Skip to main content
SpringerLink
Log in

p-Adic mathematical physics: the first 30 years

  • Review Articles
  • Published:
p-Adic Numbers, Ultrametric Analysis and Applications Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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).

    Book  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Albeverio, S. Evdokimov and M. Skopina, “p-Adic multiresolution analyses,” (2008) [arXiv:0810.1147].

    MATH  Google Scholar 

  8. S. Albeverio, S. Evdokimov and M. Skopina, “p-Adic non-orthogonal wavelet bases,” Proc. Steklov Inst. Math. 265, 1–12 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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. S. Albeverio and S. V. Kozyrev, “Multidimensional ultrametric pseudodifferential equations,” Proc. Steklov Math. Inst. 265, 19–35 (2009) [arXiv:0708.2074].

    MathSciNet  MATH  Google Scholar 

  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)].

    MATH  Google Scholar 

  15. S. Albeverio, S. Kuzhel, and S. Torba, “p-Adic Schrödinger-type operator with point interactions,” J. Math. Anal. Appl. 338, 1267–1281 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Albeverio, R. Cianci and A. Yu. Khrennikov, “p-Adic valued quantization,” p-Adic Number Ultrametric Anal. Appl. 1 (2), 91–104 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Anashin and A. Khrennikov, Applied Algebraic Dynamics, de Gruyter Expositions in Mathematics (de Gruyter, 2009).

    Book  MATH  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  Google Scholar 

  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. 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).

    Article  MathSciNet  MATH  Google Scholar 

  26. I. Ya. Arefeva and I. V. Volovich, “Cosmological daemon,” JHEP, 2011 8 (2011) 102, 32 pp. [arXiv:1103.0273].

    Article  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MATH  Google Scholar 

  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. 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].

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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].

    Article  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  40. N. Barnaby, T. Biswas and J. M. Cline, “p-Adic inflation,” (2006) [hep-th/0612230].

    Google Scholar 

  41. J. J. Benedetto and R. L. Benedetto, “A wavelet theory for local fields and related groups,” J. Geom. Anal. 3, 423–456 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Chapter  Google Scholar 

  43. A. Kh. Bikulov and I. V. Volovich, “p-Adic Brownian motion,” Izv. Math. 61 (3), 537–552 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Kh. Bikulov, “Stochastic p-adic equations of mathematical physics,” Theor. Math. Phys. 119 (2), 594–604 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  Google Scholar 

  46. K. Binder and A. P. Young, “Spin glasses: Experimental facts, theoretical concepts, and open questions,” Rev. Mod. Phys. 58, 801–976 (1986).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  48. A. Blumen, J. Klafter and G. Zumofen, “Random walks on ultrametric spaces: low temperature patterns,” J. Phys. A:Math. Gen. 19, L861 (1986).

    Article  Google Scholar 

  49. P. E. Bradley, “On p-adic classification,” p-Adic Numbers Ultrametric Anal. Appl. 1 (4), 271–283 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  52. P. E. Bradley, “Ultrametricity indices for the Euclidean and Boolean hypercubes,” p-Adic Numbers Ultrametric Anal. Appl. 8 (4), 298–311 (2016).

    Article  MathSciNet  Google Scholar 

  53. L. Brekke and P. G. O. Freund, “p-Adic numbers in physics,” Phys. Rep. 233 (1), 1–66 (1993).

    Article  MathSciNet  Google Scholar 

  54. G. Calcagni, “Cosmological tachyon from cubic string field theory,” JHEP 05 012 (2006) [hep-th/0512259].

    Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  58. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSR Series in Appl. Math. (SIAM, 1992).

    Book  MATH  Google Scholar 

  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].

    Article  Google Scholar 

  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]].

    Article  Google Scholar 

  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]].

    Article  MathSciNet  MATH  Google Scholar 

  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]].

    MathSciNet  MATH  Google Scholar 

  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. D. D. Dimitrijevic, G. S. Djordjevic and Lj. Nesic, “Quantum cosmology and tachyons,” Fortsch. Physik (Progr. Phys.) 56 (4-5), 412–417 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. G. S. Djordjevic, Lj. Nesic and D. Radovancevic, “Two-oscillator KantowskiЦSachs model of the Schwarzschild black hole interior,” Gen. Relativ. Gravit. 48, 106 (2016).

    Article  MathSciNet  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  70. B. Dragovich, “On measurements, numbers and p-adicmathematical physics,” p-Adic Numbers Ultrametric Anal. Appl. 4 (2), 102–108 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  71. B. Dragovich, “p-Adic perturbation series and adelic summability,” Phys. Lett. B 256 (3, 4), 392–39 (1991).

    Article  MathSciNet  Google Scholar 

  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].

    Article  MathSciNet  Google Scholar 

  73. B. G. Dragovich, “On p-adic aspects of some perturbation series,” Theor. Math. Phys. 93 (2), 1225–1231 (1993).

    Article  MathSciNet  Google Scholar 

  74. B. G. Dragovich, “Rational summation of p-adic series,” Theor. Math. Phys. 100 (3), 1055–1064 (1994).

    Article  MathSciNet  Google Scholar 

  75. B. Dragovich, “On p-adic series inmathematical physics,” Proc. Steklov Inst. Math. 203, 255–270 (1994).

    Google Scholar 

  76. B. Dragovich, “On p-adic series with rational sums,” Scientific Review 19–20, 97–104 (1996).

    MathSciNet  Google Scholar 

  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].

    MathSciNet  MATH  Google Scholar 

  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].

    MathSciNet  MATH  Google Scholar 

  79. B. Dragovich, “On some finite sums with factorials,” Facta Universitatis: Ser. Math. Inform. 14, 1–10 (1999) [arXiv:math/0404487 [math.NT]].

    MathSciNet  Google Scholar 

  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].

    MathSciNet  MATH  Google Scholar 

  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]].

    Article  MathSciNet  MATH  Google Scholar 

  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. B. Dragovich, “On summation of p-adic series,” accepted for publication in Contem. Math., AMS, [arXiv:1702.02569 [math. NT]] (2017).

    Google Scholar 

  84. N. J. A. Sloane, The on-line encyclopedia of integer sequences, https://oeis.org/.

  85. B. Dragovich, “Zeta strings,” [arXiv:hep-th/0703008] (2007).

    Google Scholar 

  86. B. Dragovich, “Zeta-nonlocal scalar fields,” Theor. Math. Phys. 157 (3), 1669–1675 (2008) [arXiv:0804.4114[hep-th]].

    Article  MathSciNet  MATH  Google Scholar 

  87. B. Dragovich, “Some Lagrangians with zeta function nonlocality,” [arXiv:0805.0403 [hep-th]] (2008).

    MATH  Google Scholar 

  88. B. Dragovich, “Lagrangians with Riemann zeta function,” Rom. J. Phys. 53 (9-10), 1105–1110 (2008) [arXiv:0809.1601[hep-th]].

    MathSciNet  MATH  Google Scholar 

  89. B. Dragovich, “Towards effective Lagrangians for adelic strings,” Fortschr. Phys. 57 (5-7), 546–551 (2009) [arXiv:0902.0295[hep-th]].

    Article  MathSciNet  MATH  Google Scholar 

  90. B. Dragovich, “The p-adic sector of the adelic string,” Theor. Math. Phys. 163 (3), 768–773 (2010) [arXiv:0911.3625[hep-th]].

    Article  MathSciNet  MATH  Google Scholar 

  91. B. Dragovich, “Nonlocal dynamics of p-adic strings,” Theor. Math. Phys. 164 (3), 1151–1155 (2010) [arXiv:1011.0912[hep-th]].

    Article  MathSciNet  MATH  Google Scholar 

  92. B. Dragovich, “Adeles in mathematical physics,” (2007) [arXiv:0707.3876[math-ph]].

    Google Scholar 

  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. B. Dragovich and Lj. Nešić, “p-Adic and adelic generalization of quantum cosmology,” Gravitat. Cosmol. 5, 222–228 (1999) [arXiv:gr-qc/0005103].

    MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  96. B. Dragovich, “Towards p-adic matter in the universe,” Springer Proc. Math & Stat. 36, 13–24 (2013) [arXiv:1205.4409[hep-th]].

    MathSciNet  MATH  Google Scholar 

  97. B. Dragovich, “Adelic model of harmonic oscillator,” Theor. Math. Phys. 101, 1404–1415 (1994) [arXiv:hep-th/0402193].

    Article  MathSciNet  MATH  Google Scholar 

  98. B. Dragovich, “Adelic harmonic oscillator,” Int. J. Mod. Phys. A 10, 2349–2365 (1995) [arXiv:hepth/0404160].

    Article  MathSciNet  MATH  Google Scholar 

  99. B. Dragovich, “p-Adic and adelic quantum mechanics,” Proc. Steklov Inst. Math. 245, 72–85 (2004) [arXiv:hep-th/0312046].

    MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MATH  Google Scholar 

  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]].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  104. B. Dragovich and A. Yu. Dragovich, “A p-adic model of DNA sequence and genetic code,” (2006) [arXiv:qbio.GN/0607018].

    MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MATH  Google Scholar 

  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. B. Dragovich, “p-Adic structure of the genetic code,” (2012) [arXiv:1202.2353[q-bio. OT]].

    MATH  Google Scholar 

  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. 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. B. Dragovich and D. Joksimović, “On possible uses of p-adic analysis in econometrics,” Megatrend Revija 4 (2), 5–16 (2007).

    Google Scholar 

  112. S. N. Evans, “Local field Brownian motion,” J. Theor. Probab. 6, 817–850 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  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)].

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  115. K. H. Fischer and J. A. Hertz, Spin Glasses (Cambridge Univ. Press, 1993).

    Google Scholar 

  116. H. Frauenfelder, S. G. Sligar and P. G. Wolynes, “The energy landscapes and motions of proteins,” Science 254 5038, 1598–1603 (1991).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  120. I. M. Gelfand, M. I. Graev and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions (Saunders, Philadelphia, 1969).

    Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  122. S. Haran, “Analytic potential theory over p-adics,” Ann. Inst. Fourier 43, 905–944 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  123. K. H. Hoffmann and P. Sibani, “Diffusion in hierarchies,” Phys. Rev. A 38, 4261–4270 (1988).

    Article  MathSciNet  Google Scholar 

  124. A. IlićStepić, Z. Ognjanović, N. Ikodinovićand A. Perović, “p-Adic probability logics,” p-Adic Numbers Ultrametric Anal. Appl. 8 (3), 177–203 (2016).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  Google Scholar 

  127. W. C. Lang, “Wavelet analysis on the Cantor dyadic group,” Houston J. Math. 24, 533–544 (1998).

    MathSciNet  MATH  Google Scholar 

  128. W. C. Lang, “Orthogonal wavelets on the Cantor dyadic group,” SIAM J. Math. Anal. 27, 305–312 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  130. J. W. de Jong, Graphs, spectral triples and Dirac zeta functions p-Adic Numbers Ultrametric Anal. Appl. 1 (3) 286–296 (2009).

    MATH  Google Scholar 

  131. H. Kaneko and K. Yasuda, “Capacities associated with Dirichlet space on an infinite extension of a local field,” ForumMath. 17, 1011–1032 (2005).

    MathSciNet  MATH  Google Scholar 

  132. K. Kamizono, “p-Adic Brownian motion over Qp,” Proc. Steklov Inst. Math. 265, 115–130 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  134. H. Kaneko, “Fractal theoretic aspects of local field,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 51–57 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  137. A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer, Dordrecht, 1994).

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  139. A. Yu. Khrennikov, Non-Archimedean Analysis and Its Applications (Fizmatlit, Moscow, 2003) [in Russian].

    MATH  Google Scholar 

  140. A. Yu. Khrennikov and S. V. Kozyrev, “Wavelets on ultrametric spaces,” Appl. Comput. Harm. Anal. 19, 61–76 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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].

    MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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)].

    MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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. 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).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  Google Scholar 

  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].

    Article  Google Scholar 

  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].

    Article  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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. 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].

    Article  Google Scholar 

  160. A. Yu. Khrennikov and S. V. Kozyrev, “2-Adic clustering of the PAMmatrix,” J. Theor. Biol. 261, 396–406 (2009) [arXiv:0903.0137].

    Article  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Chapter  Google Scholar 

  163. A. N. Kochubei, Pseudo-Differential Equations and Stochastics overNon-Archimedean Fields (Marcel Dekker, New York, USA, 2001).

    Book  MATH  Google Scholar 

  164. A. N. Kochubei, “A non-Archimedean wave equation,” Pacif. J. Math. 235, 245–261 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  165. A. N. Kochubei, “Analysis and probability over infinite extensions of a local field,” Potential Anal. 10, 305–325 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  167. A. N. Kochubei, Analysis in Positive Characteristic (Cambridge Univ. Press, Cambridge, 2009).

    Book  MATH  Google Scholar 

  168. A. N. Kochubei, “Parabolic equations over the field of p-adic numbers,” Math. USSR Izv. 39, 1263–1280 (1992).

    Article  MathSciNet  Google Scholar 

  169. A. N. Kochubei, “Radial solutions of non-Archimedean pseudodifferential equations,” Pacific J. Math. 269 (2), 355–369 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  170. S. V. Kozyrev, “Ultrametricity in the theory of complex systems,” Theor. Math. Phys. 185 (2), 46–360 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  175. S. V. Kozyrev, “Wavelet theory as p-adic spectral analysis,” Izvest. Math. 66 (2), 367–376 (2002) [arXiv:math-ph/0012019].

    Article  MathSciNet  MATH  Google Scholar 

  176. S. V. Kozyrev, “p-Adic pseudodifferential operators and p-adic wavelets,” Theor. Math. Phys. 138 (3), 322–332 (2004) [arXiv:math-ph/0303045].

    Article  MathSciNet  MATH  Google Scholar 

  177. S. V. Kozyrev, “Toward an ultrametric theory of turbulence,” Theor. Math. Phys. 157 (3), 1711–1720 (2008) [arXiv:0803.2719].

    Article  MATH  Google Scholar 

  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)].

    MathSciNet  MATH  Google Scholar 

  179. S. V. Kozyrev, “p-Adic pseudodifferential operators: Methods and applications,” Proc. Steklov Inst. Math. 245, 143–153 (2004).

    MATH  Google Scholar 

  180. S. V. Kozyrev, V. Al. Osipov and V. A. Avetisov, “Nondegenerate ultrametric diffusion,” J. Math. Phys. 46, 063302–063317 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  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)].

    MathSciNet  MATH  Google Scholar 

  182. S. V. Kozyrev, “Dynamics on rugged landscapes of energy and ultrametric diffusion,” p-Adic Numbers Ultrametric Anal. Appl. 2 (2), 122–132 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  Google Scholar 

  184. S. V. Kozyrev and A. Yu. Khrennikov, “2-Adic numbers in genetics and Rumer’s symmetry,” Dokl. Math. 81 (1), 128–130 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  185. S. V. Kozyrev, “Multidimensional clustering and hypergraphs,” Theor. Math. Phys. 164 (3), 1163–1168 (2010).

    Article  MATH  Google Scholar 

  186. S. V. Kozyrev, “Cluster networks and Bruhat-Tits buildings,” Theor. Math. Phys. 180 (2), 958–966 (2014) [arXiv:1404.6960].

    Article  MathSciNet  MATH  Google Scholar 

  187. A. V. Kosyak, A. Khrennikov and V. M. Shelkovich, “Wavelet bases on adele rings,” Dokl. Math. 85, 75–79 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  188. A. V. Kosyak, A. Khrennikov and V. M. Shelkovich, “Pseudodifferential operators on adele rings and wavelet bases,” Dokl. Math. 85, 358–362 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  189. Yu. I. Manin, “Numbers as functions,” p-Adic Numbers Ultrametric Anal. Appl. 5 (4), 313–325 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  190. Yu. I. Manin, “Painleve VI equations in p-adic time,” p-Adic Numbers Ultrametric Anal. Appl. 8 (3), 217–224 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  192. M. Marcolli and N. Tedeschi, “Multifractals, Mumford curves and eternal inflation,” p-Adic Numbers Ultrametric Anal. Appl. 6 (2), 135–154 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  193. J. Marcinek and M. Marcolli, “KMS weights on higher rank buildings,” p-Adic Numbers Ultrametric Anal. Appl. 8 (1), 45–67 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  194. M. Marcolli, “Cyclotomy and endomotives,” p-Adic Numbers Ultrametric Anal. Appl. 1 (3), 217–263 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  195. Y. Meyer, Wavelets and Operators (Cambridge Univ. Press, Cambridge, 1992).

    MATH  Google Scholar 

  196. M. Mezard, G. Parisi and M. Virasoro, Spin-Glass Theory and Beyond (World Scientific, Singapore, 1987).

    MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  200. A. Morozov, “Are there p-adic knot invariants?,” Theor. Math. Phys. 187 (1), 447–454 (2016) [arXiv:1509.04928].

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  208. F. Murtagh and A. Heck, Multivariate Data Analysis (Springer Science & BusinessMedia, 2012).

    MATH  Google Scholar 

  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. F. Murtagh and P. Legendre, “Ward’s hierarchical agglomerative clustering method: which algorithms implementWard’s criterion?,” J. Classif. 31 (3), 274–295 (2014).

    Article  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  212. F. Murtagh, Multidimensional Clustering Algorithms (Physica-Verlag, Heidelberg, 1985).

    MATH  Google Scholar 

  213. F. Murtagh, “From data to the p-adic or ultrametric model,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 58–68 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  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].

    MathSciNet  MATH  Google Scholar 

  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).

    Article  Google Scholar 

  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. I. Ya. Novikov and M. A. Skopina, “Why are Haar bases in various structures the same?,”Mat. Zametki 91 (6), 950–953 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  218. A. T. Ogielski and D. L. Stein, “Dynamics on ultrametric spaces,” Phys. Rev. Lett. 55, 1634–1637 (1985).

    Article  MathSciNet  Google Scholar 

  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].

    Article  MathSciNet  Google Scholar 

  220. G. Rammal, M. A. Toulouse and M. A. Virasoro, “Ultrametricity for physicists,” Rev. Mod. Phys. 58, 765–788 (1986).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  224. M. Talagrand, Spin glasses, a Challenge forMathematicians (Springer-Verlag, 2003).

    Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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. M. Imakaev, K. Tchourine, S. Nechaev and L. Mirny, “Effects of topological constraints on globular polymers,” Soft Matter 11, 665–671 (2015).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  231. B. Bonev and G. Cavalli, “Organization and function of the 3D genome,” Nature Rev. Genet. 17, 661–678 (2016).

    Article  Google Scholar 

  232. G. E. Hinton and R. R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” Science 313, 504–507 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  233. Y. Bengio, “Learning deep architectures for AI,” Found. TrendsMach. Learn. 2 (1), (2009).

    Google Scholar 

  234. C. Hennig, M. Meila, F. Murtagh and R. Rocci, Handbook of Cluster Analysis (CRC Press, 2015).

    MATH  Google Scholar 

  235. C. Linnaeus, Systema naturae (Leiden, 1735).

    Google Scholar 

  236. V. A. Lemin, “Finite ultrametric spaces and computer science,” pp. 219–241, in Categorical Perspectives, Trends in Mathematics (Springer, 2001).

    MATH  Google Scholar 

  237. F. Bruhat and J. Tits, “Groupes reductifs sur un corps local, I. Donnees radicielles valuees,” Publ. Math. IHES 41, 5–251 (1972).

    Article  MATH  Google Scholar 

  238. P. B. Garrett, Buildings and Classical Groups (Chapman and Hall, London, 1997).

    Book  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  240. E. Bauer and R. Kohavi, “An empirical comparison of voting classification algorithms: bagging, boosting, and variants,” Mach. Learn. 36, 105–139 (1999).

    Article  Google Scholar 

  241. D. H. Huson, R. Rupp and C. Scornavacca, Phylogenetic Networks (Cambridge Univ. Press, Cambridge, 2010).

    Book  Google Scholar 

  242. A. Dress, K. T. Huber, J. Koolen, V. Moulton and A. Spillner, Basic Phylogenetic Combinatorics (Cambridge Univ. Press, Cambridge, 2012).

    MATH  Google Scholar 

  243. E. V. Koonin, The Logic of Chance: The Nature and Origin of Biological Evolution (FT Press Science, 2012).

    Google Scholar 

  244. J.-L. Starck, F. Murtagh and J. Fadili, Sparse Image and Signal Processing: Wavelets and Related Geometric Multiscale Analysis (Cambridge Univ. Press, 2015).

    Book  MATH  Google Scholar 

  245. M. N. Khokhlova and I. V. Volovich, “Modeling theory and hypergraph of classes,” Proc. Steklov Inst. Math. 245, 266–272 (2004).

    MATH  Google Scholar 

  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. S. Albeverio and W. Karwowski, Diffusion on p-Adic Numbers (Bielefeld-Bochum Stochastik, 1990).

    MATH  Google Scholar 

  248. A. Khrennikov and M. Nilsen. p-Adic Deterministic and Random Dynamics, Math. Appl. 574 (Kluwer Acad. Publishers, Dordrecht, 2004).

    Google Scholar 

  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.

  250. G. Parisi, “Infinite number of order parameters for spin-glasses,” Phys. Rev. Lett. 43, 1754 (1979).

    Article  Google Scholar 

  251. M. Mezard, G. Parisi, N. Sourlas, G. Toulouse and M. Virasoro, “Nature of the spin-glass phase,” Phys. Rev. Lett. 52, 1156 (1984).

    Article  MATH  Google Scholar 

  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).

    MATH  Google Scholar 

  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. P. P. Varju, “Random walks in compact groups,” DocumentaMath. 18, 1137–1175 (2013).

    MathSciNet  MATH  Google Scholar 

  255. S. Mustafa, “Random walks on unimodular p-adic groups,” Stoch. Proc. Appl. 115, 927–937 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  256. V. Anashin, A. Khrennikov and E. Yurova, “T-functions revisited: new criteria for bijectivity/transitivity,” Designs Codes Crypt. 7, 383–407 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  257. V. Anashin, “Uniformly distributed sequences of p-adic integers,” Math. Notes 55, 109–133 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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. H. Qusay, “Demystifying cloud computing,” J. Defense Softw. Engin., CrossTalk N 1/2, 16–21 (2011).

    Google Scholar 

  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. A. Khrennikov, “Human subconscious as the p-adic dynamical system’,’ J. Theor. Biol. 193, 179–196 (1998).

    Article  Google Scholar 

  267. A. Khrennikov, Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena, Ser. Fundamental Theories of Physics (Kluwer, Dordreht, 2004).

    Book  MATH  Google Scholar 

  268. A. Khrennikov, Classical and Quantum Mental Models and Freud’s Theory of Unconscious Mind (VäxjöUniv. Press, Växjö, 2002).

    Google Scholar 

  269. F. Murtagh, “Ultrametric model of mind, I: Review,” p-Adic Numbers Ultrametric Anal. Appl. 4, 193–206 (2012).

    Article  MathSciNet  Google Scholar 

  270. F. Murtagh, “Ultrametric model of mind, II: Application to text content analysis,” p-Adic Numbers Ultrametric Anal. Appl. 4, 207–221 (2012).

    Article  MathSciNet  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  275. A. Khrennikov and N. Kotovich, “Towards ultrametric modeling of unconscious creativity,” Int. J. Cogn. Inform. Natural Intell. 8, 98–109 (2014).

    Article  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  280. L. A. Richards, “Capillary conduction of liquids through porous mediums,” Physics 1 (5), 318–333 (1931).

    Article  MATH  Google Scholar 

  281. J. Richter, The Soil as a Reactor (Catena, 1987).

    Google Scholar 

  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).

    Article  Google Scholar 

  283. S. V. Kozyrev, “Ultrametric analysis and interbasin kinetics,” AIP Conf. Proc. 826, 121–128 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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. A. Wiles, “Modular elliptic curves and Fermat’s last theorem, Annals Math. 141, 443–551 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  287. R. Taylor and A. Wiles, “Ring-theoretic properties of certain Hecke algebras,” Annals Math. 141, 553–572 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  288. Yves Hellegouarch, Invitation to the Mathematics of Fermat-Wiles (Academic Press, 2001).

    MATH  Google Scholar 

  289. R. P. Langlands, “Problems in the theory of automorphic forms,” Lect. Notes Math. 170, 18–61 (Springer Verlag, 1970).

    Google Scholar 

  290. L. Hormander, The Analysis of Linear Partial Differential Operators, III Pseudo-Differential Operators, IV Fourier Integral Operators (Springer-Verlag, 1985)

    MATH  Google Scholar 

  291. M. A. Shubin, Pseudodifferential Operators and Spectral Theory (Nauka, 1978).

    MATH  Google Scholar 

  292. N. N. Bogoliubov and D. V. Schirkov, Introduction to the Theory of Quantized Fields (Springer, 1984).

    Google Scholar 

  293. 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. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function (Clarendon Press, Oxford, 1986)

    MATH  Google Scholar 

  295. A. A. Karatsuba and S. M. Voronin, The Riemann Zeta-Function (Walter de Gruyter Publishers, Berlin-New York 1992

    Book  MATH  Google Scholar 

  296. K. Chandrasekharan, Arithmetic Functions (Springer, 1972).

    Google Scholar 

  297. A. Kapustin and E. Witten, “Electric-magnetic duality and the geometric Langlands program,” (2006) [hep-th/0604151]

    MATH  Google Scholar 

  298. S. Gukov and E. Witten, “Gauge theory, ramification, and the geometric Langlands program,” (2006) [hep-th/0612073]

    MATH  Google Scholar 

  299. E. Frenkel, “Lectures on the Langlands program and conformal field theory,” (2005) [hep-th/0512172].

    MATH  Google Scholar 

  300. N. M. Katz and P. Sarnak, “Zeroes of zeta-functions and symmetry,” Bull. Amer. Math. Soc. (N. Y.) 36, 1–26 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  301. J. P. Keating and N. C. Snaith, “Random matrix theory and L-functions at s = 1/2,” Comm. Math. Phys. 214, 91–110 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  302. D. Harlow, S. Shenker, D. Stanford and L. Susskind, “Eternal symmetree,” Phys. Rev. D 85, 063516 (2012) [arXiv:1110.0496].

    Article  Google Scholar 

  303. Yu. I. Manin and M. Marcolli, “Big Bang, blowup, and modular curves: algebraic geometry in cosmology,” SIGMA 10, 073 (2014) [arXiv:1402.2158].

    MathSciNet  MATH  Google Scholar 

  304. Yu. I. Manin and M. Marcolli, “Symbolic dynamics, modular curves, and Bianchi IX cosmologies,” (2015) [arXiv:1504.04005].

    MATH  Google Scholar 

  305. 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. Zhi Hu and Sen Hu, “Symplectic group and Heisenberg group in p-adic quantum mechanics,” [arXiv:1502.01789].

  307. 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].

    Article  MathSciNet  Google Scholar 

  308. 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. 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. 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. Spin Glasses and Biology, Ed. D. L. Stein (World Scientific, Singapore, 1992).

  312. W. Schikhof, Ultrametric Calculus: an introduction to p-adic analysis (Cambridge Univ. Press, 1985).

    Book  MATH  Google Scholar 

  313. 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).

    Article  Google Scholar 

  314. V. S. Varadarajan, “Path integrals for a class of p-adic Schrödinger equations,” Lett. Math. Phys. 39, 97–106 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  315. 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. V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).

    Book  MATH  Google Scholar 

  317. V. S. Vladimirov, “Generalized functions over the field of p-adic numbers,” Russ. Math. Surv. 43, 19–64 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  318. 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).

    Article  MathSciNet  MATH  Google Scholar 

  319. V. S. Vladimirov and I. V. Volovich, “p-Adic quantum mechanics,” Comm. Math. Phys. 123, 659–676 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  320. 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).

    MATH  Google Scholar 

  321. V. S. Vladimirov and Ya. I. Volovich, “Nonlinear dynamics equation in p-adic string theory,” Theor. Math. Phys. 138, 297 (2004) [math-ph/0306018].

    Article  MATH  Google Scholar 

  322. 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).

    Article  MATH  Google Scholar 

  323. V. S. Vladimirov, “Nonlinear equations for p-adic open, closed, and open-closed strings,” Theor. Math. Phys. 149 (3), 1604–1616 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  324. V. S. Vladimirov, “On the equations for p-adic closed and open strings,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 79–87 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  325. V. S. Vladimirov, “Solutions of p-adic string equations,” Theor. Math. Phys. 167 (2), 539–546 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  326. V. S. Vladimirov, “Nonexistence of solutions of the p-adic strings,” Theor. Math. Phys. 174 (2), 178–185 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  327. I. V. Volovich, “D-branes, black holes and SU(∞) gauge theory,” (1996) [hep-th/9608137]

    Google Scholar 

  328. I. V. Volovich, “From p-adic strings to etale strings,” Proc. SteklovMath. Inst. 203, 41–48 (1994).

    MATH  Google Scholar 

  329. I. V. Volovich, “p-Adic string,” Class. Quant. Grav. 4 (4), L83–L87 (1987).

    Article  MathSciNet  Google Scholar 

  330. I. V. Volovich, “Number theory as the ultimate physical theory,” p-Adic Numbers Ultrametric Anal. Appl. 2 (1), 77–87 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  331. I. V. Volovich, “Time irreversibility problem and functional formulation of classical mechanics,” (2009) [arXiv:0907.2445].

    Google Scholar 

  332. A. S. Trushechkin and I. V. Volovich, “Functional classical mechanics and rational numbers,” p-Adic Numbers Ultrametric Anal. Appl. 1 (4), 361–367 (2009.

    Article  MathSciNet  MATH  Google Scholar 

  333. I. V. Volovich, “Functional stochastic classical mechanics,” p-Adic Numbers Ultrametric Anal. Appl. 7 (1), 56–70 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  334. I. V. Volovich, “Bogolyubov’s equations and functional mechanics,” Theor. Math. Phys. bf 164 (3), 1128–1135 (2010).

    Article  MATH  Google Scholar 

  335. I. V. Volovich, “Randomness in classical mechanics and quantum mechanics,” Found. Phys. 41 (3), 516–528 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  336. K. Yasuda, “Extension of measures to infinite dimensional spaces over p-adic field,” Osaka J. Math. 37, 967–985 (2000).

    MathSciNet  MATH  Google Scholar 

  337. K. Yasuda, “Additive processes on local fields,” J. Math. Sci. Univ. Tokyo 3, 629–654 (1996).

    MathSciNet  MATH  Google Scholar 

  338. K. Yasuda, “Limit theorems for p-adic valued asymmetric semistable laws and processes,” p-Adic Numbers Ultrametric Anal. Appl. 9 (1), 62–77 (2017).

    Article  MathSciNet  Google Scholar 

  339. E. Zelenov, “p-Adic law of large numbers,” Izv. RAN Ser. Math. 80 (3), 31–42 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  340. E. Zelenov, “p-Adic Brownian motion,” Izv. RAN Ser. Math. 80 (6), 92–102 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  341. E. I. Zelenov, “Adelic decoherence,” p-Adic Numbers Ultrametric Anal. Appl. 4 (1), 84–87 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  342. A. P. Zubarev, “On stochastic generation of ultrametrics in high-dimensional Euclidean spaces,” p-Adic Numbers Ultrametric Anal. Appl. 6 (2), 155–165 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  343. W. A. Zuniga-Galindo, “Parabolic equations and Markov processes over p-adic fields,” Potential Anal. 28, 185–200 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  344. 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).

    Article  MathSciNet  MATH  Google Scholar 

  345. W. A. Zuniga-Galindo, “Fundamental solutions of pseudo-differential operators over p-adic fields,” Rend. Sem. Mat. Univ. Padova 109, 241–245 (2003).

    MathSciNet  MATH  Google Scholar 

  346. 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).

    Article  MathSciNet  MATH  Google Scholar 

  347. W. A. Zuniga-Galindo, Pseudodifferential Equations Over Non-Archimedean Spaces, Lecture Notes in Mathematics 2174 (Springer, 2017).

    MATH  Google Scholar 

  348. 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).

    Article  MathSciNet  MATH  Google Scholar 

  349. 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).

    Article  MathSciNet  MATH  Google Scholar 

  350. W. A. Zuniga-Galindo, “The non-Archimedean stochastic heat equation driven by Gaussian noise,” J. Fourier Anal. Appl. 21 (3), 600–627 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  351. W. A. Zuniga-Galindo, “Non-Archimedean white noise, pseudodifferential stochastic equations, and massive Euclidean fields,” J. Fourier Anal. Appl. 23 (2), 288–323 (2017).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Physics, University of Belgrade, 11080, Belgrade, Serbia

    B. Dragovich

  2. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    B. Dragovich

  3. International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science, Linnaeus University, S-35195, Växjö, Sweden

    A. Yu. Khrennikov

  4. National Research University of Information Technologies, Mechanics and Optics (ITMO), St. Petersburg, 197101, Russia

    A. Yu. Khrennikov

  5. Steklov Mathematical Institute of the Russian Academy of Sciences, Gubkina Str. 8, Moscow, 119991, Russia

    S. V. Kozyrev, I. V. Volovich & E. I. Zelenov

Authors
  1. B. Dragovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Yu. Khrennikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. V. Kozyrev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. I. V. Volovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. E. I. Zelenov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev or I. V. Volovich.

Additional information

The text was submitted by the authors in English.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dragovich, B., Khrennikov, A.Y., Kozyrev, S.V. et al. p-Adic mathematical physics: the first 30 years. P-Adic Num Ultrametr Anal Appl 9, 87–121 (2017). https://doi.org/10.1134/S2070046617020017

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070046617020017

Key words

Search

Navigation