## The golden mean, scale free extension of real number system, fuzzy sets and $$1/f$$ spectrum in physics and biology.(English)Zbl 1032.26502

Summary: We show that the generic $$1/f$$ spectrum problem acquires a natural explanation in a class of scale-free solutions to the ordinary differential equations. We prove the existence and uniqueness of this class of solutions and show how this leads to a nonstandard, fuzzy extension of the ordinary framework of calculus, and hence, that of the classical dynamics and quantum mechanics. The exceptional role of the golden mean irrational number is also explained.

### MSC:

 26E50 Fuzzy real analysis 00A79 Physics 26E35 Nonstandard analysis 34A99 General theory for ordinary differential equations 81P05 General and philosophical questions in quantum theory 92B99 Mathematical biology in general
Full Text:

### References:

 [1] Press, W.H., Flicker noises in astronomy and elsewhere, Comments astrophys., 7, 103-119, (1978) [2] Bak, P.; Tang, C.; Wiesenfeld, W., Self-organized criticality, Phys. rev. A, 38, 364-374, (1988) · Zbl 1230.37103 [3] Datta DP. A new class of scale free solutions to linear ordinary differential equations and the universality of the golden mean $$5−12=0.618033…$$. Chaos, Solitons, & Fractals, in press [4] Robinson, A., Nonstandard analysis, (1966), North-Holland Amsterdam [5] Wolf, M., 1/f noise in the distribution of prime numbers, Physcia A, 241, 493-499, (1997) [6] Planat, M., 1/f noise, the measurement of time and number theory, Fluctuat. noise lett., 1, R65-R74, (2001) [7] Selvam, A.M., Universal quantification for deterministic chaos, Appl. math. model., 17, 642-649, (1993) · Zbl 0795.58033 [8] El Naschie, M.S., Fisher’s scaling and dualities at high energy in $$E\^{}\{(∞)\}$$ spaces, Chaos, solitons, & fractals, 12, 1557-1561, (2001) · Zbl 1021.81050 [9] El Naschie, M.S., On ’t Hooft dimensional regularisation in $$E\^{}\{(∞)\}$$ spaces, Chaos, solitons, & fractals, 12, 851-858, (2001) · Zbl 1012.83008 [10] El Naschie, M.S., On the irreducibility of spatial ambiguity in quantum physics, Chaos, solitons, & fractals, 6, 913-919, (1998) · Zbl 0938.81007 [11] El Naschie, M.S., Remarks on superstrings, fractal gravity, nagasawa’s diffusion and Cantorian spacetime, Chaos, solitons, & fractals, 8, 1873-1886, (1997) · Zbl 0934.83049 [12] El Naschie, M.S., Penrose tiling, semi-conduction and Cantorian 1/fα spectra in four and five dimensions, Chaos, solitons, & fractals, 3, 489-491, (1993) · Zbl 0795.58046 [13] Datta, D.P., Duality and scaling in quantum mechanics, Phys. lett. A, 233, 274-280, (1997) · Zbl 1044.81509 [14] Simmons, G.F., Differential equations with applications and historical notes, (1972), McGraw Hill New York · Zbl 0231.34001 [15] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic, (2000), Prentice-Hall of India New Delhi [16] Ott, E., Chaos in dynamical systems, (1993), Cambridge University Press Cambridge · Zbl 0792.58014 [17] Guilini, D.; Joos, E.; Kiefer, C.; Kupsch, J.; Stamatescu, I.-O.; Zeh, H.D., Decoherence and the appearance of a classical world in quantum theory, (1996), Springer Berlin · Zbl 0855.00003 [18] El Naschie, M.S., Chaos, solitons, & fractals, 14, 7, 1121, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.