On p-adic mathematical physics

  • B. Dragovich
  • A. Yu. Khrennikov
  • S. V. Kozyrev
  • I. V. Volovich
Review Articles


A brief review of some selected topics in p-adic mathematical physics is presented.

Key words

p-adic numbers p-adic mathematical physics complex systems hierarchical structures adeles ultrametricity string theory quantum mechanics quantum gravity probability biological systems cognitive science genetic code wavelets 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. P. Serre, A Course in Arithmetics (Springer GTM7, 1973).Google Scholar
  2. 2.
    I. V. Volovich, “Number theory as the ultimate physical theory,” Preprint No. TH 4781/87, CERN, Geneva, (1987).Google Scholar
  3. 3.
    I. V. Volovich, “p-Adic string,” Class. Quant. Grav. 4, L83–L87 (1987).MathSciNetCrossRefGoogle Scholar
  4. 4.
    V. S. Vladimirov and I. V. Volovich, “Superanalysis. I. Differential calculus,” Theor. Math. Phys. 59, 317–335 (1984).MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    V. S. Vladimirov and I. V. Volovich, “Superanalysis. II. Integral calculus,” Theor. Math. Phys. 60, 743–765 (1985).CrossRefGoogle Scholar
  6. 6.
    Yu. I. Manin, “Reflections on arithmetical physics,” in Conformal Invariance and String Theory, pp. 293–303 (Academic Press, Boston, 1989).Google Scholar
  7. 7.
    V. S. Varadarajan, “Arithmetic Quantum Physics: Why, What, and Whither,” Proc. Steklov Inst. Math. 245, 258–265 (2004).MathSciNetGoogle Scholar
  8. 8.
    N. N. Bogolyubov, “On a new method in the theory of superconductivity,” J. Exp. Theor. Phys. 34(1), 58 (1958).Google Scholar
  9. 9.
    Broken Symmetry. Selected Papers of Y. Nambu Eds. T. Eguchi and K. Nishijima, (World Scientific, Singapore, 1995).Google Scholar
  10. 10.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).Google Scholar
  11. 11.
    A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer, Dordrecht, 1994).MATHGoogle Scholar
  12. 12.
    L. Brekke and P. G. O. Freund, “p-Adic numbers in physics,” Phys. Rep. 233(1), 1–66 (1993).MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields (Marcel Dekker, New York, USA, 2001).MATHGoogle Scholar
  14. 14.
    S. V. Kozyrev, Methods and Applications of Ultrametric and p-Adic Analysis: From Wavelet Theory to Biophysics, Modern Problems of Mathematics 12 (Steklov Math. Inst., Moscow, 2008) [in Russian].Google Scholar
  15. 15.
    M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory: Volumes 1, 2 (Cambridge Univ. Press, Cambridge, 1987).Google Scholar
  16. 16.
    P. G. O. Freund and M. Olson, “Non-archimedean strings,” Phys. Lett. B 199, 186–190 (1987).MathSciNetCrossRefGoogle Scholar
  17. 17.
    I. M. Gelfand, M. I. Graev and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions (Saunders, Philadelphia, 1969).Google Scholar
  18. 18.
    P. H. Frampton and Y. Okada, “Effective scalar field theory of p-adic string,” Phys. Rev. D 37, 3077–3084 (1988).MathSciNetCrossRefGoogle Scholar
  19. 19.
    L. Brekke, P. G. O. Freund, M. Olson and E. Witten, “Nonarchimedean string dynamics,” Nucl. Phys. B 302(3), 365–402 (1988).MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. G. O. Freund and E. Witten, “Adelic string amplitudes,” Phys. Lett. B 199, 191–194 (1987).MathSciNetCrossRefGoogle Scholar
  21. 21.
    I. V. Volovich, “Harmonic analysis and p-adic strings,” Lett. Math. Phys. 16, 61–67 (1988).MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Z. I. Borevich and I. R. Shafarevich, Number Theory (AP, 1966).Google Scholar
  23. 23.
    V. S. Vladimirov, “Adelic formulas for four-particle string and superstring tree amplitudes in one-class quadratic fields,” Proc. Steklov Inst. Math. 245, 3–21 (2004).Google Scholar
  24. 24.
    I. Ya. Aref’eva, B. Dragovich and I. V. Volovich, “On the adelic string amplitudes,” Phys. Lett. B 209, 445–450 (1988).MathSciNetCrossRefGoogle Scholar
  25. 25.
    I. Ya. Aref’eva, B. Dragovich and I. V. Volovich, “Open and closed p-adic strings and quadratic extensions of number fields,” Phys. Lett. B 212, 283–289 (1988).MathSciNetCrossRefGoogle Scholar
  26. 26.
    I. Ya. Aref’eva, B. Dragovich and I. V. Volovich, “p-Adic superstrings,” Phys. Lett. B 214, 339–346 (1988).MathSciNetCrossRefGoogle Scholar
  27. 27.
    D. R. Lebedev and A. Yu. Morozov, “p-Adic single-loop calculations,” Theor. Math. Phys. 82(1), 1–6 (1990).MathSciNetCrossRefGoogle Scholar
  28. 28.
    L. Chekhov, A. Mironov and A. Zabrodin, “Multiloops calculations in p-adic string theory and Bruhat-Tits trees,” Commun. Math. Phys. 125, 675 (1989).MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    P. H. Frampton and Y. Okada, “p-Adic string N-point function,” Phys. Rev. Lett. 60, 484–486 (1988).MathSciNetCrossRefGoogle Scholar
  30. 30.
    P. H. Frampton and H. Nishino, “Theory of p-adic closed strings,” Phys. Rev. Lett. 62, 1960–1964 (1989).MathSciNetCrossRefGoogle Scholar
  31. 31.
    I. V. Volovich, “p-Adic space-time and string theory,” Theor. Math. Phys. 71(3), 574–576 (1987).MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    I. V. Volovich, “From p-adic strings to etale ones,” Trudy Steklov Math. Inst. 203, 41–47 (1994); arXiv:hepth/9608137.Google Scholar
  33. 33.
    A. Connes and M. Marcolli, “Quantum fields and motives,” J. Geom. Phys. 56(1), 55–85 (2005).MathSciNetCrossRefGoogle Scholar
  34. 34.
    A. Connes, C. Consani and M. Marcolli, “Noncommutative geometry and motives: the thermodynamics of endomotives,” Advances in Mathematics 214(2), 761–831 (2007); arXiv:math.QA/0512138.MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    A. Connes, C. Consani and M. Marcolli, “The Weil proof and the geometry of the adeles class space,” arXiv:math/0703392.Google Scholar
  36. 36.
    A. Connes and M. Marcolli, “Noncommutative geometry, quantum fields, and motives,” Colloquium Publications 55 (AmericanMath. Society, 2008).Google Scholar
  37. 37.
    C. Consani and M. Marcolli, “Spectral triples from Mumford curves,” Int. Math. Research Notices 36, 1945–1972 (2003).MathSciNetCrossRefGoogle Scholar
  38. 38.
    G. Cornelissen, M. Marcolli, K. Reihani and A. Vdovina, “Noncommutative geometry on trees and buildings,” in Traces in Geometry, Number Theory, and Quantum Fields, pp. 73–98 (Vieweg Verlag, 2007).Google Scholar
  39. 39.
    V. Voevodsky, “Motives over simplicial schemes,” arXiv:0805.4431.Google Scholar
  40. 40.
    M. Baker, J. Teitelbaum, B. Conrad, K. S. Kedlaya and D. S. Thakur, p-Adic Geometry: Lectures from the 2007 Arizona Winter School (American Mathematical Society, 2008).Google Scholar
  41. 41.
    A. N. Kochubei and M. R. Sait-Ametov, “Interaction measures on the space of distributions over the field of p-adic numbers,” Infin. Dimens. Anal. Quantum Probab. Related Topics 6, 389–411 (2003).MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    M. D. Missarov, “Random fields on the adele ring and Wilson’s renormalization group,” Annales de l’institut Henri Poincare (A): Physique Theorique 50(3), 357–367 (1989).MATHMathSciNetGoogle Scholar
  43. 43.
    V. A. Smirnov, “Calculation of general p-adic Feynman amplitude,” Comm. Math. Phys. 149(3), 623–636 (1992).MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    D. Ghoshal and A. Sen, “Tachyon condensation and brane descent relations in p-adic string theory,” Nucl. Phys. B 584, 300–312 (2000).MATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    N. Moeller and B. Zwiebach, “Dynamics with infinitely many time derivatives and rolling tachyons,” JHEP 10, 034 (2002).MathSciNetCrossRefGoogle Scholar
  46. 46.
    I. Ya. Aref’eva, D. M. Belov, A. A. Giryavets, A. S. Koshelev and P. B. Medvedev, “Noncommutative field theories and (super)string field theories,” arXiv:hep-th/0111208.Google Scholar
  47. 47.
    I.Ya. Aref’eva, D.M. Belov, A. S. Koshelev and P.B. Medvedev, “Tachyon condensation in cubic superstring field theory,” Nucl. Phys. B 638, 3–20 (2002); arXiv:hep-th/0011117.MATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Ya. I. Volovich, “Numerical study of nonlinear equations with infinite number of derivatives,” J. Phys. A: Math. Gen 36, 8685 (2003); arXiv:math-ph/0301028.MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    V. S. Vladimirov and Ya. I. Volovich, “Nonlinear dynamics equation in p-adic string theory,” Theor. Math. Phys. 138, 297–307 (2004); arXiv:math-ph/0306018.MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    I. Ya. Aref’eva, L. V. Joukovskaya and A. S. Koshelev, “Time evolution in superstring field theory on non-BPS brane. Rolling tachyon and energy-momentum conservation,” JHEP 09, 012 (2003); arXiv:hepth/0301137.MathSciNetCrossRefGoogle Scholar
  51. 51.
    V. S. Vladimirov, “On the non-linear equation of a p-adic open string for a scalar field,” Russ. Math. Surv. 60, 1077–1092 (2005).MATHCrossRefGoogle Scholar
  52. 52.
    V. S. Vladimirov, “On the equations for p-adic closed and open strings,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 79–87 (2009).CrossRefGoogle Scholar
  53. 53.
    V. Forini, G. Grignani, G. Nardelli, “A new rolling tachyon solution of cubic string field theory,” JHEP 0503, 079 (2005); arXiv:hep-th/0502151.MathSciNetCrossRefGoogle Scholar
  54. 54.
    D. V. Prokhorenko, “On some nonlinear integral equation in the (super)string theory,” arXiv:mathph/0611068.Google Scholar
  55. 55.
    I. Ya. Aref’eva, “Nonlocal string tachyon as a model for cosmological dark energy,” AIP Conf. Proc. 826, 301–311 (2006); arXiv:astro-ph/0410443.MathSciNetCrossRefGoogle Scholar
  56. 56.
    I. Ya. Aref’eva and L. V. Joukovskaya, “Time lumps in nonlocal stringy models and cosmological applications,” JHEP 10, 087 (2005); arXiv:hep-th/0504200.MathSciNetCrossRefGoogle Scholar
  57. 57.
    I. Ya. Aref’eva and A. S. Koshelev, “Cosmic acceleration and crossing of w = − 1 barrier in non-local cubic superstring field theory model,” JHEP 0702, 041 (2007); arXiv:hep-th/0605085.CrossRefGoogle Scholar
  58. 58.
    G. Calcagni, “Cosmological tachyon from cubic string field theory,” JHEP 05, 012 (2006); arXiv:hepth/0512259.MathSciNetCrossRefGoogle Scholar
  59. 59.
    B. Dragovich, “p-Adic and adelic quantum cosmology: p-Adic origin of dark energy and dark matter,” in p-Adic Mathematical Physics, Amer. Inst. Phys. Conf. Series 826, 25–42 (2006); arXiv:hep-th/0602044.MathSciNetGoogle Scholar
  60. 60.
    N. Barnaby, T. Biswas and J.M. Cline, “p-Adic inflation,” JHEP 0704, 056 (2007); arXiv:hep-th/0612230.MathSciNetCrossRefGoogle Scholar
  61. 61.
    A. S. Koshelev, Non-local SFT tachyon and cosmology,” JHEP 0704, 029 (2007); arXiv:hep-th/0701103.MathSciNetCrossRefGoogle Scholar
  62. 62.
    L. V Joukovskaya, “Dynamics in nonlocal cosmological models derived from string field theory,” Phys. Rev. D 76, 105007 (2007); arXiv:0707.1545.Google Scholar
  63. 63.
    G. Calcagni, M. Montobbio and G. Nardelli, “Route to nonlocal cosmology,” Phys. Rev. D 76, 126001 (2007).Google Scholar
  64. 64.
    I.Ya. Aref’eva, L. V. Joukovskaya and S.Yu. Vernov, “Dynamics in nonlocal linear models in the Friedmann-Robertson-Walker metric,” J. Phys. A:Math. Theor 41, 304003 (2008).Google Scholar
  65. 65.
    N. Barnaby and J. M. Cline, “Predictions for nongaussianity from nonlocal inflation,” JCAP 0806, 030 (2008); arXiv:0802.3218.Google Scholar
  66. 66.
    N. J. Nunes, D. J. Mulryne, “Non-linear non-local cosmology,” arXiv:0810.5471.Google Scholar
  67. 67.
    N. Barnaby, “Nonlocal inflation,” arXiv:0811.0814.Google Scholar
  68. 68.
    I.Ya. Aref’eva and I.V. Volovich, “Quantization of the Riemann zeta-function and cosmology,” Int. J. Geom. Meth. Mod. Phys. 4, 881–895 (2007); arXiv:hep-th/0701284v2.MATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    B. Dragovich, “Zeta-nonlocal scalar fields,” Theor. Math. Phys. 157(3), 1671–1677 (2008); arXiv:0804.4114.CrossRefMATHMathSciNetGoogle Scholar
  70. 70.
    M. J. Shai Haran, The Mysteries of the Real Prime (Oxford University Press, USA, 2001).MATHGoogle Scholar
  71. 71.
    I. Ya. Aref’eva and I. V. Volovich, “Quantum group particles and non-Archimedean geometry,” Phys. Lett. B 268, 179–187 (1991).MathSciNetCrossRefGoogle Scholar
  72. 72.
    S. V. Kozyrev, “The space of free coherent states is isomorphic to space of distributions on p-adic numbers,” Infin. Dimens. Anal. Quantum Prob. 1(2), 349–355 (1998); arXiv:q-alg/9706020.MATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    A. Connes and M. Marcolli, “From physics to number theory via noncommutative geometry. Part I: Quantum statistical mechanics of Q-lattices,” in Frontiers in Number Theory, Physics, and Geometry, I pp. 269–350 (Springer Verlag, 2006).Google Scholar
  74. 74.
    A. Connes and M. Marcolli, “From physics to number theory via noncommutative geometry. Part II: Renormalization, the Riemann-Hilbert correspondence, andmotivic Galois theory,” in Frontiers in Number Theory, Physics, and Geometry, II pp. 617–713 (Springer Verlag, 2006).Google Scholar
  75. 75.
    R. Schmidt, “Arithmetic gravity and Yang-Mills theory: An approach to adelic physics via algebraic spaces,” arXiv:0809.3579v1.Google Scholar
  76. 76.
    V. S. Varadarajan, Geometry of Quantum Theory (Springer Verlag, 2007).Google Scholar
  77. 77.
    V. S. Vladimirov and I. V. Volovich, “p-Adic quantum mechanics,” Comm. Math. Phys. 123, 659–676 (1989).MATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    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
  79. 79.
    E. I. Zelenov, “The infinite-dimensional p-adic symplectic group,” Russian Acad. Sci. Izv.Math. 43, 421–441 (1994).MathSciNetCrossRefGoogle Scholar
  80. 80.
    Harish-Chandra, “Harmonic analysis on reductive p-adic groups,” in Proc. of Symposia in Pure Mathematics, Vol. XXVI, Amer. Math. Soc. Providence, R. I., pp. 167–192 (1973).MathSciNetGoogle Scholar
  81. 81.
    A. Weil, “Sur certains groupes d’operateurs unitaires,” Acta Mathematica 111, 143–211 (1964).MATHMathSciNetCrossRefGoogle Scholar
  82. 82.
    P. Schneider, “Continuous representation theory of p-adic Lie groups,” Proc. ICM Madrid 2006, Vol. II, pp. 1261–1282 (2006).Google Scholar
  83. 83.
    E. I. Zelenov, “Quantum approximation theorem,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 88–90 (2009).CrossRefGoogle Scholar
  84. 84.
    V. S. Varadarajan, “Multipliers for the symmetry groups of p-adic spacetime,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 69–78 (2009).CrossRefMathSciNetGoogle Scholar
  85. 85.
    E. I. Zelenov, “p-Adic Heisenberg group and Maslov index,” Commun. Math. Phys. 155, 489–502 (1993).MATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    V. S. Vladimirov and I. V. Volovich, “A p-adic Schrödinger-type equation,” Lett. Math. Phys. 18, 43–53 (1989).MATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    A. N. Kochubei, “A Schrödinger type equation over the field of p-adic numbers,” J. Math. Phys. 34, 3420–3428 (1993).MATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    W. A. Zuniga-Galindo, “Decay of solutions of wave-type pseudo-differential equations over p-adic fields,” Publ. Res. Inst. Math. Sci. 42, 461–479 (2006); 44, 45–48 (2008).MATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    T. Digernes, V. S. Varadarajan and D. Weisbart, “Matrix valued Schrödinger operators on local fields,” to be published in Proc. SteklovMath. Inst. 265, (2009).Google Scholar
  90. 90.
    B. Dragovich, “Adelic model of harmonic oscillator,” Theor. Math. Phys. 101, 1404–1415 (1994); arXiv:hep-th/0402193.MATHMathSciNetCrossRefGoogle Scholar
  91. 91.
    B. Dragovich, “Adelic harmonic oscillator,” Int. J. Mod. Phys. A 10, 2349–2365 (1995); arXiv:hepth/0404160.MATHMathSciNetCrossRefGoogle Scholar
  92. 92.
    B. Dragovich, “p-Adic and adelic quantum mechanics,” Proc. Steklov Inst. Math. 245, 72–85 (2004); arXiv:hep-th/0312046v1.MathSciNetGoogle Scholar
  93. 93.
    G. Djordjević, B. Dragovich and Lj. Nešić, “p-Adic and adelic free relativistic particle,” Mod. Phys. Lett. A 14, 317–325 (1999); arXiv:hep-th/0005216.CrossRefGoogle Scholar
  94. 94.
    B. Dragovich, “On generalized functions in adelic quantum mechanics,” Integral Transform. Spec. Funct. 6, 197–203 (1998); arXiv:math-ph/0404076.MATHMathSciNetCrossRefGoogle Scholar
  95. 95.
    E.M. Radyna and Ya. V. Radyno, “Distributions and mnemofunctions on adeles,” Proc. Steklov. Inst.Math. 245, 215–227 (2004).Google Scholar
  96. 96.
    G. Parisi, “p-Adic functional integral,” Mod. Phys. Lett. A 4, 369–374 (1988).MathSciNetGoogle Scholar
  97. 97.
    E. I. Zelenov, “p-Adic path integrals,” J. Math. Phys. 32, 147–153 (1991).MATHMathSciNetCrossRefGoogle Scholar
  98. 98.
    V. S. Varadarajan, “Path integrals for a class of p-adic Schrödinger equations,” Lett. Math. Phys. 39, 97–106 (1997).MATHMathSciNetCrossRefGoogle Scholar
  99. 99.
    O. G. Smolyanov and N. N. Shamarov, “Feynman and Feynman-Kac formulas for evolution equations with Vladimirov operator,” Doklady Mathematics 77(3), 345–350 (2008).MATHCrossRefGoogle Scholar
  100. 100.
    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
  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.
    G. Djordjević and B. Dragovich, “p-Adic path integrals for quadratic actions,” Mod. Phys. Lett. A 12(20), 1455–1463 (1997); arXiv:math-ph/0005026.MATHCrossRefGoogle Scholar
  103. 103.
    R. N. Fernandez, V. S. Varadarajan and D. Weisbart, “Airy functions over local fields,” to be publ. in Lett. Math. Phys.Google Scholar
  104. 104.
    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).MATHMathSciNetCrossRefGoogle Scholar
  105. 105.
    B. Dragovich, “Adelic generalization of wave function of the Universe,” The First Hungarian-Yugoslav Astronomical Conference, Hungary, Baja, April 26–27, 1995. Publ. Obs. Astron. Belgrade 49 143–144 (1995).Google Scholar
  106. 106.
    G. S. Djordjević, B. Dragovich, Lj. D. Nešić and I.V. Volovich, “p-Adic and adelic minisuperspace quantum cosmology,” Int. J. Mod. Phys. A 17(10), 1413–1433 (2002); arXiv:gr-qc/0105050.MATHCrossRefGoogle Scholar
  107. 107.
    B. Dragovich and Lj. Nešić, “p-Adic and adelic generalization of quantum cosmology,” Gravitation and Cosmology 5, 222–228 (1999); arXiv:gr-qc/0005103.MATHMathSciNetGoogle Scholar
  108. 108.
    R. S. Ismagilov, “On the spectrum of the self-adjoint operator in L 2(K) where K is a local field; an analog of the Feynman-Kac formula,” Theor. Math. Phys. 89, 1024–1028 (1991).MathSciNetCrossRefGoogle Scholar
  109. 109.
    A. N. Kochubei, “Parabolic equations over the field of p-adic numbers,” Math. USSR Izv. 39, 1263–1280 (1992).MathSciNetCrossRefGoogle Scholar
  110. 110.
    S. Haran, “Analytic potential theory over p-adics,” Ann. Inst. Fourier 43, 905–944 (1993).MATHMathSciNetGoogle Scholar
  111. 111.
    A. Kh. Bikulov and I. V. Volovich, “p-Adic Brownian motion,” Izv. Math. 61(3), 537–552 (1997).MATHMathSciNetCrossRefGoogle Scholar
  112. 112.
    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.MATHMathSciNetCrossRefGoogle Scholar
  113. 113.
    K. Kamizono, “p-Adic Brownian motion over Qp,” to be published.Google Scholar
  114. 114.
    S. N. Evans, “Local field Brownian motion,” J. Theor. Probab. 6, 817–850 (1993).MATHCrossRefGoogle Scholar
  115. 115.
    S. Albeverio and W. Karwowski, “A random walk on p-adic numbers - generator and its spectrum,” Stochastic processes. Theory and Appl. 53, 1–22 (1994).MATHMathSciNetCrossRefGoogle Scholar
  116. 116.
    K. Yasuda, “Additive processes on local fields,” J.Math. Sci. Univ. Tokyo 3, 629–654 (1996).MATHMathSciNetGoogle Scholar
  117. 117.
    S. Albeverio and Ya. Belopolskaya, “Stochastic processes in Qp associated with systems of nonlinear PDEs,” p-Adic Numbers, Ultrametric Analysis and Applications, to be published (2009).Google Scholar
  118. 118.
    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).MATHMathSciNetCrossRefGoogle Scholar
  119. 119.
    H. Kaneko, “Fractal theoretic aspects of local field,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 51–57 (2009).CrossRefGoogle Scholar
  120. 120.
    V. S. Vladimirov, “Generalized functions over the field of p-adic numbers,” Russ. Math. Surv. 43, 19–64 (1988).MATHCrossRefGoogle Scholar
  121. 121.
    A. N. Kochubei, “A non-Archimedean wave equation,” Pacif. J. Math. 235, 245–261 (2008).MATHMathSciNetCrossRefGoogle Scholar
  122. 122.
    W. A. Zuniga-Galindo, “Parabolic equations and Markov processes over p-adic fields,” Potential Anal. 28, 185–200 (2008).MATHMathSciNetCrossRefGoogle Scholar
  123. 123.
    S. Albeverio, S. Kuzhel and S. Torba, “p-Adic Schrödinger-type operator with point interactions,” J. Math. Anal. Appl. 338, 1267–1281 (2008).MATHMathSciNetCrossRefGoogle Scholar
  124. 124.
    G. Rammal, M. A. Toulouse and M. A. Virasoro, ”Ultrametricity for physicists,” Rev. Mod. Phys. 58, 765–788 (1986).MathSciNetCrossRefGoogle Scholar
  125. 125.
    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.MATHMathSciNetCrossRefGoogle Scholar
  126. 126.
    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.MATHMathSciNetCrossRefGoogle Scholar
  127. 127.
    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).MathSciNetGoogle Scholar
  128. 128.
    V. A. Avetisov and A. Kh. Bikulov, “Protein ultrametricity and spectral diffusion in deeply frozen proteins,” in press, Biophys. Rev. and Lett. 3(3), (2008); arXiv:0804.4551.Google Scholar
  129. 129.
    V. A. Avetisov, A. Kh. Bikulov and A. P. Zubarev, “First passage time distribution and number of returns for ultrametric random walk,” in press, J. Phys. A: Math. Theor. 42 (2009); arXiv:0808.3066.Google Scholar
  130. 130.
    V. A. Avetisov and Yu. N. Zhuravlev, “An evolutionary interpretation of the p-adic ultrametric diffusion equation,” Doklady Mathematics 75(3), 435–455 (2007); arXiv:0808.3066.MathSciNetCrossRefGoogle Scholar
  131. 131.
    A. Yu. Khrennikov, “Mathematical methods of the non-archimedean physics,” Uspekhi Mat. Nauk 45(4), 79–110 1990.MathSciNetGoogle Scholar
  132. 132.
    A. Yu. Khrennikov, “p-Adic quantum mechanics with p-adic valued functions,” J. Math. Phys. 32, 932–937 (1991).MATHMathSciNetCrossRefGoogle Scholar
  133. 133.
    A. Yu. Khrennikov, “Real-non-Archimedean structure of space-time,” Theor. Math. Phys. 86(2), 177–190 (1991).MathSciNetCrossRefGoogle Scholar
  134. 134.
    A. Yu. Khrennikov, “p-Adic probability theory and its applications. The principle of statistical stabization of frequencies,” Theor. Math. Phys. 97(3), 348–363 (1993).MathSciNetCrossRefGoogle Scholar
  135. 135.
    S. Albeverio and A. Yu. Khrennikov, “Representation of the Weyl group in spaces of square integrable functions with respect to p-adic valued Gaussian distributions,” J. Phys. A: Math. Gen. 29, 5515–5527 (1996).MATHMathSciNetCrossRefGoogle Scholar
  136. 136.
    S. Albeverio, R. Cianci and A. Yu. Khrennikov, “On the spectrum of the p-adic position operator,” J. Phys. A: Math. Gen. 30, 881–889 (1997).MATHMathSciNetCrossRefGoogle Scholar
  137. 137.
    S. Albeverio, R. Cianci and A. Yu. Khrennikov, “On the Fourier transform and the spectral properties of the p-adic momentum and Schrodinger operators,” J. Phys. A:Math. Gen. 30, 5767–5784 (1997).MATHMathSciNetCrossRefGoogle Scholar
  138. 138.
    W. H. Schikhof, Ultrametric Calculus (Cambridge University Press, Cambridge, 1984).MATHGoogle Scholar
  139. 139.
    A. Escassut, Ultrametric Banach Algebras (World Scientific, Singapore, 2003).MATHGoogle Scholar
  140. 140.
    A. Escassut, Analytic Elements in p-Adic Analysis (World Scientific, Singapore 1995).MATHGoogle Scholar
  141. 141.
    P.-C. Hu and C.-C. Yang, Meromorphic Functions over non-Archimedean Fields (Kluwer Academic Publishers, 2001).Google Scholar
  142. 142.
    B. Dwork, G. Gerotto and F. J. Sullivan, An Introduction to G-Functions (Princeton University Press, 1994).Google Scholar
  143. 143.
    M. L. Gorbachuk and V. I. Gorbachuk, “On the Cauchy problem for differential equations in a Banach space over the field of p-adic numbers, I, II,” Meth. Funct. Anal. Topology 9, 207–212 (2003); Proc. Steklov Inst. Math. 245, 91–97 (2004).MATHMathSciNetGoogle Scholar
  144. 144.
    I. Ya. Aref’eva, B. Dragovich and I. V. Volovich, “On the p-adic summability of the anharmonic oscillator,” Phys. Lett. B 200, 512–514 (1988).MathSciNetCrossRefGoogle Scholar
  145. 145.
    B. Dragovich, “p-Adic perturbation series and adelic summability,” Phys. Lett. B 256, 392–396 (1991).MathSciNetCrossRefGoogle Scholar
  146. 146.
    B. Dragovich, “On some p-adic series with factorials,” in p-Adic Functional Analysis, Proc. Fourth Int. Conf. p-Adic Analysis, Eds. W.H. Schikhof et al., Lecture Notes on Pure and Appl. Math. 192, pp. 95–105 (Marcel Dekker, N.Y., 1997).Google Scholar
  147. 147.
    A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, Dordrecht, 1997).MATHGoogle Scholar
  148. 148.
    A. Yu. Khrennikov, Interpretations of Probability (VSP, Utrecht, 1999).MATHGoogle Scholar
  149. 149.
    G. K. Kalisch, “On p-adic Hilbert spaces,” Ann. Math. 48, 180–192 (1947).MathSciNetCrossRefGoogle Scholar
  150. 150.
    S. Albeverio, J. M. Bayod, C. Perez-Garcia, A. Yu. Khrennikov and R. Cianci, “Non-Archimedean analogues of orthogonal and symmetric operators,” Izv. Akad. Nauk 63(6), 3–28 (1999).Google Scholar
  151. 151.
    A. N. Kochubei, “p-Adic commutation relations,” J. Phys. A: Math. Gen. 29, 6375–6378 (1996).MATHMathSciNetCrossRefGoogle Scholar
  152. 152.
    H. Keller, H. Ochsenius and W. H. Schikhof, “On the commutation relation AB − BA = I for operators on nonclassical Hilbert spaces,” in p-Adic Functional Analysis, Eds. A.K. Katseras, W.H. Schikhof and L. van Hamme. Lecture Notes in Pure and Appl. Math. 222, 177–190 (2003).Google Scholar
  153. 153.
    A. Yu. Khrennikov, “Human subconscious as the p-adic dynamical system,” J. Theor. Biology 193, 179–196 (1998).CrossRefGoogle Scholar
  154. 154.
    A. Yu. Khrennikov, “p-Adic dynamical systems: description of concurrent struggle in biological population with limited growth,” Dokl. Akad. Nauk 361, 752 (1998).MATHMathSciNetGoogle Scholar
  155. 155.
    S. Albeverio, A. Yu. Khrennikov and P. Kloeden, “Memory retrieval as a p-adic dynamical system,” Biosystems 49, 105–115 (1999).CrossRefGoogle Scholar
  156. 156.
    D. Dubischar, V. M. Gundlach, O. Steinkamp and A. Yu. Khrennikov, “A p-adic model for the process of thinking disturbed by physiological and information noise,” J. Theor. Biology 197, 451–467 (1999).CrossRefGoogle Scholar
  157. 157.
    A. Yu. Khrennikov, “Description of the operation of the human subconscious by means of p-adic dynamical systems,” Dokl. Akad. Nauk 365, 458–460 (1999).MATHMathSciNetGoogle Scholar
  158. 158.
    A. Yu. Khrennikov, “p-Adic discrete dynamical systems and collective behaviour of information states in cognitive models,” Discrete Dynamics in Nature and Society 5, 59–69 (2000).CrossRefGoogle Scholar
  159. 159.
    S. Albeverio, A. Yu. Khrennikov and B. Tirozzi, “p-Adic neural networks,” Math. Models and Meth. in Appl. Sciences 9(9), 1417–1437 (1999).MATHMathSciNetCrossRefGoogle Scholar
  160. 160.
    A. Yu. Khrennikov, “Classical and quantum mechanics on p-adic trees of ideas,” BioSystems 56, 95–120 (2000).CrossRefGoogle Scholar
  161. 161.
    A. Yu. Khrennikov, Classical and Quantum Mental Models and Freud’s Theory of Unconscious Mind, Series Math. Modelling in Phys., Engineering and Cognitive Sciences 1 (Växjö Univ. Press, Växjö, 2002).Google Scholar
  162. 162.
    A. Yu. Khrennikov, Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena (Kluwer, Dordreht, 2004).MATHGoogle Scholar
  163. 163.
    A. Yu. Khrennikov, “Probabilistic pathway representation of cognitive information,” J. Theor. Biology 231, 597–613 (2004).CrossRefGoogle Scholar
  164. 164.
    F. Murtagh, “On ultrametric algorithmic information,” Computer Journal, in press. (Online, Advance Access, 9 Oct. 2007,
  165. 165.
    J. Benois-Pineau, A. Yu. Khrennikov, and N. V. Kotovich, “Segmentation of images in p-adic and Euclidean metrics,” Doklady Mathematics 64(3), 450–455 (2001).Google Scholar
  166. 166.
    A. Yu. Khrennikov and N. V. Kotovich, “Representation and compression of images with the aid of the madic coordinate system,” Dokl. Akad. Nauk 387(2), 159–163 (2002).MathSciNetGoogle Scholar
  167. 167.
    A. Yu. Khrennikov, N. V. Kotovich and E. L. Borzistaya, “Compression of images with the aid of representation by p-adic maps and approximation by Mahler’s polynomials,” Doklady Mathematics 69(3), 373–377 (2004).MATHMathSciNetGoogle Scholar
  168. 168.
    S. V. Kozyrev, “Wavelet theory as p-adic spectral analysis,” Izvestiya: Mathematics 66(2), 367–376 (2002); arXiv:math-ph/0012019.MATHMathSciNetCrossRefGoogle Scholar
  169. 169.
    S. V. Kozyrev, “p-Adic pseudodifferential operators and p-adic wavelets,” Theor. Math. Phys. 138(3), 322–332 (2004); arXiv:math-ph/0303045.MathSciNetCrossRefMATHGoogle Scholar
  170. 170.
    S. Albeverio and S. V. Kozyrev, “Frames of p-adic wavelets and orbits of the affine group,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 18–33 (2009); arxiv:0801.4713.CrossRefGoogle Scholar
  171. 171.
    J. J. Benedetto and R. L. Benedetto, “A wavelet theory for local fields and related groups,” J. Geom. Analysis 3, 423–456 (2004).MathSciNetGoogle Scholar
  172. 172.
    S. Albeverio, S. Evdokimov and M. Skopina, “p-Adic multiresolution analysis and wavelet frames,” (2008); arXiv:0802.1079v1.Google Scholar
  173. 173.
    S. Albeverio, S. Evdokimov and M. Skopina, “p-Adic multiresolution analyses,” arXiv:0810.1147.Google Scholar
  174. 174.
    A. Yu. Khrennikov and V.M. Shelkovich, “Non-Haar p-adic wavelets and their application to pseudodifferential operators and equations,” (2006); arXiv:0808.3338v1.Google Scholar
  175. 175.
    A. Yu. Khrennikov, V. M. Shelkovich and M. Skopina, “p-Adic refinable functions and MRA-based wavelets,” J. Appr. Theory (2008); arXiv:0711.2820.Google Scholar
  176. 176.
    V. M. Shelkovich and M. Skopina, “p-Adic Haar multiresolution analysis and pseudo-differential operators,” J. Fourier Anal. Appl. (2008); arXiv:0705.2294.Google Scholar
  177. 177.
    A. Yu. Khrennikov and V. M. Shelkovich, “Distributional asymptotics and p-adic Tauberian and Shannon-Kotelnikov theorems,” Asymptotical Analysis 46(2), 163–187 (2006).MATHMathSciNetGoogle Scholar
  178. 178.
    S. V. Kozyrev and A. Yu. Khrennikov, “Pseudodifferential operators on ultrametric spaces and ultrametric wavelets,” Izvestiya: Mathematics 69(5), 989–1003 (2005).MATHMathSciNetCrossRefGoogle Scholar
  179. 179.
    A. Yu. Khrennikov and S. V. Kozyrev, “Wavelets on ultrametric spaces,” Appl. Comp. Harmonic Analysis 19, 61–76 (2005).MATHMathSciNetCrossRefGoogle Scholar
  180. 180.
    S. V. Kozyrev, “Wavelets and spectral analysis of ultrametric pseudodifferential operators,” Sbornik: Mathematics 198(1), 97–116 (2007); arXiv:math-ph/0412082.MATHMathSciNetCrossRefGoogle Scholar
  181. 181.
    A. N. Kochubei, “Analysis and probability over infinite extensions of a local field,” Potential Anal. 10, 305–325 (1999).MATHMathSciNetCrossRefGoogle Scholar
  182. 182.
    K. Yasuda, “Extension of measures to infinite dimensional spaces over p-adic field,” Osaka J. Math. 37, 967–985 (2000).MATHMathSciNetGoogle Scholar
  183. 183.
    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).MATHMathSciNetCrossRefGoogle Scholar
  184. 184.
    H. Kaneko and K. Yasuda, “Capacities associated with Dirichlet space on an infinite extension of a local field,” Forum Math. 17, 1011–1032 (2005).MATHMathSciNetCrossRefGoogle Scholar
  185. 185.
    A. N. Kochubei, Analysis in Positive Characteristic (Cambridge University Press, Cambridge, 2009).MATHGoogle Scholar
  186. 186.
    S. Fischenko and E. I. Zelenov, “p-Adic models of turbulence,” in p-Adic Mathematical Physics, Eds. A. Yu. Khrennikov, Z. Rakić and I. V. Volovich, AIP Conference Proceedings 286 pp. 174–191 (Melville, New York, 2006).Google Scholar
  187. 187.
    S.V. Kozyrev, “Toward an ultrametric theory of turbulence,” Theor.Math. Phys. 157(3), 1711–1720 (2008); arXiv:0803.2719.CrossRefMathSciNetGoogle Scholar
  188. 188.
    M. Mezard, G. Parisi and M. Virasoro, Spin-Glass Theory and Beyond (World Scientific, Singapore, 1987).MATHGoogle Scholar
  189. 189.
    G. Parisi and N. Sourlas, “p-Adic numbers and replica symmetry breaking,” European Phys. J. B 14, 535–542 (2000); arXiv:cond-mat/9906095.MathSciNetCrossRefGoogle Scholar
  190. 190.
    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
  191. 191.
    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
  192. 192.
    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
  193. 193.
    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).MATHMathSciNetCrossRefGoogle Scholar
  194. 194.
    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).MATHMathSciNetCrossRefGoogle Scholar
  195. 195.
    A. Yu. Khrennikov and M. Nilsson, p-Adic Deterministic and Random Dynamical Systems (Kluwer, Dordreht, 2004).Google Scholar
  196. 196.
    D. K. Arrowsmith and F. Vivaldi, “Geometry of p-adic Siegel discs,” Physica D 71, 222–236 (1994).MATHMathSciNetCrossRefGoogle Scholar
  197. 197.
    F. Vivaldi and S. Hatjyspyros, “Galois theory of periodic orbits of polynomialmaps,” Nonlinearity D 5, 961–978 (1992).MATHCrossRefGoogle Scholar
  198. 198.
    V. Anashin and A. Khrennikov, Applied Algebraic Dynamics, De Gruyter Expositions in Mathematics (Walter De Gruyter Inc, Berlin, 2009).Google Scholar
  199. 199.
    M. R. Herman and J. C. Yoccoz, “Generalization of some theorem of small divisors to non-archimedean fields,” in Geometric Dynamics, Lecture Notes Math. 1007, pp. 408–447 (Springer-Verlag, New York- Berlin-Heidelberg, 1983).CrossRefGoogle Scholar
  200. 200.
    J. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics 241 (Springer-Verlag, New York, 2007).Google Scholar
  201. 201.
    V. I. Arnold, “Number-theoretic turbulence in Fermat-Euler arithmetics and large Young diagrams geometry statistics,” J. Math. Fluid Mech. 7, 4–50 (2005).MathSciNetCrossRefGoogle Scholar
  202. 202.
    B. Dragovich, A. Khrennikov and D. Mihajlović, “Linear fractional p-adic and adelic dynamical systems,” Rep. Math. Phys. 60, 55–68 (2007); arXiv:math-ph/0612058.MATHMathSciNetCrossRefGoogle Scholar
  203. 203.
    V. S. Anashin, “Uniformly distributed sequences over p-adic integers,” in Number Theoretic and Algebraic Methods in Computer Science, Eds. I. Shparlinsky A. J. van der Poorten and H. G. Zimmer, Proc. Int. Conf. (Moscow, June-July, 1993), pp. 1–18 (World Scientific, Singapore 1995).Google Scholar
  204. 204.
    V. S. Anashin, “Uniformly distributed sequences of p-adic integers,” Mathematical Notes 55(2), 109–133 (1994).MathSciNetCrossRefGoogle Scholar
  205. 205.
    V. Anashin, “Ergodic transformations in the space of p-adic integers,” in p-Adic Mathematical Physics, Eds. A. Yu. Khrennikov, Z. Rakić and I. V. Volovich, AIP Conf. Proc. 826, pp. 3–24 (Melville, New York, 2006).Google Scholar
  206. 206.
    B. Dragovich and A. Yu. Dragovich, “A p-adic model of DNA sequence and genetic code,” arXiv:qbio. GN/0607018.Google Scholar
  207. 207.
    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
  208. 208.
    M. Pitkänen, “Could genetic code be understood number theoretically?” Electronic preprint: (2006).
  209. 209.
    B. Dragovich and A. Yu. Dragovich, “A p-adicmodel of DNA sequence and genetic code,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 34–41 (2009); arXiv:q-bio.GN/0607018.CrossRefGoogle Scholar
  210. 210.
    B. Dragovich and A. Yu. Dragovich, “p-Adic modelling of the genome and the genetic code,” Computer Journal, doi:10.1093/comjnl/bxm083, to appear (2009); arXiv:0707.3043.Google Scholar
  211. 211.
    A. Yu. Khrennikov, 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
  212. 212.
    M. N. Khokhlova and I. V. Volovich, “Modeling theory and hypergraph of classes,” Proc. Steklov Inst.Math. 245, 266–272 (2004).Google Scholar
  213. 213.
    J. Q. Trelewicz and I. V. Volovich, “Analysis of business connections utilizing theory of topology of random graphs,” in p-AdicMathematical Physics, Eds. A. Yu. Khrennikov, Z. Rakić and I. V. Volovich. AIP Conf. Proc. 826, pp. 330–344 (Melville, New York, 2006).Google Scholar
  214. 214.
    A. Kh. Bikulov, A. P. Zubarev and L. V. Kaidalova, “Hierarchical dynamical model of financial market near the crash point and p-adic analysis,” Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya “Fiziko-Matematicheskie Nauki” 42, 135–141 (2006) [in Russian].Google Scholar
  215. 215.
    B. Dragovich and D. Joksimović, “On possible uses of p-adic analysis in econometrics,” Megatrend Revija 4(2), 5–16 (2007).Google Scholar
  216. 216.
    F. Murtagh, “From data to the p-adic or ultrametric model,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 58–68 (2009).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  • B. Dragovich
    • 1
  • A. Yu. Khrennikov
    • 2
  • S. V. Kozyrev
    • 3
  • I. V. Volovich
    • 3
  1. 1.Institute of PhysicsBelgradeSerbia
  2. 2.Växjö UniversityVäxjöSweden
  3. 3.Steklov Mathematical InstituteMoscowRussia

Personalised recommendations