Discrete-scale invariance and complex dimensions

Physics Report

Volumn 297, Issue 5, 1998, Pages 239-270

  Sornette, Didier ^a,b  

^a INSTITUT NON LINÉAIRE DE NICE   (France)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

0.5.70.Jk; 11.30. j; 47.53.+n; 47.54.+r; 64.60.Hr

Indexed keywords

EID: 0000448397     PISSN: 03701573     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0370-1573(97)00076-8     Document Type: Article

References (146)

1. B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, San Francisco, 1982.
2. G.A. Edgar (Ed.), Classics on Fractals, Addison-Wesley, Readings, MA, 1993.
3. E.A. Novikov, Dokl. Akad. Nauk SSSR 168/6, 1279 (1966).
4. E.A. Novikov, Phys. Fluids A 2 (1990) 814.
5. G. Jona-Lasinio, Nuovo Cimento 26B (1975) 99.
6. Nauenberg M. J. Phys. A. 8:1975;925.
7. Th. Niemeijer, J.M.J. van Leeuwen, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, vol. 6, Academic Press, London, 1976, p. 425.
8. Saleur H., Sornette D. J. Phys. I France. 6:1996;327.
9. P. Bak, How Nature Works: the Science of Self-organized Criticality, Copernicus, New York, NY, USA, 1996.
10. D.J. Wallace, R.K.P. Zia, Phys. Lett. 48A (1974) 325.
11. D. Sornette, A. Johansen, A. Arnéodo, J.-F. Muzy, H. Saleur, Phys. Rev. Lett. 76 (1996) 251.
12. J. Cardy, J. Phys. A 17 (1984) L385.
13. S. Weinberg, preprint hep-th/9702027.
14. S. Weinberg, The Quantum Theory of Fields, Cambridge University Press, Cambridge, 1995.
15. Sornette D., Sammis C.G. J. Phys. I France. 5:1995;607.
16. H. Saleur, C.G. Sammis, D. Sornette, Nonlinear Processes Geophys. 3 (2) (1996) 102.
17. H. Saleur, C.G. Sammis, D. Sornette, J. Geophys. Res. 101 (1996) 17661.
18. A. Johansen, D. Sornette, H. Wakita, U. Tsunogai, W.I. Newman, H. Saleur, Japan, J. Phys. I France 6 (1996) 1391.
19. J.-C. Anifrani, C. Le Floc'h, D. Sornette, B. Souillard, J. Phys. I France 5 (6) (1995) 631.
20. J.-C. Anifrani, A. Johansen, C. Le Floc'h, G. Ouillon, D. Sornette, C. Vanneste, B. Souillard, Proc. 6th European Conf. on Non-Destructive Testing, 24-28 October 1994, Nice, Presentation N72.
21. D. Sornette, A. Johansen, J.-P. Bouchaud, J. Phys. I France 6 (1996) 167.
22. J.A. Feigenbaum, P.G.O. Freund, Int. J. Mod. Phys. 10 (27) (1996) 3737.
23. Sornette D., Johansen A. Physica A. 245:1997;411.
24. D. Bessis, J.-D. Fournier, G. Servizi, G. Tourchetti, S. Vaienti, Phys. Rev. A 36 (1987) 920.
25. Fournier J.-D., Tourchetti G., Vaienti S. Phys. Lett. A. 140:1989;331.
26. Orlandini E., Tesi M.C., Turchetti G. Europhys. Lett. 21:1993;719.
27. A. Kapitulnik, A. Aharony, G. Deutscher, D. Stauffer, J. Phys. A 16 (1983) L269.
28. Vlad M.O., Mackey M.C. Phys. Scripta. 50:1994;615.
29. D. Bessis, J.S. Geronimo, P. Moussa, J. Phys. Lett. 44 (1983) L977.
30. B. Douçot, W. Wang, J. Chaussy, B. Pannetier, R. Rammal, Phys. Rev. Lett. 57 (1986) 1235.
31. B. Derrida, L. De Seze, C. Itzykson, J. Stat. Phys. 33 (1983) 559.
32. Derrida B., Itzykson C., Luck J.M. Commun. Math. Phys. 94:1984;115.
33. Meurice Y., Ordaz G., Rodgers V.G.J. Phys. Rev. Lett. 75:1995;4555.
34. Newman W.I., Turcotte D.L., Gabrielov A.M. Phys. Rev. E. 52:1995;4827.
35. B. Kutjnak-Urbanc, S. Zapperi, S. Milošević, H.E. Stanley, Phys. Rev. E 54 (1996) 272.
36. Karevski D., Turban L. J. Phys. A. 29:1996;3461.
37. Wu F.Y. Rev. Mod. Phys. 54:1982;235.
38. P. Flajolet, M. Golin, in: A. Lingas, R. Karlsson, S. Carlsson (Eds.), Automata, Languages and Programming, 20th Int. Coll., ICALP 93 Proc. Springer, Berlin, 1993, pp. 137-149.
39. P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger, R.F. Tichy, Theoret. Comput. Sci. 123 (1994) 291.
40. J. Bernasconi, W.R. Schneider, J. Phys. A 15 (1983) L729.
41. Vlad M.O. Int. J. Mod. Phys. B. 6:1992;417.
42. Vlad M.O. J. Phys. A. 25:1992;749.
43. Dee D., Ghil M. SIAM J. Appl. Math. 44:1984;111.
44. Ghil M., Mullhaupt A. J. Stat. Phys. 41:1985;125.
45. Arneodo A., Argoul F., Bacry E., Muzy J.F. Phys. Rev. Lett. 68:1992;3456.
46. Arneodo A., Argoul F., Muzy J.F., Tabard M. Phys. Lett. A. 171:1992;31.
47. Arneodo A., Argoul F., Muzy J.F., Tabard M., Bacry E. Fractals. 1:1993;629.
48. Huang Y., Ouillon G., Saleur H., Sornette D. Phys. Rev. E. 55:1997;6433.
49. Ouillon G., Sornette D., Genter A., Castaing C. J. Phys. I France. 6:1996;1127.
50. J.M. DeGraff, A. Aydin, JGR 98 (B4) (1993) 6411.
51. Pollard D.D., Aydin A. Geol. Soc. Am. Bull. 100:1988;1181.
52. S. Nemat-Nasser, L.M. Keer, K.S. Parihar, Technical Report No. 77-9-2, Earthquake Research and Engineering lab., Department of Civil Engineering, Northwestern University, September 1977.
53. S. Nemat-Nasser, L.M. Keer, K.S. Parihar, Int. J. Solids Struct. 14 (1978) 409.
54. S. Nemat-Nasser, A. Oranratnachai, Trans. ASME 101 (1979) 34.
55. M.A. Sadovskiy, Dokl. Akad. Nauk SSSR, 247 (4) (1979) 829.
56. Sadovskiy M.A., Bolkhovitinov L.G., Pisarenko V.F. Izv. Akad. Nauk SSSR Fizika Zemli. 12:1982;3.
57. M.A. Sadovskiy, V.F. Pisarenko, Inst. Fiz. Zemli Akad. Nauk SSSR 2 (1982).
58. Sadovskiy M.A., Pisarenko V.F., Rodionov V.N. Vestn. Akad. Nauk SSSR. 1:1983;82.
59. Sadovskiy M.A., Golubeva T.V., Pisarenko V.F., Shnirman M.G. Izv. Earth Phys. 20:1984;87.
60. Borodich F.M. J. Mech. Phys. Solids. 45:1997;239.
61. Sahimi M., Arbabi S. Phys. Rev. Lett. 77:1996;3689.
62. Ball R.C., Blumenfeld R. Phys. Rev. Lett. 65:1990;1784.
63. Ball R.C., Blumenfeld R. Phys. Rev. Lett. 68:1992;2254.
64. Varnes D.J., Bufe C.G. Geophys. J. Int. 124:1996;149.
65. Bowman D.D., Sammis C.G. EOS Trans. Am. Geophys. 77:1996;486.
66. S.W. Smith, C.G. Sammis, EOS Trans. Am. Geophys. U. 77 (1996) F480.
67. Y. Huang, H. Saleur, C.G. Sammis, D. Sornette, Europhys. Lett., in press; cond-mat/9612065.
68. Feigenbaum M.J. J. Stat. Phys. 19:1978;25.
69. Feigenbaum M.J. J. Stat. Phys. 21:1979;669.
70. P. Coullet, C. Tresser, J. Phys. Coll. 39 (1978) C5.
71. Coullet P., Tresser C. C.R. Acad. Sci. 287:1978;577.
72. P. Collet, J.P. Eckmann, Iterated Maps of the Interval and Dynamical Systems, Birkhauser, Boston, 1980.
73. B. Derrida, J. Phys. Colloques C5 (1978) 49.
74. B. Derrida, A. Gervois, Y. Pomeau, Ann. Inst. H. Poincaré AXXIX (1978) 305.
75. Derrida B., Gervois A., Pomeau Y. J. Phys. A. 12:1979;269.
76. Argoul F., Arnéodo A., Collet P., Lesne A. Europhys. Lett. 3:1987;643.
77. West B.J., Fan X. Fractals. 1:1993;21.
78. Vollmer J., Breymann W. Europhys. Lett. 27:1994;23.
79. Lubensky T.C., Isaacson J. Phys. Rev. Lett. 41:1978;829.
80. D. Stauffer, A. Aharony, Introduction to Percolation Theory, 2nd ed., Taylor and Francis, London, 1992.
81. Parisi G., Sourlas N. Phys. Rev. Lett. 46:1981;453.
82. Aharony A. Phys. Rev. B. 12:1975;1049.
83. Chen J.-H., Lubensky T.C. Phys. Rev. B. 16:1977;2106.
84. Khmelnitskii D.E. Phys. Lett. A. 67:1978;59.
85. Boyanovsky D., Cardy J.L. Phys. Rev. B. 26:1982;154.
86. Weinrib A., Halperin B.I. Phys. Rev. B. 27:1983;413.
87. Derrida B., Hilhorst H. J. Phys. A. 16:1983;2641.
88. M. Mézard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987.
89. Rammal R., Toulouse G., Virasoro M.A. Rev. Mod. Phys. 58:1986;765.
90. D.L. Stein, in: J. Souletie, J. Vannimenus, R. Stora (Eds.), Chance and Matter, North-Holland, Amsterdam, 1987.
91. Lässig M. Phys. Lett. B. 278:1992;439.
92. Lässig M. Nucl. Phys. B. 380:1992;601.
93. Ludwig A.W.W. Nucl. Phys. B. 330:1990;639.
94. West B.J. Int. J. Mod. Phys. B. 4:1990;1629.
95. West B.J. Ann. Biomed. Eng. 18:1990;135.
96. Schlesinger M.F., West B.J. Phys. Rev. Lett. 67:1991;2106.
97. West B.J., Deering W. Phys. Rep. 246:1994;1.
98. Anselmet F., Gagne Y., Hopfinger E.J., Antonia R.A. J. Fluid Mech. 140:1984;63.
99. U. Frisch, Turbulence, the Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995, pp. 130-131.
100. Yamada M., Okhitani K. Phys. Rev. Lett. 60:1988;983.
101. Okhitani K., Yamada M. Prog. Theor. Phys. 81:1989;329.
102. Jensen M.H., Paladin G., Vulpiani A. Phys. Rev. A. 43:1991;7798.
103. Nakano T. Prog. Theor. Phys. 79:1988;569.
104. T. Dombre, J.-L. Gilson, preprint, October 1995.
105. Graner F., Dubrulle B. Astron. Astrophys. 282:1994;262.
106. Graner F., Dubrulle B. Astron. Astrophys. 282:1994;269.
107. Choptuik M.W. Phys. Rev. Lett. 70:1993;9.
108. Abrahams A.M., Evans C.R. Phys. Rev. Lett. 70:1993;2980.
109. A.M. Abrahams, C.R. Evans, General Relativity Gravitation 26 (1994) 379.
110. Hirschmann E.W., Eardley D.M. Phys. Rev. D. 52:1995;5850.
111. Maier R.S., Stein D.L. Phys. Rev. Lett. 777:1996;4860.
112. Williams M.L. Bull. Seismol. Soc. Am. 49:1959;199.
113. Rice J.R. Trans. ASME. 55:1988;98.
114. J.R. Rice, Z. Suo, J.-S. Wang, in: M. Ruhle, A.G. Evans, M.F. Ashby, J.P. Hirth (Eds.), Metal-Ceramic Interfaces, Acta-Scripta Metallurgica Proc. Ser., vol. 4, Pergamon Press, Oxford, 1990, pp. 269-294.
115. K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985, p. 108.
116. Borodich F.M. Int. J. Solids Struct. 30:1993;1513.
117. Borodich F.M. PMM J. Appl. Math. Mech. 56:1992;681.
118. F.M. Borodich, Comptes Rendus Acad. Sci. Paris II 316 (1993) 281.
119. Hayata K., Koshiba M. Opt. Rev. 2:1995;331.
120. McWhirter J.G., Pike E.R. J. Phys. A. 11:1978;1729.
121. Ostrowsky N., Sornette D., Parker P., Pike E.R. Opt. Acta. 28:1981;1059.
122. C.E. Smith (Ed.), Log Periodic Antenna Design Handbook, 1st ed., Smith Electronics, Cleveland, OH, 1966 (1979 printing).
123. D.C. Baker, T.G. Reuss, IEEE Trans.- Broadcasting 36 (1990) 89.
124. D.R. Dykaar, B.I. Greene, J.F. Federici, A.F.J. Levi et al., Appl. Phys. Lett. 59 (1991) 262.
125. H.K. Smith, P.E. Mayes, IEEE Trans. Antennas Propagation 39 (1991) 1659.
126. P.S. Excell, N.N. Jackson, K.T. Wong, IEE Proc. H Microwaves Antennas and Propagation 140 (1993) 101.
127. R.R. Delyser, D.C. Chang, E.F. Kuester, Int. J. Microwave Millimeter-Wave Computer-Aided Eng. 3 (1993) 143.
128. M.M. Gitin, F.W. Wise, G. Arjavalingam, Y. Pastol et al., IEEE Trans. Antennas Propagation 42 (1994) 335.
129. Derrida B., Eckmann J.P., Erzan A. J. Phys. A. 16:1983;893.
130. Falconer K.J. J. Theoret. Probab. 7:1994;681.
131. M.F. Barnsley, Fractals Everywhere, 2nd ed. (rev. with the assistance of Hawley Rising III), Academic Press Professional, Boston, 1993.
132. Erzan A., Eckmann J.-P. Phys. Rev. Lett. 78:1997;3245.
133. Solis F.J., Tao L. Phys. Lett. A. 228:1997;351.
134. Sornette A., Sornette D. Europhys. Lett. 9:1989;197.
135. D. Sornette, Ph. Davy, A. Sornette, J. Geophys. Res. 95 (1990) 17353.
136. F. Pazmandi, R.T. Scalettar, G.T. Zimanyi, preprint cond-mat/9704155.
137. Sornette D., Vanneste C. Phys. Rev. Lett. 68:1992;612.
138. Sornette D., Vanneste C., Knopoff L. Phys. Rev. A. 45:1992;8351.
139. Vanneste C., Sornette D. J. Phys. I France. 2:1992;1621.
140. L. Lamaignère, F. Carmona, D. Sornette, Phys. Rev. Lett. 77 (1996) 2738.
141. A. Johansen, D. Sornette, preprint 1997.
142. Moffatt H.K. J. Fluid Mech. 18:1964;1.
143. Dean W.R., Montagnon P.E. Proc. Cambridge Phil. Soc. 45:1949;389.
144. M.P. Brenner, X.D. Shi, S.R. Nagel, Phys. Rev. Lett. 73 (1994) 3391; X.D. Shi, M.P. Brenner, S.R. Nagel, Science 265 (1994) 219.
145. M.P. Brenner, X.D. Shi, S.R. Nagel, Phys. Rev. Lett. 73 (1994) 3391; X.D. Shi, M.P. Brenner, S.R. Nagel, Science 265 (1994) 219.
146. D. Stauffer, D. Sornette, Physica A, in press; preprint cond-mat/9712085.