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Scale Relativity and Fractal Space-Time: Theory and Applications

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Abstract

In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations. In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-periodic law for species punctuated evolution, human development and society evolution), to Earth sciences (log-periodic deceleration of the rate of California earthquakes and of Sichuan earthquake replicas, critical law for the arctic sea ice extent) and tentative applications to systems biology.

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References

  • Abbott L. F., Wise M. B. (1981) Dimension of a quantum-mechanical path. American Journal of Physics 49: 37–39

    Article  Google Scholar 

  • Agnese A. G., Festa R. (1997) Clues to discretization on the cosmic scale. Physics Letters A 227: 165–171

    Article  Google Scholar 

  • Allègre C., Le Mouel J., Provost A. (1982) Scaling rules in rock fracture and possible implications for earthquake prediction. Nature 297: 47–49

    Article  Google Scholar 

  • Amelino-Camelia G. (2001) Testable scenario for relativity with minimum length. Physics Letters B 510: 255–263

    Article  Google Scholar 

  • Amelino-Camelia G. (2002) Doubly-special relativity: First results and key open problems. International Journal of Modern Physics D 11: 1643–1669

    Article  Google Scholar 

  • Auffray, Ch., & Nottale, L. (2008). Scale relativity theory and integrative systems biology: 1: Founding principles and scale laws. Progress in Biophysics and Molecular Biology, 97, 79–114. http://luth.obspm.fr/~uthier/nottale/arPBMB08AN.pdf.

  • Ben Adda F., Cresson J. (2000) Divergence d’échelle et différentiabilité. Comptes Rendus de l’Acad謩e des Sciences Paris 330: 261–264

    Google Scholar 

  • Ben Adda F., Cresson J. (2004) Quantum derivatives and the Schrödinger equation. Chaos, Solitons & Fractals 19: 1323–1334

    Article  Google Scholar 

  • Ben Adda F., Cresson J. (2005) Fractional differential equations and the Schrödinger equation. Applied Mathematics and Computation 161: 323–345

    Article  Google Scholar 

  • Berry M. V. (1996) Quantum fractals in boxes. Journal of Physics A: Mathematical and General 29: 6617–6629

    Article  Google Scholar 

  • Cafiero R., Loreto V., Pietronero L., Vespignani A., Zapperi S. (1995) Local rigidity and self-organized criticality for avalanches. Europhysics letters 29: 111–116

    Article  Google Scholar 

  • Campagne, J. E., & Nottale, L. (2003). unpublished preprint.

  • Carpinteri A., Chiaia B. (1996) Power scaling laws and dimensional transitions in solid mechanics. Chaos, Solitons & Fractals 7: 1343

    Article  Google Scholar 

  • Cash, R., Chaline, J., Nottale, L., & Grou, P. (2002). Développement humain et loi log-périodique. Comptes Rendus Biologies, 325, 585–590. http://luth.obspm.fr/~uthier/nottale/arcash.pdf.

  • Castro C. (1997) String theory, scale relativity and the generalized uncertainty principle. Foundations of Physics Letters 10: 273–293

    Article  Google Scholar 

  • Castro C., Granik A. (2000) How the new scale relativity theory resolves some quantum paradoxes. Chaos, Solitons & Fractals 11: 2167–2178

    Article  Google Scholar 

  • Célérier, M. N., & Nottale, L. (2004). Quantum-classical transition in scale relativity. Journal of Physics A: Mathematical and General, 37, 931–955. http://arXiv.org/abs/quant-ph/0609161.

  • Célérier, M. N., & Nottale L. (2006). The Pauli equation in scale relativity. Journal of Physics A: Mathematical and General, 39, 12565–12585. http://arXiv.org/abs/quant-ph/0609107.

    Google Scholar 

  • Chaline, J., Nottale, L., & Grou, P., (1999). L’arbre de la vie a-t-il une structure fractale? Comptes Rendus de l’AcadéMie des Sciences Paris, 328, 717–726. http://luth.obspm.fr/~uthier/nottale/arCNGcra.pdf.

  • Connes A. (1994) Noncommutative Geometry. Academic Press, New York

    Google Scholar 

  • Connes, A., Douglas, M. R., & Schwarz, A. (1998). Noncommutative geometry and matrix theory. Journal of High Energy Physics, 02,003 (hep-th/9711162).

  • Cresson J. (2001) Mémoire d’habilitation à diriger des recherches. Université de Franche-Comté, Besançon

    Google Scholar 

  • Cresson J. (2002) Scale relativity theory for one-dimensional non-differentiable manifolds. Chaos, Solitons & Fractals 14: 553–562

    Article  Google Scholar 

  • Cresson J. (2003) Scale calculus and the Schrödinger equation. Journal of Mathematical Physics 44: 4907–4938

    Article  Google Scholar 

  • Cresson J. (2007) Fractional embedding of differential operators and Lagrangian systems. Journal of Mathematical Physics 48(033504): 1–34

    Google Scholar 

  • da Rocha, D., & Nottale, L. (2003). Gravitational structure formation in scale-relativity. Chaos, Solitons & Fractals, 16, 565–595. http://arXiv.org/abs/astro-ph/0310036.

  • Dubois, D. (2000). Computational derivation of quantum relativist electromagnetic systems with forward- backward space-time shifts. In Proceedings of CASYS’1999, 3rd international conference on computing anticipatory systems, Liège, Belgium, American Institute of physics conference proceedings (Vol. 517, pp. 417–429).

  • Dubrulle, B., Graner, F., & Sornette, D. (Eds.). (1997). Scale invariance and beyond, Proceedings of Les Houches school (p. 275). Berlin, New York: EDP Sciences, Les Ullis/Springer.

  • El Naschie M. S. (1992) Multi-dimensional Cantor sets in classical and quantum mechanics. Chaos, Solitons & Fractals 2: 211–220

    Article  Google Scholar 

  • El Naschie M. S. (2000) Scale relativity in Cantorian space-time. Chaos, Solitons & Fractals 11: 2391–2395

    Article  Google Scholar 

  • El Naschie, M. S., Rössler, O., Prigogine, I. (eds) (1995) Quantum mechanics, diffusion and chaotic Fractals. Pergamon, New York

    Google Scholar 

  • Feynman R. P., Hibbs A. R. (1965) Quantum Mechanics and Path Integrals. MacGraw-Hill, New York

    Google Scholar 

  • Forriez, M. (2005a). Etude de la Motte de Boves. Geography and History Master I report, Artois University.

  • Forriez, M. (2005b). Construction d’un espace géographique fractal. Geography Master II report, Avignon University.

  • Forriez, M., & Martin, P. (2006). De l’utilité de la théorie de la relativité d’échelle de L. Nottale en géographie. 2: Application d’un modèle scalaire spatio-temporel. In Géopoint 2006. Demain la géographie. Permanences, dynamiques, mutations: Pourquoi? Comment? (pp. 305–312). Avignon, Groupe Dupont-UMR ESPACE du CNRS Editeurs.

  • Galopeau, P., Nottale, L., Ceccolini, D., Da Rocha, D., Schumacher, G. & Tran-Minh, N. (2004). Distribution of orbital elements of planets and exoplanets in scale relativity. In F. Combes, D. Barret, T. Contini, F. Meynadier & L. Pagani (Eds.), Scientific Highlights 2004, Proceedings of the Journées de la SF2A, 14–18 June 2004 (pp. 75–76). Paris: EDP Sciences.

  • Gould S. J., Eldredge N. (1977) Punctuated equilibria: the tempo and mode of evolution reconsidered. Paleobiology 3(2): 115–151

    Google Scholar 

  • Grabert H., Hänggi P., Talkner P. (1979) Is quantum mechanics equivalent to a classical stochastic process?. Physical Review A 19: 2440–2445

    Article  Google Scholar 

  • Green M. B., Schwarz J. H., Witten E. (1987) Superstring theory (Vol. 2). Cambridge University Press, Cambridge

    Google Scholar 

  • Grou P. (1987) L’aventure économique. L’Harmattan, Paris

    Google Scholar 

  • Grou, P., Nottale, L., & Chaline, J. (2004). Log-periodic laws applied to geosciences. In Zona Arqueologica, Miscelanea en homenaje a Emiliano Aguirre (pp. 230–237), IV Arqueologia, 230, Museo Arquelogico Regional, Madrid. http://luth.obspm.fr/~uthier/nottale/arGNCaix.pdf.

  • Hall M. J. W. (2004) Incompleteness of trajectory-based interpretations of quantum mechanics. Journal of Physics A: Mathematical and General 37: 9549–9556

    Article  Google Scholar 

  • Hermann R. (1997) Numerical simulation of a quantum particle in a box. Journal of Physics A: Mathematical and General 30: 3967–3975

    Article  Google Scholar 

  • Hermann R., Schumacher G., Guyard R. (1998) Scale relativity and quantization of the solar system, Orbit quantization of the planets satellites. Astronomy and Astrophysics 335: 281–286

    Google Scholar 

  • Hinshaw G., et al. (2008). Five-Years WMAP observations, arXiv:08030732 [astro-ph].

  • Johansen A., Sornette D. (2001) Finite-time singularity in the dynamics of the world population, economic and financial indices. Physica A 294: 465–502

    Article  Google Scholar 

  • Jumarie G. (2001) Schrödinger equation for quantum fractal space-time of order n via the complex valued fractional Brownian motion. International Journal of Modern Physics A 16: 5061–5084

    Article  Google Scholar 

  • Jumarie G. (2006a) Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions: Further results. Computer and Mathematics 51: 1367–1376

    Google Scholar 

  • Jumarie G. (2006b) Fractionalization of the complex-valued Brownian motion of order n using Riemann-Liouville derivative. Applications to mathematical finance and stochastic mechanics. Chaos, Solitons & Fractals 28: 1285–1305

    Article  Google Scholar 

  • Jumarie G. (2007) The Minkowski’s spacetime is consistent with differential geometry of fractional order. Physical Letters A 363: 5–11

    Article  Google Scholar 

  • Kröger H. (2000) Fractal geometry in quantum mechanics, field theory and spin systems. Physics Reports 323: 81–181

    Article  Google Scholar 

  • Lang K. R. (1980) Astrophysical formulae. Springer, Berlin

    Google Scholar 

  • Laperashvili, L. V., & Ryzhikh, D. A. (2001). Phase transition in Gauge theories and the Planck Scale physics. arXiv: hep-th/0110127. Moscow: Institute for Theoretical and Experimental Physics.

  • Levy-Leblond J. M. (1976) One more derivation of the Lorentz transformation. American Journal of Physics 44: 271–277

    Article  Google Scholar 

  • Losa, G., Merlini, D., Nonnenmacher, T. & Weibel, E. (Eds.). (2002). Fractals in biology and medicine, Proceedings of Fractal 2000 3rd international symposium(Vol. III). Birkhäuser Verlag.

  • Mandelbrot B. (1982) The fractal geometry of nature. Freeman, San Francisco

    Google Scholar 

  • Martin, P., & Forriez, M. (2008). De l’utilité de la théorie de la relativité d’échelle de L. Nottale en géographie. 1: Recherche d’un modèle scalaire spatio-temporel. In Géopoint 2006. Demain la géographie. Permanences, dynamiques, mutations: Pourquoi? Comment? (pp. 297–303). Avignon, Groupe Dupont-UMR ESPACE du CNRS Editeurs.

  • McKeon D. G. C., Ord G. N. (1992) Time reversal in stochastic processes and the Dirac equation. Physical Review Letters 69: 3–4

    Article  Google Scholar 

  • Nelson E. (1966) Derivation of the Schrödinger equation from Newtonian mechanics. Physical Review 150: 1079–1085

    Article  Google Scholar 

  • Noble D. (2002) Modelling the heart: insights, failures and progress. BioEssays 24: 1155–1163

    Article  Google Scholar 

  • Noble D. (2006) The music of life: Biology beyond the genome. Oxford University Press, Oxford

    Google Scholar 

  • Nottale L. (1989) Fractals and the quantum theory of space-time. International Journal of Modern Physics A 4: 5047–5117

    Article  Google Scholar 

  • Nottale L. (1992). The theory of scale relativity. International Journal of Modern Physics A, 7, 4899–4936. http://luth.obspm.fr/~uthier/nottale/arIJMP2.pdf.

    Google Scholar 

  • Nottale L. (1993) Fractal space-time and microphysics: Towards a theory of scale relativity. World Scientific, Singapore

    Google Scholar 

  • Nottale, L. (1994a). Scale-relativity: First steps toward a field theory. In J. Diaz Alonso & M. Lorente Paramo (Eds.), Relativity in General, (Spanish Relativity Meeting 1993) (pp. 121–132). Paris: Editions Frontières. http://luth.obspm.fr/~uthier/nottale/arSalas.pdf.

  • Nottale, L. (1994b). New formulation of stochastic mechanics. Application to Chaos. In D. Benest & C. Froeschlé (Eds.), Chaos and diffusion in Hamiltonian systems, Proceedings of the 4th workshop in astronomy and astrophysics of Chamonix (France), 7–12 February 1994 (pp. 173–198). Editions Frontières. http://luth.obspm.fr/~uthier/nottale/arChamonix.pdf.

  • Nottale, L. (1996a). Scale relativity and fractal space-time: Application to quantum physics, cosmology and chaotic Systems. Chaos, Solitons & Fractals, 7, 877–938. http://luth.obspm.fr/~uthier/nottale/arRevFST.pdf.

  • Nottale, L. (1996b). Scale-relativity and quantization of extra-solar planetary systems. Astronomy and Astrophysics Letters, 315, L9–L12. http://luth.obspm.fr/~uthier/nottale/arA&A315.pdf.

  • Nottale, L. (1997a). Scale relativity and quantization of the universe. 1. Theoretical framework. Astronomy and Astrophysics, 327, 867–889. http://luth.obspm.fr/~uthier/nottale/arA&A327.pdf.

  • Nottale, L. (1997b). Scale relativity. In B. Dubrulle, F. Graner, & D. Sornette (Eds.), Scale invariance and beyond, Proceedings of Les Houches school (pp. 249–261). Berlin, New York: EDP Sciences, Les Ullis/Springer. http://luth.obspm.fr/~uthier/nottale/arhouche.pdf.

  • Nottale, L. (1998a). Scale relativity and quantization of planet obliquities. Chaos, Solitons & Fractals, 9, 1035–1041. http://luth.obspm.fr/~uthier/nottale/arobliq.pdf.

  • Nottale, L. (1998b). Scale relativity and quantization of the planetary system around the pulsar PSR B1257 + 12. Chaos, Solitons & Fractals, 9, 1043–1050. http://luth.obspm.fr/~uthier/nottale/arPSR.pdf.

  • Nottale L. (1998c) La relativité dans tous ses états. Hachette, Paris, p 319

    Google Scholar 

  • Nottale, L. (1999). The scale-relativity program. Chaos, Solitons & Fractals, 10, 459–468. http://luth.obspm.fr/~uthier/nottale/arSRprogr.pdf.

  • Nottale, L. (2001a). Scale relativity and non-differentiable fractal space-time. In B. G. Sidharth & M. V. Altaisky (Eds.), Frontiers of fundamental physics, Proceedings of Birla Science Center 4th international symposium, 11–13 Dec 2000 (pp. 65–79). Dordrecht: Kluwer. http://luth.obspm.fr/~uthier/nottale/arBirla00.pdf.

  • Nottale, L. (2001b). Relativité d’échelle et morphogenèse. Revue de Synthèse, T. 122, 4e S., No 1, January–March 2001, 93–116. http://luth.obspm.fr/~uthier/nottale/arSynthese2.pdf.

  • Nottale, L. (2001c). Scale relativity and gauge invariance. Chaos, Solitons & Fractals, 12, 1577–1583. http://luth.obspm.fr/~uthier/nottale/arCSF2001.pdf.

  • Nottale, L. (2002). Relativité, être et ne pas être. In Penser les limites. Ecrits en l’honneur d’André Green (pp. 157–165). Delachiaux et Niestlé. http://luth.obspm.fr/~uthier/nottale/arGreen.pdf.

  • Nottale, L. (2003). Scale-relativistic cosmology. Chaos, Solitons & Fractals, 16, 539–564. http://luth.obspm.fr/~uthier/nottale/arScRCosmo.pdf.

  • Nottale, L. (2004a). The theory of scale relativity: Non-differentiable geometry and fractal space-time. In D. M. Dubois (Ed.), “Computing anticipatory systems. CASYS’03—6th international conference” (Liège, Belgique, 11–16 Aug 2003), American Institute of Physics Conference Proceedings (Vol. 718, pp. 68–95). http://luth.obspm.fr/~uthier/nottale/arcasys03.pdf.

  • Nottale, L. (2004b). Scale relativity and dark potential. In F. Combes, D. Barret, T. Contini, F. Meynadier & L. Pagani (Eds.), Scientific highlights 2004, Proceedings of the Journées de la SF2A, Paris, 14–18 June 2004 (pp. 699–702). EDP Sciences. http://luth.obspm.fr/~uthier/nottale/arSF2A04.pdf

  • Nottale, L. (2006). Astrophysical applications of the theory of scale relativity. In B. G. Sidharth, F. Honsell, & A. de Angelis (Eds.), Proceedings of the 6th international symposium frontiers of fundamental physics, Udine, Italy, 26–29 Sep 2004 (pp. 107–114). Springer. http://luth.obspm.fr/~uthier/nottale/arUdine.pdf.

  • Nottale, L. (2007a). Scale relativity: A fractal matrix for organization in nature. Special issue (July 2007) on “Physics of emergence and organization”, Electronic Journal of Theoretical Physics, 4 No. 16(II), 187–274. http://luth.obspm.fr/~uthier/nottale/arEJTP.pdf.

  • Nottale, L. (2007b). Un nouveau paradigme pour la physique? Nouvelles perspectives. In P. Bourgeois & P. Grou (Eds.), Les Grands Défis Technologiques et Scientifiques au XXIè siècle (Chap. 9, pp. 121–137), Ellipse. http://luth.obspm.fr/~uthier/nottale/arNottaleNT.pdf.

  • Nottale, L. (2008). Origin of complex and quaternionic wavefunctions in quantum mechanics: the scale- relativistic view. In Proceedings of 7th international colloquium on clifford algebra, Toulouse, France, 19–29 May 2005, Advances in Applied Clifford Algebra (Vol. 18, pp. 917–944). http://luth.obspm.fr/~uthier/nottale/arICCA7.pdf.

  • Nottale, L. (2010) The theory of scale relativity (600 pp.). submitted for publication.

  • Nottale, L., & Auffray, C. (2008). Scale relativity theory and integrative systems biology: 2. Macroscopic quantum-type mechanics. Progress in Biophysics and Molecular Biology, 97, 115–157. http://luth.obspm.fr/~uthier/nottale/arPBMB08NA.pdf.

  • Nottale, L., & Célérier, M. N. (2007). Derivation of the postulates of quantum mechanics form the first principles of scale relativity. Journal of Physics A, 40, 14471–14498. http://arXiv.org/abs/0711.2418.

    Google Scholar 

  • Nottale, L., Célérier, M. N., & Lehner, T. (2006). Non-Abelian gauge field theory in scale relativity. Journal of Mathematical Physics, 47, 032303, 1–19. http://arXiv.org/abs/hep-th/0605280.

  • Nottale, L., Chaline, J. & Grou, P. (2000). Les arbres de l’évolution: Univers, Vie, Sociétés (379 pp.). Hachette, Paris.

  • Nottale, L., Chaline, J. & Grou, P. (2002). On the fractal structure of evolutionary trees. In G. Losa, D. Merlini, T. Nonnenmacher & E. Weibel (Eds.), Fractals in biology and medicine, Vol. III, Proceedings of Fractal 2000, 3rd international symposium (pp. 247–258). Birkhäuser Verlag. http://luth.obspm.fr/~uthier/nottale/arbiomed.pdf.

  • Nottale, L., Héliodore, F., & Dubois, J. (2007). Log-periodic evolution of earthquake rate, in preparation.

  • Nottale L., Schneider J. (1984) Fractals and non-standard analysis. Journal of Mathematical Physics 25: 1296–1300

    Article  Google Scholar 

  • Nottale, L., & Schumacher, G. (1998). Scale relativity, fractal space-time and gravitational structures. In M. M. Novak (Ed.), Fractals and beyond: Complexities in the sciences (pp. 149–160). World Scientific. http://luth.obspm.fr/~uthier/nottale/arFrac98.pdf.

  • Nottale, L., Schumacher, G., & Gay, J. (1997). Scale relativity and quantization of the solar system. Astronomy and Astrophysics, 322, 1018–1025. http://luth.obspm.fr/~uthier/nottale/arA&A322.pdf.

  • Nottale, L., Schumacher, G., & Lefèvre, E. T. (2000). Scale-relativity and quantization of exoplanet orbital semi-major axes. Astronomy and Astrophysics, 361, 379–387. http://luth.obspm.fr/~uthier/nottale/arA&A361.pdf.

  • Nottale, L., & Timar, P. (2006). De l’objet à l’espace psychique. Psychanalyse & Psychose, 6 195–212. http://luth.obspm.fr/~uthier/nottale/arpsy2.pdf.

  • Nottale, L., & Timar, P. (2008). Relativity of scales: application to an endo-perspective of temporal structures. In S. Vrobel, O. E. Rssler, T. Marks-Tarlow, (Eds.), Simultaneity: Temporal structures and observer perspectives (Chap. 14, pp. 229–242). Singapore: World Scientific. http://www.luth.obspm.fr/~uthier/nottale/arNotTim08.pdf.

  • Novak, M. (Ed.). (1998). Fractals and beyond: Complexities in the sciences, Proceedings of the Fractal’ 98 conference. World Scientific.

  • Ord G. N. (1983) Fractal space time: a geometric analog of relativistic quantum mechanics. Journal of Physics A: Mathematical and General 16: 1869–1884

    Article  Google Scholar 

  • Ord G. N. (1996) The Schrödinger and Dirac free particle equations without quantum mechanics. Annals of Physics 250: 51–62

    Article  Google Scholar 

  • Ord G. N., Galtieri J. A. (2002) The Feynman propagator from a single path. Physical Review Letters 89(250403): 1–4

    Google Scholar 

  • Pecker J. C., Schatzman E. (1959) Astrophysique générale. Masson, Paris

    Google Scholar 

  • Peterson M. A. (1979) Analogy between thermodynamics and mechanics. American Journal of Physics 47: 488–490

    Article  Google Scholar 

  • Pissondes J. C. (1999) Quadratic relativistic invariant and metric form in quantum mechanics. Journal of physics A: Mathematical and general 32: 2871–2885

    Article  Google Scholar 

  • Polchinski J. (1998) String theories. Cambridge University Press, Cambridge, UK

    Book  Google Scholar 

  • Queiros-Condé D. (2000) Principle of flux entropy conservation for species evolution, Series IIA - Earth and Planetary Science. Comptes Rendus de l’Acad謩e des Sciences Paris 330: 445–449

    Google Scholar 

  • Rovelli C., Smolin L. (1988) Knot theory and quantum gravity. Physical Review Letters 61: 1155–1158

    Article  Google Scholar 

  • Rovelli C., Smolin L. (1995) Spin networks and quantum gravity. Physical Review D 52: 5743–5759

    Article  Google Scholar 

  • Saar S. H., Brandenburg A. (1999) Time evolution of the magnetic activity cycle period, II. Results for an expanded stellar sample. The Astrophysical Journal 524: 295–310

    Article  Google Scholar 

  • Schneider, J. (2008). The Extrasolar Planets Encyclopaedia. http://exoplanet.eu/index.php.

  • Smith N. P, Nickerson D. P., Crampin E. J., Hunter P. J. (2004) Multiscale computational modelling of the heart, Acta numerica. Cambridge University Press, Cambridge, pp 371–431

    Google Scholar 

  • Sornette D., Johansen A., Bouchaud J. P. (1996) Stock market crashes, precursors and replicas. The Journal of Physics I France 6: 167–175

    Article  Google Scholar 

  • Sornette D., Sammis C. G. (1995) Complex critical exponents from renormalization group theory of Earthquakes: Implications for earthquake predictions. The Journal of Physics I France 5: 607–619

    Article  Google Scholar 

  • Sornette D. (1998) Discrete-scale invariance and complex dimensions. Physics Reports 297: 239–270

    Article  Google Scholar 

  • Spergel, D. N., et al. (2007). Wilkinson microwave anisotropy probe (WMAP) three year results: Implications for cosmology. Astrophysics Journal, 170(Suppl), 377–408.

    Google Scholar 

  • Tegmark, M., et al. (2006). Cosmological constraints from the SDSS luminous red galaxies. Physical Review, D, 74, 123507, 1–34 (arXiv: astro-ph/0308632).

    Google Scholar 

  • Timar, P. (2002). La transitionnalité de l’espace-temps psychique. http://www.spp.asso.fr/main/PsychanalyseCulture/SciencesDeLaComplexite/Items/3.htm.

  • Wang M. S., Liang W. K. (1993) Comment on “Repeated measurements in stochastic mechanics. Physical Review D 48: 1875–1877

    Google Scholar 

  • Yao, W.-M., et al. (Particle Data Group). (2006). Review of particle physics, Journal of Physics, G, 33, 1–1232.

    Google Scholar 

  • Zeldovich Ya. B., Ruzmaikin A. A., Sokoloff D. D. (1983) Magnetic fields in astrophysics. Gordon and Breach, New York

    Google Scholar 

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    Laurent Nottale

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Nottale, L. Scale Relativity and Fractal Space-Time: Theory and Applications. Found Sci 15, 101–152 (2010). https://doi.org/10.1007/s10699-010-9170-2

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