During the last several decades, a great variety of irregular timedependent phenomena and spatial morphologies have been shown to possess stochastic scale-invariance. This led to the development of models based on random fractal processes and, in general, (multi-dimensional) random fractal fields. In contrast to the ideal fractals, commonly assumed to be “scale-free” (reflected for example in the assumption of a simple power-law type correlation functions), the real scale-invariant hierarchies have a finite extend, limited by both a smallest and a largest scales.
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References
Chen, Y.T., Alan H.Cook, MetherellA.J.F.: An experimental test of the inverse square law of gravitation at range of 0.1 m. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 394, 47-68 (1984).
Long, J.C., Chan, H.W. and Price, J.C.: Experimental status of gravitationalstrength forces in the sub-centimeter regime. Nucl. Phys. B, 539, 23-34 (1999); available at arXiv:hep-ph/9805217.
Berry, M.V.: Diffractals. J. Phys. A: Math. Gen. 12, 781-797 (1979).
Panchev, S.: Random Functions and Turbulence. Pergamon Press, Oxford (1971); Monin, A.S. and Yaglom, A.M.: Statistical Fluid Mechanics. MIT Press, Boston (1971).
Priestley, M.A.: Spectral Analysis and Time Series. Academic Press, London (1981).
Yordanov, O.I. and Atanasov, I.: Self-affine random surfaces. European Phys. J. B, 29, 211-215 (2002).
Orey, S.: Gaussian sample functions and the Hausdorff dimension of level crossings. Z. Wahrsch’theorie verw. Geb. 15, 249-256 (1970).
Falconer, K.J.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985).
Mandelbrot, B.B.: Fractals, W.H. Freeman and Company, San Francisco (1979); Mandelbrot, B.B.: The Fractal Geometry of Nature, W.H. Freeman and Company, San Francisco (1982).
Sayles, R.S. and Thomas, T.R.: Surface topography as a nonstationary random process, Nature (London) 271, 431-434 (1978); Berry M.V. and Hannay, J.R.: Nature (London) 273, 573 (1978).
Yordanov, O.I. and Nickolaev, N.I.: Self-affinity of time series with a power-law power spectrum. Phys. Rev. E, Rapid Commun., 49, R2517-R2520 (1994).
Yordanov, O.I.: Properties of a special function pertaining to approximately scale-invariant random fields. In: Dobrev V.K. (ed) Proc. 4th Int. Symp. Quantum Theory and Symmetries IV, Heron Press Int., Sofia, Bulgaria, p. 809. (2005).
Greis, N.P. and Greenside, H.S.: Implication of a power-law power-spectrum for self-affinity Phys. Rev. A, 44, 2324-2334 (1991).
Yordanov, O.I. and Ivanova, K.: Description of surface roughness as an approximate self-affine random structure, Surf. Sci., 331-333, 1043-1049 (1995).
Bender, C. M. and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York (1978).
Holzwarth, M., Wißing M., Simeonova D.S., Tzanev, S., Snowdon, K.J. and Yordanov, O.I.: Preparation of atomically amooth surfaces via sputtering under glancing incidence conditions, Surf. Sci., 331-333, 1093-1098 (1995).
Yordanov, O.I. and Guissard, A.: Approximate self-affine model for cultivated soil roughness, Physica A, 238/1-4, 49-65 (1997).
Yordanov, O.I. and Nickolaev, N.I.: Approximate, saturated and blurred selfaffinity of random processes with finite domain power-law power spectrum, Physica D, 101, 116-130 (1997).
Dimitrova, E.S. and Yordanov, O.I.: Statistics of some low-dimensional chaotic flows, Int. J. Chaos Bifur., 11, 2675-2682 (2001).
Vulkova, L.A., Atanasov, I.S. and Yordanov, O.I.: Approximate Self-Affine Characterization of Two-Point Rough Surface Statistics Simulated by Eden Algorithm, Vacuum, 58, 158-165 (2000).
Vulkova, L.A., Atanasov, I.S. and Yordanov, O.I.: Comparison of Numerical Algorithms for Simulation of Molecular Beam epitaxy, Vacuum, 69, 69-73 (2003).
Press W.H., Teukolsky, S.A., Vetterling W.T and Flannery, B.P.: Numerical Recipes in C, Second edition. Cambridge University Press, Cambridge (1992). Book and codes available on-line on http://www.nr.com.
See Wikipeadia: http://en.wikipedia.org/wiki/Spectraldensity.
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Yordanov, O.I. (2008). Approximate Scale-Invariant Random Fields: Review and Current Developments. In: Konaté, D. (eds) Mathematical Modeling, Simulation, Visualization and e-Learning. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74339-2_16
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