin-glass phase,” Phys. Rev. Lett. 52, 1156 (1984).MATHCrossRefGoogle Scholar
  252. 252.
    P. Diaconis, “Random walk on groups: characters and geometry,” C. M. Campbell et al. (eds) (Groups St. Andrews, 2001) Cambridge Univ. Press 1 120–142, (2003).MATHGoogle Scholar
  253. 253.
    S. Evans, “Local field U-statistics,” Algebraic Methods in Statistics and Probability (Marlos A. G. Viana and Donald St. P. Richards eds.) Contemp. Math. 287, (2001).Google Scholar
  254. 254.
    P. P. Varju, “Random walks in compact groups,” DocumentaMath. 18, 1137–1175 (2013).MathSciNetMATHGoogle Scholar
  255. 255.
    S. Mustafa, “Random walks on unimodular p-adic groups,” Stoch. Proc. Appl. 115, 927–937 (2005).MathSciNetMATHCrossRefGoogle Scholar
  256. 256.
    V. Anashin, A. Khrennikov and E. Yurova, “T-functions revisited: new criteria for bijectivity/transitivity,” Designs Codes Crypt. 7, 383–407 (2014).MathSciNetMATHCrossRefGoogle Scholar
  257. 257.
    V. Anashin, “Uniformly distributed sequences of p-adic integers,” Math. Notes 55, 109–133 (1994).MathSciNetMATHCrossRefGoogle Scholar
  258. 258.
    V. Anashin, “Ergodic transformations in the space of p-adic integers,” Proc. Int. Conf. on p-adic Mathematical Physics, AIP Conference Proceedings 826, 3–24 (2006).MathSciNetMATHCrossRefGoogle Scholar
  259. 259.
    V. Anashin, A. Khrennikov and E. Yurova, “Ergodicity criteria for non-expanding transformations of 2-adic spheres,” Disc. Cont. Dyn. Syst. 34, 367–377 (2014).MathSciNetMATHGoogle Scholar
  260. 260.
    A. Khrennikov and E. Yurova, “Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis,” J. Number Theor. 133 (2), 484–491 (2013).MathSciNetMATHCrossRefGoogle Scholar
  261. 261.
    E. Yurova Axelsson, “On recent results of ergodic property for p-adic dynamical systems,” p-Adic Numbers Ultrametric Anal. Appl. 6, 235–257 (2014).MathSciNetCrossRefGoogle Scholar
  262. 262.
    A. Khrennikov and E. Yurova, “Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions,” Chaos Solit. Fract. 60, 11–30 (2014).MathSciNetMATHCrossRefGoogle Scholar
  263. 263.
    A. Khrennikov and E. Yurova, “Secure cloud computations: Description of (fully)homomorphic ciphers within the p-adic model of encryption,” (2016) [arXiv:1603.07699 [cs. CR]].Google Scholar
  264. 264.
    H. Qusay, “Demystifying cloud computing,” J. Defense Softw. Engin., CrossTalk N 1/2, 16–21 (2011).Google Scholar
  265. 265.
    P. Mell and T. Grance, “The NIST definition of cloud computing,” Technical report, National Institute of Standards and Technology: U. S. Department of Commerce. Special publication 800–145 (2011).Google Scholar
  266. 266.
    A. Khrennikov, “Human subconscious as the p-adic dynamical system’,’ J. Theor. Biol. 193, 179–196 (1998).CrossRefGoogle Scholar
  267. 267.
    A. Khrennikov, Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena, Ser. Fundamental Theories of Physics (Kluwer, Dordreht, 2004).MATHCrossRefGoogle Scholar
  268. 268.
    A. Khrennikov, Classical and Quantum Mental Models and Freud’s Theory of Unconscious Mind (VäxjöUniv. Press, Växjö, 2002).Google Scholar
  269. 269.
    F. Murtagh, “Ultrametric model of mind, I: Review,” p-Adic Numbers Ultrametric Anal. Appl. 4, 193–206 (2012).MathSciNetCrossRefGoogle Scholar
  270. 270.
    F. Murtagh, “Ultrametric model of mind, II: Application to text content analysis,” p-Adic Numbers Ultrametric Anal. Appl. 4, 207–221 (2012).MathSciNetCrossRefGoogle Scholar
  271. 271.
    F. Murtagh, “The new science of complex systems through ultrametric analysis: Application to search and discovery, to narrative and to thinking,” p-Adic Numbers Ultrametric Anal. Appl. 5, 326–337 (2013).CrossRefGoogle Scholar
  272. 272.
    G. Iurato and A. Khrennikov, “Hysteresis model of unconscious-conscious interconnection: Exploring dynamics on m-adic trees,” p-Adic Numbers Ultrametric Anal. Appl. 7, 312–321 (2015).MathSciNetMATHCrossRefGoogle Scholar
  273. 273.
    G. Iurato, A. Khrennikov and F. Murtagh, “Formal foundations for the origins of human consciousness,” p-Adic Numbers Ultrametric Anal. Appl. 8, 249–279 (2016).MathSciNetCrossRefGoogle Scholar
  274. 274.
    G. Iurato and A. Khrennikov, “On the topological structure of a mathematical model of human unconscious,” p-Adic Numbers Ultrametric Anal. Appl. 9, 78–81 (2017).MathSciNetCrossRefGoogle Scholar
  275. 275.
    A. Khrennikov and N. Kotovich, “Towards ultrametric modeling of unconscious creativity,” Int. J. Cogn. Inform. Natural Intell. 8, 98–109 (2014).CrossRefGoogle Scholar
  276. 276.
    A. Khrennikov, K. Oleschko and M. de Jesus Correa Lopez, “Modeling fluid’s dynamics with master equations in ultrametric spaces representing the treelike structure of capillary networks,” Entropy 18, 249 (2016).MathSciNetCrossRefGoogle Scholar
  277. 277.
    A. Yu. Khrennikov, K. Oleschko and M. de Jesus Correa Lopez, “Applications of p-adic numbers: from physics to geology,” Contemp. Math. 665, 121–131 (2016).MathSciNetMATHCrossRefGoogle Scholar
  278. 278.
    A. Khrennikov, K. Oleschko and M. de Jesus Correa Lopez, “Application of p-adic wavelets to model reaction-diffusion dynamics in random porous media,” J. Fourier Anal. Appl. 22, 809–822 (2016).MathSciNetMATHCrossRefGoogle Scholar
  279. 279.
    K. Oleschko and A. Khrennikov, “Applications of p-adics to geophysics: Linear and quasilinear diffusion of water-in-oil and oil-in-water emulsions,” Theor. Math. Phys. 190, 154–163 (2017).MathSciNetCrossRefGoogle Scholar
  280. 280.
    L. A. Richards, “Capillary conduction of liquids through porous mediums,” Physics 1 (5), 318–333 (1931).MATHCrossRefGoogle Scholar
  281. 281.
    J. Richter, The Soil as a Reactor (Catena, 1987).Google Scholar
  282. 282.
    M. Th. van Genuchten, “A closed-form equation for predicting the hydraulic conductivity of unsaturated soils,” Soil Sc. Soc. America J. 44 (5), 892–898 (1980).CrossRefGoogle Scholar
  283. 283.
    S. V. Kozyrev, “Ultrametric analysis and interbasin kinetics,” AIP Conf. Proc. 826, 121–128 (2006).MathSciNetMATHCrossRefGoogle Scholar
  284. 284.
    S. Albeverio, A. Yu. Khrennikov and V. M. Shelkovich, “The Cauchy problems for evolutionary pseudodifferential equations over p-adic field and the wavelet theory,” J. Math. Anal. Appl. 375, 82–98 (2011).MathSciNetMATHCrossRefGoogle Scholar
  285. 285.
    A. Yu. Khrennikov and A. N. Kochubei, “p-Adic analogue of the porous medium equation,” (2016) [arXiv:1611.08863 [math. AP]].Google Scholar
  286. 286.
    A. Wiles, “Modular elliptic curves and Fermat’s last theorem, Annals Math. 141, 443–551 (1995)MathSciNetMATHCrossRefGoogle Scholar
  287. 286a.
    R. Taylor and A. Wiles, “Ring-theoretic properties of certain Hecke algebras,” Annals Math. 141, 553–572 (1995)MathSciNetMATHCrossRefGoogle Scholar
  288. 286b.
    Yves Hellegouarch, Invitation to the Mathematics of Fermat-Wiles (Academic Press, 2001).MATHGoogle Scholar
  289. 287.
    R. P. Langlands, “Problems in the theory of automorphic forms,” Lect. Notes Math. 170, 18–61 (Springer Verlag, 1970).Google Scholar
  290. 288.
    L. Hormander, The Analysis of Linear Partial Differential Operators, III Pseudo-Differential Operators, IV Fourier Integral Operators (Springer-Verlag, 1985)MATHGoogle Scholar
  291. 288a.
    M. A. Shubin, Pseudodifferential Operators and Spectral Theory (Nauka, 1978).MATHGoogle Scholar
  292. 289.
    N. N. Bogoliubov and D. V. Schirkov, Introduction to the Theory of Quantized Fields (Springer, 1984).Google Scholar
  293. 290.
    Yu. I. Manin, “Reflections on arithmetical physics,” in Poiana Brasov 1987, Proceedings, Conformal Invariance and String Theory, 293–303 (Acad. Press, Boston 1989).Google Scholar
  294. 291.
    E. C. Titchmarsh, The Theory of the Riemann Zeta-Function (Clarendon Press, Oxford, 1986)MATHGoogle Scholar
  295. 291a.
    A. A. Karatsuba and S. M. Voronin, The Riemann Zeta-Function (Walter de Gruyter Publishers, Berlin-New York 1992MATHCrossRefGoogle Scholar
  296. 291b.
    K. Chandrasekharan, Arithmetic Functions (Springer, 1972).Google Scholar
  297. 292.
    A. Kapustin and E. Witten, “Electric-magnetic duality and the geometric Langlands program,” (2006) [hep-th/0604151]MATHGoogle Scholar
  298. 292a.
    S. Gukov and E. Witten, “Gauge theory, ramification, and the geometric Langlands program,” (2006) [hep-th/0612073]MATHGoogle Scholar
  299. 292b.
    E. Frenkel, “Lectures on the Langlands program and conformal field theory,” (2005) [hep-th/0512172].MATHGoogle Scholar
  300. 293.
    N. M. Katz and P. Sarnak, “Zeroes of zeta-functions and symmetry,” Bull. Amer. Math. Soc. (N. Y.) 36, 1–26 (1999)MathSciNetMATHCrossRefGoogle Scholar
  301. 293a.
    J. P. Keating and N. C. Snaith, “Random matrix theory and L-functions at s = 1/2,” Comm. Math. Phys. 214, 91–110 (2000).MathSciNetMATHCrossRefGoogle Scholar
  302. 294.
    D. Harlow, S. Shenker, D. Stanford and L. Susskind, “Eternal symmetree,” Phys. Rev. D 85, 063516 (2012) [arXiv:1110.0496].CrossRefGoogle Scholar
  303. 295.
    Yu. I. Manin and M. Marcolli, “Big Bang, blowup, and modular curves: algebraic geometry in cosmology,” SIGMA 10, 073 (2014) [arXiv:1402.2158].MathSciNetMATHGoogle Scholar
  304. 296.
    Yu. I. Manin and M. Marcolli, “Symbolic dynamics, modular curves, and Bianchi IX cosmologies,” (2015) [arXiv:1504.04005].MATHGoogle Scholar
  305. 297.
    W. Fan, F. Fathizadeh and M. Marcolli, “Modular forms in the spectral action of Bianchi IX gravitational instantons,” (2015) [arXiv:1511.05321].Google Scholar
  306. 298.
    Zhi Hu and Sen Hu, “Symplectic group and Heisenberg group in p-adic quantum mechanics,” [arXiv:1502.01789].Google Scholar
  307. 299.
    S. S. Gubser, J. Knaute, S. Parikh, A. Samberg and P. Witaszczyk, “p-adic AdS/CFT,” Comm. Math. Phys. 352 (3), 1019–1059 (2017) [arXiv:1605.01061].MathSciNetCrossRefGoogle Scholar
  308. 300.
    M. Heydeman, M. Marcolli, I. Saberi and B. Stoica, “Tensor networks, “p-Adic fields, and algebraic curves: arithmetic and the AdS3/CFT2 correspondence,” (2016) [arXiv:1605.07639].Google Scholar
  309. 301.
    S. S. Gubser, M. Heydeman, C. Jepsen, M. Marcolli, S. Parikh, I. Saberi, B. Stoica and B. Trundy, “Edge length dynamics on graphs with applications to p-adic AdS/CFT,” (2016) [arXiv:1612.09580].Google Scholar
  310. 302.
    S. S. Gubser, C. Jepsen, S. Parikh and B. Trundy, “O(N) and O(N) and O(N),” (2017) [arXiv:1703.04202].Google Scholar
  311. 303.
    Spin Glasses and Biology, Ed. D. L. Stein (World Scientific, Singapore, 1992).Google Scholar
  312. 304.
    W. Schikhof, Ultrametric Calculus: an introduction to p-adic analysis (Cambridge Univ. Press, 1985).MATHCrossRefGoogle Scholar
  313. 305.
    R. Unger, D. Harel, S. Wherland and J. L. Sussman, “A 3D building blocks approach to analyzing and predicting structure of proteins,” Proteins: Struct. Funct. Genet. 5, 355–373 (1989).CrossRefGoogle Scholar
  314. 306.
    V. S. Varadarajan, “Path integrals for a class of p-adic Schrödinger equations,” Lett. Math. Phys. 39, 97–106 (1997).MathSciNetMATHCrossRefGoogle Scholar
  315. 307.
    V. S. Varadarajan, “Arithmetic quantum physics: why, what, and whither,” in Selected Topics of p-Adic Mathematical Physics and Analysis, Proc. V. A. Steklov Inst. Math. 245, 273–280 (2005).Google Scholar
  316. 308.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).MATHCrossRefGoogle Scholar
  317. 309.
    V. S. Vladimirov, “Generalized functions over the field of p-adic numbers,” Russ. Math. Surv. 43, 19–64 (1988).MathSciNetMATHCrossRefGoogle Scholar
  318. 310.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, “Spectral theory in p-adic quantum mechanics, and representation theory,” Math. USSR-Izv. 36 (2), 281–309 (1991).MathSciNetMATHCrossRefGoogle Scholar
  319. 311.
    V. S. Vladimirov and I. V. Volovich, “p-Adic quantum mechanics,” Comm. Math. Phys. 123, 659–676 (1989).MathSciNetMATHCrossRefGoogle Scholar
  320. 312.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, “The spectral theory in the p-adic quantum mechanics,” Izvestia Akad. Nauk SSSR, Ser. Mat. 54, 275–302 (1990).MATHGoogle Scholar
  321. 313.
    V. S. Vladimirov and Ya. I. Volovich, “Nonlinear dynamics equation in p-adic string theory,” Theor. Math. Phys. 138, 297 (2004) [math-ph/0306018].MATHCrossRefGoogle Scholar
  322. 314.
    V. S. Vladimirov, “On the non-linear equation of a p-adic open string for a scalar field,” Russian Math. Surv. 60 (6), 1077–1092 (2005).MATHCrossRefGoogle Scholar
  323. 315.
    V. S. Vladimirov, “Nonlinear equations for p-adic open, closed, and open-closed strings,” Theor. Math. Phys. 149 (3), 1604–1616 (2006).MathSciNetMATHCrossRefGoogle Scholar
  324. 316.
    V. S. Vladimirov, “On the equations for p-adic closed and open strings,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 79–87 (2009).MathSciNetMATHCrossRefGoogle Scholar
  325. 317.
    V. S. Vladimirov, “Solutions of p-adic string equations,” Theor. Math. Phys. 167 (2), 539–546 (2011).MathSciNetMATHCrossRefGoogle Scholar
  326. 318.
    V. S. Vladimirov, “Nonexistence of solutions of the p-adic strings,” Theor. Math. Phys. 174 (2), 178–185 (2013).MathSciNetMATHCrossRefGoogle Scholar
  327. 319.
    I. V. Volovich, “D-branes, black holes and SU(∞) gauge theory,” (1996) [hep-th/9608137]Google Scholar
  328. 319a.
    I. V. Volovich, “From p-adic strings to etale strings,” Proc. SteklovMath. Inst. 203, 41–48 (1994).MATHGoogle Scholar
  329. 320.
    I. V. Volovich, “p-Adic string,” Class. Quant. Grav. 4 (4), L83–L87 (1987).MathSciNetCrossRefGoogle Scholar
  330. 321.
    I. V. Volovich, “Number theory as the ultimate physical theory,” p-Adic Numbers Ultrametric Anal. Appl. 2 (1), 77–87 (2010).MathSciNetMATHCrossRefGoogle Scholar
  331. 322.
    I. V. Volovich, “Time irreversibility problem and functional formulation of classical mechanics,” (2009) [arXiv:0907.2445].Google Scholar
  332. 323.
    A. S. Trushechkin and I. V. Volovich, “Functional classical mechanics and rational numbers,” p-Adic Numbers Ultrametric Anal. Appl. 1 (4), 361–367 (2009.MathSciNetMATHCrossRefGoogle Scholar
  333. 324.
    I. V. Volovich, “Functional stochastic classical mechanics,” p-Adic Numbers Ultrametric Anal. Appl. 7 (1), 56–70 (2015).MathSciNetMATHCrossRefGoogle Scholar
  334. 325.
    I. V. Volovich, “Bogolyubov’s equations and functional mechanics,” Theor. Math. Phys. bf 164 (3), 1128–1135 (2010).MATHCrossRefGoogle Scholar
  335. 326.
    I. V. Volovich, “Randomness in classical mechanics and quantum mechanics,” Found. Phys. 41 (3), 516–528 (2011).MathSciNetMATHCrossRefGoogle Scholar
  336. 327.
    K. Yasuda, “Extension of measures to infinite dimensional spaces over p-adic field,” Osaka J. Math. 37, 967–985 (2000).MathSciNetMATHGoogle Scholar
  337. 328.
    K. Yasuda, “Additive processes on local fields,” J. Math. Sci. Univ. Tokyo 3, 629–654 (1996).MathSciNetMATHGoogle Scholar
  338. 329.
    K. Yasuda, “Limit theorems for p-adic valued asymmetric semistable laws and processes,” p-Adic Numbers Ultrametric Anal. Appl. 9 (1), 62–77 (2017).MathSciNetCrossRefGoogle Scholar
  339. 330.
    E. Zelenov, “p-Adic law of large numbers,” Izv. RAN Ser. Math. 80 (3), 31–42 (2016).MathSciNetMATHCrossRefGoogle Scholar
  340. 331.
    E. Zelenov, “p-Adic Brownian motion,” Izv. RAN Ser. Math. 80 (6), 92–102 (2016).MathSciNetMATHCrossRefGoogle Scholar
  341. 332.
    E. I. Zelenov, “Adelic decoherence,” p-Adic Numbers Ultrametric Anal. Appl. 4 (1), 84–87 (2012).MathSciNetMATHCrossRefGoogle Scholar
  342. 333.
    A. P. Zubarev, “On stochastic generation of ultrametrics in high-dimensional Euclidean spaces,” p-Adic Numbers Ultrametric Anal. Appl. 6 (2), 155–165 (2014).MathSciNetMATHCrossRefGoogle Scholar
  343. 334.
    W. A. Zuniga-Galindo, “Parabolic equations and Markov processes over p-adic fields,” Potential Anal. 28, 185–200 (2008).MathSciNetMATHCrossRefGoogle Scholar
  344. 335.
    W. A. Zuniga-Galindo, “Pseudo-differential equations connected with p-adic forms and local zeta functions,” Bull. Austral. Math. Soc., 70 (1), 73–86 (2004).MathSciNetMATHCrossRefGoogle Scholar
  345. 336.
    W. A. Zuniga-Galindo, “Fundamental solutions of pseudo-differential operators over p-adic fields,” Rend. Sem. Mat. Univ. Padova 109, 241–245 (2003).MathSciNetMATHGoogle Scholar
  346. 337.
    W. A. Zuniga-Galindo, “Local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields,” p-Adic Numbers Ultrametric Anal. Appl. 3 (4), 344–358 (2011).MathSciNetMATHCrossRefGoogle Scholar
  347. 338.
    W. A. Zuniga-Galindo, Pseudodifferential Equations Over Non-Archimedean Spaces, Lecture Notes in Mathematics 2174 (Springer, 2017).MATHGoogle Scholar
  348. 339.
    J. Galeano-Penaloza and W. A. Zuniga-Galindo, “Pseudo-differential operators with semi-quasielliptic symbols over p-adic fields,” J. Math. Anal. Appl. 386 (1), 32–49 (2012).MathSciNetMATHCrossRefGoogle Scholar
  349. 340.
    W. A. Zuniga-Galindo, “The Cauchy problem for non-Archimedean pseudodifferential equations of Klein–Gordon type,” J. Math. Anal. Appl. 420 (2), 1033–1050 (2014).MathSciNetMATHCrossRefGoogle Scholar
  350. 341.
    W. A. Zuniga-Galindo, “The non-Archimedean stochastic heat equation driven by Gaussian noise,” J. Fourier Anal. Appl. 21 (3), 600–627 (2015).MathSciNetMATHCrossRefGoogle Scholar
  351. 342.
    W. A. Zuniga-Galindo, “Non-Archimedean white noise, pseudodifferential stochastic equations, and massive Euclidean fields,” J. Fourier Anal. Appl. 23 (2), 288–323 (2017).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Institute of PhysicsUniversity of BelgradeBelgradeSerbia
  2. 2.Mathematical InstituteSerbian Academy of Sciences and ArtsBelgradeSerbia
  3. 3.International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive ScienceLinnaeus UniversityVäxjöSweden
  4. 4.National Research University of Information Technologies, Mechanics and Optics (ITMO)St. PetersburgRussia
  5. 5.Steklov Mathematical Institute of the Russian Academy of SciencesMoscowRussia

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