Abstract

We review some of the ways in which the fractal concept has found application in wave-propagation contexts. The scaling properties of fractals in both geometrical and statistical situations are reviewed and the relation to inverse power laws discussed. The relationship among the self-similar scaling properties of fractals, Lévy distributions, and renormalized group theory is explored to provide a simple picture of wave propagation through multiscale media. Finally, the notion of using a wavelet transform in the processing of fractal time series is considered.

© 1990 Optical Society of America

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  60. M. Marians, “Computed scintillation spectra for strong turbulence,” Radio Sci. 10, 115–119 (1975).
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  61. E. Jakeman, J. G. McWhirter, “Correlation function dependence of scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
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  62. G. Bourgois, “About the ergodicity hypothesis in random propagation studies,” Astron. Astrophys. 102, 218–222 (1981).
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  65. J. Morlet, G. Arens, E. Fourgeau, D. Giord, “Wave propagation and sampling theory—part I: complex signal and scattering in multilayered media,” Geophysica 47, 203–221 (1982);“part II: sampling theory and complex waves,” Geophysica 47, 222–236 (1982).
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    [CrossRef]
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  73. B. B. Mandelbrot, “Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of carrier,” J. Fluid Mech. 62, 331–358 (1974).
    [CrossRef]
  74. K. Sreenivasan, C. Meneveau, “The fractal facets of turbulence,” J. Fluid Mech. 173, 357–386 (1986).
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    [CrossRef]
  80. B. D. Hughes, M. F. Shlesinger, E. W. Montroll, “Random walks with self-similar clusters,” Proc. Nat. Acad. Sci. USA 78, 3287–3291 (1981)
    [CrossRef] [PubMed]

1989 (3)

R. E. Glazman, P. B. Weichman, “Statistical geometry of a small patch in a developed sea,” J. Geophys. Res. 94, 4998–5010 (1989).
[CrossRef]

B. J. West, M. F. Shlesinger, “On the ubiquity of 1/f-noise,” Int. J. Mod. Phys. B 3s, 795–820 (1989).
[CrossRef]

F. Argoul, A. Arneodo, G. Grasseau, Y. Gagne, E. J. Hopfinger, U. Frisch, “Wavelet analysis of turbulence reveals the multifractal nature of the Richardson cascade,” Nature (London) 338, 51–53 (1989).
[CrossRef]

1988 (3)

A. Arneodo, G. Grasseau, M. Holschneider, “Wavelet transform of multifractals,” Phys. Rev. Lett. 61, 2281–2284 (1988).
[CrossRef] [PubMed]

P. L. Turcotte, “Fractals in fluid mechanics,” Annu. Rev. Fluid Mech. 20, 5–16 (1988).
[CrossRef]

M. Holschneider, “On the wavelet transformation of fractal objects,” J. Stat. Phys. 50, 963 (1988).
[CrossRef]

1987 (3)

B. J. West, A. L. Goldberger, “Physiology in fractal dimensions,” Am. Sci. 75, 354–365 (1987).

A. L. Goldberger, B. J. West, “Fractals: a contemporary mathematical concept with applications to physiology and medicine,” Yale J. Biol. Med. 60, 104–119 (1987).

M. F. Shlesinger, B. J. West, J. Klafter, “Lévy dynamics of enhanced diffusion; application to turbulence,” Phys. Rev. Lett. 58, 1100–1103 (1987).
[CrossRef] [PubMed]

1986 (6)

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Prococcia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141–1150 (1986).
[CrossRef] [PubMed]

B. J. West, V. Bhargava, A. L. Goldberger, “Beyond the principle of similitude: renormalization in the bronchial tree,” J. Appl. Physiol. 60, 1089–1097 (1986).
[PubMed]

K. Sreenivasan, C. Meneveau, “The fractal facets of turbulence,” J. Fluid Mech. 173, 357–386 (1986).
[CrossRef]

R. D. Mauldin, S. C. Williams, “On the Hausdorff dimension of some graphs,” Trans. Am. Math. Soc. 298, 793–803 (1986).
[CrossRef]

E. E. Underwood, K. Banerji, “Fractals in fractography,” Mater. Sci. Eng. 80, 1–14 (1986).
[CrossRef]

R. S. Wu, “Heterogeneity spectrum, wave scattering response of a fractal random medium and the rupture processes in the medium,” J. Wave Mater. Int. 1, 79–96 (1986).

1985 (3)

A. L. Goldberger, V. Bhargava, B. J. West, A. J. Mandel, “On a mechanism of cardiac electrical stability: the fractal hypothesis,” Biophys. J. 48, 525–528 (1985).
[CrossRef] [PubMed]

M. Ausloos, D. H. Berman, “A multivariate Weierstrass–Mandelbrot function,” Proc. R. Soc. London Ser. A 400, 331–350 (1985).
[CrossRef]

C. A. Aviles, C. H. Scholz, “Fractal analysis of characteristic fault segments of the San Andres fault system,” Eos 66, 314–321 (1985).

1984 (3)

J. E. Gubernatis, E. Domany, “Effects of microstructure on the speed and attenuation of elastic waves in porous materials,” Wave Motion 6, 579–589 (1984).
[CrossRef]

B. J. West, M. F. Shlesinger, “The fractal interpretation of the weak scattering of elastic waves,” J. Stat. Phys. 36, 779–786 (1984).
[CrossRef]

H. Takayasu, “Stable distribution and Levy process in fractal turbulence,” Prog. Theor. Phys. 72, 471–479 (1984).
[CrossRef]

1983 (1)

S. Kitaigorodskii, “On the theory of the equilibrium range in the spectrum of wind-generated gravity waves,” J. Phys. Oceanogr. 13, 816–827 (1983).
[CrossRef]

1982 (2)

E. W. Montroll, M. F. Shlesinger, “On 1/f-noise and distributions with long tails,” Proc. Natl. Acad. Sci. USA 79, 3380–3387 (1982).
[CrossRef]

J. Morlet, G. Arens, E. Fourgeau, D. Giord, “Wave propagation and sampling theory—part I: complex signal and scattering in multilayered media,” Geophysica 47, 203–221 (1982);“part II: sampling theory and complex waves,” Geophysica 47, 222–236 (1982).
[CrossRef]

1981 (4)

G. Bourgois, “About the ergodicity hypothesis in random propagation studies,” Astron. Astrophys. 102, 218–222 (1981).

B. D. Hughes, M. F. Shlesinger, E. W. Montroll, “Random walks with self-similar clusters,” Proc. Nat. Acad. Sci. USA 78, 3287–3291 (1981)
[CrossRef] [PubMed]

G. I. Barenblatt, I. A. Leykin, “On the self-similar spectra of wind waves in the high frequency range,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 17, 35–41 (1981).

M. F. Shlesinger, B. D. Hughes, “Analogs of renormalization group transformations in random processes,” Physica 109A, 597–608 (1981).

1980 (2)

M. V. Berry, Z. V. Lewis, “On the Weierstrauss–Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[CrossRef]

R. L. Fante, “Some physical insights into beam propagation in strong turbulence,” Radio Sci. 15, 757–762 (1980).
[CrossRef]

1979 (4)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

C. L. Rino, “A power law phase screen model for ionospheric scintillations 1. Weak scatter,” Radio Sci. 14, 1135–1146 (1979);“2. Strong scatter,” Radio Sci. 14, 1147–1155 (1979).
[CrossRef]

K. G. Wilson, “Problems in physics with many scales of length,” Sci. Am. 241(2), 158–179 (1979).
[CrossRef]

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

1978 (2)

H. J. Morris, L. Kadanoff, “Teaching the renormalization group,” Am. J. Phys. 46, 652–657 (1978).
[CrossRef]

U. Frisch, P. Sulem, M. Nelkin, “A simple dynamical model of intermittent fully developed turbulence,” J. Fluid Mech. 87, 719–736 (1978).
[CrossRef]

1977 (3)

E. Jakeman, J. G. McWhirter, “Correlation function dependence of scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

R. K. Crane, “Ionospheric scintillation,” Proc. IEEE 65, 180–199 (1977).
[CrossRef]

R. Uimeki, C. H. Lin, K. C. Yeh, “Multifrequency spectra of ionospheric amplitude scintillations,” J. Geophys. Res. 82, 2752–2760 (1977).
[CrossRef]

1975 (3)

M. Nauenberg, “Scaling representation for critical phenomena,” J. Phys. A Gen. Phys. 8, 925–928 (1975).
[CrossRef]

G. Jona-Lasinio, “The renormalization group: a probabilistic view,” Nuovo Cimento 26B, 99–119 (1975).

M. Marians, “Computed scintillation spectra for strong turbulence,” Radio Sci. 10, 115–119 (1975).
[CrossRef]

1974 (2)

B. B. Mandelbrot, “Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of carrier,” J. Fluid Mech. 62, 331–358 (1974).
[CrossRef]

B. B. Mandelbrot, “Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier,” J. Fluid Mech. 62, 331–358 (1974).
[CrossRef]

1971 (1)

D. N. Matheson, L. T. Little, “Radio scintillation due to plasma irregularities with power law spectra: the interplanetary medium,” Planet. Space Sci. 19, 1615–1624 (1971).
[CrossRef]

1966 (2)

E. A. Novikov, Soviet Phys. Dokl. 11, 497–501 (1966).

V. E. Zakharov, N. N. Filonenki, “The energy spectrum for stochastic oscillation of a fluid’s surface,” Dokl. Akad. Nauk SSSR 170, 1292–1295 (1966).

1964 (1)

A. Hewish, P. F. Scott, D. Wills, “Interplanetary scintillation of small diameter radio sources,” Nature (London) 203, 1214–1217 (1964).
[CrossRef]

1961 (1)

L. F. Richardson, “The problem of continuity: an appendix of statistics of deadly quarrels,” Gen. Syst. Yearbook 6, 139–187 (1961).

1955 (1)

W. J. Pierson, “Wind generated gravity waves,” Adv. Geophys. 2, 93–178 (1955).
[CrossRef]

1952 (1)

M. Lax, “Multiple scattering of waves: II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

1950 (1)

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionosphere problems,” Philos. Trans. R. Soc. London A 242, 579–591 (1950).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (London) 93, 429–457 (1946).

1945 (1)

L. L. Foldy, “The multiple scattering of waves: I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

1941 (1)

A. N. KolmogorovC. R. Dokl. Acad. Sci. URSS 30, 301 (1941).

1926 (1)

L. F. Richardson, “Atmospheric diffusion shown on a distance–neighbor graph,” Proc. R. Soc. London Ser. A 110, 709–725 (1926).
[CrossRef]

Albowitz, M. J.

M. J. Albowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981).

Arens, G.

J. Morlet, G. Arens, E. Fourgeau, D. Giord, “Wave propagation and sampling theory—part I: complex signal and scattering in multilayered media,” Geophysica 47, 203–221 (1982);“part II: sampling theory and complex waves,” Geophysica 47, 222–236 (1982).
[CrossRef]

Argoul, F.

F. Argoul, A. Arneodo, G. Grasseau, Y. Gagne, E. J. Hopfinger, U. Frisch, “Wavelet analysis of turbulence reveals the multifractal nature of the Richardson cascade,” Nature (London) 338, 51–53 (1989).
[CrossRef]

A. Arneodo, F. Argoul, J. Elezgaray, G. Gresseau, Centre de Recherche Paul Pascal, Domaine Universitaire, 33405 Talence Cedex, France, “Wavelet transform analysis of fractals: application to nonequilibrium phase transitions,” (preprint).

Arneodo, A.

F. Argoul, A. Arneodo, G. Grasseau, Y. Gagne, E. J. Hopfinger, U. Frisch, “Wavelet analysis of turbulence reveals the multifractal nature of the Richardson cascade,” Nature (London) 338, 51–53 (1989).
[CrossRef]

A. Arneodo, G. Grasseau, M. Holschneider, “Wavelet transform of multifractals,” Phys. Rev. Lett. 61, 2281–2284 (1988).
[CrossRef] [PubMed]

A. Arneodo, F. Argoul, J. Elezgaray, G. Gresseau, Centre de Recherche Paul Pascal, Domaine Universitaire, 33405 Talence Cedex, France, “Wavelet transform analysis of fractals: application to nonequilibrium phase transitions,” (preprint).

Ausloos, M.

M. Ausloos, D. H. Berman, “A multivariate Weierstrass–Mandelbrot function,” Proc. R. Soc. London Ser. A 400, 331–350 (1985).
[CrossRef]

Aviles, C. A.

C. A. Aviles, C. H. Scholz, “Fractal analysis of characteristic fault segments of the San Andres fault system,” Eos 66, 314–321 (1985).

Banerji, K.

E. E. Underwood, K. Banerji, “Fractals in fractography,” Mater. Sci. Eng. 80, 1–14 (1986).
[CrossRef]

Barenblatt, G. I.

G. I. Barenblatt, I. A. Leykin, “On the self-similar spectra of wind waves in the high frequency range,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 17, 35–41 (1981).

Berman, D. H.

M. Ausloos, D. H. Berman, “A multivariate Weierstrass–Mandelbrot function,” Proc. R. Soc. London Ser. A 400, 331–350 (1985).
[CrossRef]

Berry, M. V.

M. V. Berry, Z. V. Lewis, “On the Weierstrauss–Mandelbrot fractal function,” Proc. R. Soc. London Ser. A 370, 459–484 (1980).
[CrossRef]

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

Bhargava, V.

B. J. West, V. Bhargava, A. L. Goldberger, “Beyond the principle of similitude: renormalization in the bronchial tree,” J. Appl. Physiol. 60, 1089–1097 (1986).
[PubMed]

A. L. Goldberger, V. Bhargava, B. J. West, A. J. Mandel, “On a mechanism of cardiac electrical stability: the fractal hypothesis,” Biophys. J. 48, 525–528 (1985).
[CrossRef] [PubMed]

Booker, H. G.

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionosphere problems,” Philos. Trans. R. Soc. London A 242, 579–591 (1950).
[CrossRef]

Bourgois, G.

G. Bourgois, “About the ergodicity hypothesis in random propagation studies,” Astron. Astrophys. 102, 218–222 (1981).

Crane, R. K.

R. K. Crane, “Ionospheric scintillation,” Proc. IEEE 65, 180–199 (1977).
[CrossRef]

Dashen, R.

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
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T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Prococcia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141–1150 (1986).
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B. J. West, A. L. Goldberger, “Physiology in fractal dimensions,” Am. Sci. 75, 354–365 (1987).

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C. R. Dokl. Acad. Sci. URSS (1)

A. N. KolmogorovC. R. Dokl. Acad. Sci. URSS 30, 301 (1941).

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V. E. Zakharov, N. N. Filonenki, “The energy spectrum for stochastic oscillation of a fluid’s surface,” Dokl. Akad. Nauk SSSR 170, 1292–1295 (1966).

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C. A. Aviles, C. H. Scholz, “Fractal analysis of characteristic fault segments of the San Andres fault system,” Eos 66, 314–321 (1985).

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L. F. Richardson, “The problem of continuity: an appendix of statistics of deadly quarrels,” Gen. Syst. Yearbook 6, 139–187 (1961).

Geophysica (1)

J. Morlet, G. Arens, E. Fourgeau, D. Giord, “Wave propagation and sampling theory—part I: complex signal and scattering in multilayered media,” Geophysica 47, 203–221 (1982);“part II: sampling theory and complex waves,” Geophysica 47, 222–236 (1982).
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B. J. West, M. F. Shlesinger, “On the ubiquity of 1/f-noise,” Int. J. Mod. Phys. B 3s, 795–820 (1989).
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J. Appl. Physiol. (1)

B. J. West, V. Bhargava, A. L. Goldberger, “Beyond the principle of similitude: renormalization in the bronchial tree,” J. Appl. Physiol. 60, 1089–1097 (1986).
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B. B. Mandelbrot, “Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier,” J. Fluid Mech. 62, 331–358 (1974).
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B. B. Mandelbrot, “Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of carrier,” J. Fluid Mech. 62, 331–358 (1974).
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K. Sreenivasan, C. Meneveau, “The fractal facets of turbulence,” J. Fluid Mech. 173, 357–386 (1986).
[CrossRef]

U. Frisch, P. Sulem, M. Nelkin, “A simple dynamical model of intermittent fully developed turbulence,” J. Fluid Mech. 87, 719–736 (1978).
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Figures (11)

Fig. 1
Fig. 1

Fractal plots of various coastlines in which the apparent length L(η) is graphed versus the measuring unit η26: plotted as log10 [total length (km)] versus log10 length of side (km).

Fig. 2
Fig. 2

On a line segment of unit length a kink is formed, giving rise to four line segments, each of length 1/3. The total length of this wave is 4/3. On each of these line segments a kink is formed, giving rise to 16 line segments each of length 1/9. The total length of this curve is (4/3)2. This process is continued through n = 5.

Fig. 3
Fig. 3

Experimental fractal plots for Koch quadric island (top curve) and a Koch triadic island (bottom curve). The apparent length L(η) is graphed versus the measuring unit η.27(Taken from Ref. 27; used with permission.)

Fig. 4
Fig. 4

The extended Weierstrass function [Eq. (2.32) with random phases. L is the level of an artificial floor, which as it is lowered reveals more of the surface (M = 1, D = 2.5, b = 1.5). (Taken from Ref. 40; used with permission.)

Fig. 5
Fig. 5

Surfaces for M = 2 with random phases (D = 2.5, b = 1.2). The upper surface is the sum of the two lower surfaces. (Taken from Ref.40; used with permission.)

Fig. 6
Fig. 6

Four magnifications of the surface for D = 2.05, showing self-similarity (M = 8, b = 1.2). The upper-right-hand surface is the fivefold magnification of a section of the upper-left-hand surface. Similarly, the lower-left-hand surface is a fivefold magnification of a piece of the upper-right-hand surface, and the lower-righthand surface is a fivefold magnification of the lower-left-hand surface. The vertical extent is magnified by 53−D. (Taken from Ref. 40; used with permission.)

Fig. 7
Fig. 7

Power-spectral density for the differential phase fluctuations (Sϕ). The experimental and theoretical curves are taken from Ref. 7. The dashed curve has a κ−1 slope, whereas the high frequency asymptotes all have a κ−3 slope.

Fig. 8
Fig. 8

Wavelet transform [sign(TgX)|Tg(a, b)X|1/2] of the triadic Cantor set: (a) uniform measure p1 = p2 = 1/2; (b) two distinct measures p1 = 3/4, p2 = 1/2 [note that the scales in (a) and (b) are different]. A black ( T g X < T ) and a white ( T g X T ) coding of Tg(a, b)X is also shown: (c) uniform Cantor set, (d) nonuniform Cantor set. This coding is obtained by defining on each a = constant line a threshold T = δ max T g ( a , b ) X ( δ > 0 ). In the limit a → 0+, the white regions point to the singularities that are located at the points of the triadic Cantor set.23 The function g(x) is defined by Eq. (4.7) and n = 2 in relation (4.23). (Taken from Ref. 68; used with permission).

Fig. 9
Fig. 9

(a) Renyi dimension Dq graphed versus q [relations (4.28) and (4.29)] for the uniform (long-dashed line) and the nonuniform (solid curves) triadic Cantor set. (b) Spectrum of singularities f(α) is plotted versus the strength α for the uniform (dashed line) and the nonuniform (solid curve) triadic Cantor set. (Taken from Ref. 68; used with permission.)

Fig. 10
Fig. 10

(a) Logarithm of the wavelet transform graphed as a function of ln(a) for a uniform Cantor set, using an arbitrary scale. (b) Logarithm of the wavelet transform graphed as a function of ln(a) for the nonuniform Cantor set, where b* corresponds to the kneading sequence RRRRRRRR LLL… LL…. Here the wavelet function g(x) is defined by the second derivative of the Gaussian function [Eq. (4.7)] and n = 2 in relation (4.23). (Taken from Ref. 68; used with permission.)

Fig. 11
Fig. 11

Wavelet analysis of fully developed turbulence from wind-tunnel data. The coordinates and intensity are the same as in Fig. 8. The uppermost graphs show the signals being analyzed. (a) 852-m sample depicting the large scale (28l0 = l0/10) analyzed using a French top hat wavelet. (b) A Mexican hat wavelet for 20× magnification of the central portion is indicated by the arrow in the uppermost graph of (a). (c) Analogous to (b), again with the Mexican hat wavelet. The successive forkings reveal the fractal nature of the Richardson cascade. (Taken from Ref. 71; used with permission.)

Equations (208)

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α ( ω ) ω s ,
α ( ω ) = B ω μ , 1 μ 4
L ( η ) = L 0 η 1 d .
ln L ( η ) = ln L 0 + ( 1 d ) ln η .
L ( η ) = ( 4 / 3 ) n ,
η = 1 / 3 n .
n = ln η / ln 3 ,
L ( η ) = ( 4 / 3 ) ( ln η / ln 3 ) = exp [ ln η ln 3 ( ln 4 ln 3 ) ] = η 1 d .
d = ln 4 / ln 3 1.2628
N ( η ) = 1 / η d
L ( η ) = ¾ L ( η / 3 ) .
L ( η ) = A ( η ) / η α ,
α = ln 4 / ln 3 1 = d 1 ,
A ( η ) = A ( η / 3 ) ,
M ( r ) = M 0 r d .
M ( r ) = M ( r / b n ) / a n = M 0 r d ( a n / b n d ) ,
d = ln a / ln b .
W ( t ) = n = 0 1 a n cos ( b n ω 0 t ) ,
W ( t ) = 1 a W ( b t ) + cos ω 0 t .
W ( t ) 1 a W ( b t ) .
W ( t ) = A ( t ) t α ,
α = ln a / ln b ,
X ( t ) = n = 1 a n ( 1 cos b n ω 0 t ) ,
d = 2 ln a / ln b ,
F ( K ) = 1 l E F ( K ) + G ( K ) ,
F ( u ) = 1 l E F ( λ u ) + G ( u ) ,
F ( u ) = 1 l N E F ( λ N u ) + n = 0 N 1 1 l n E G ( λ n u ) .
lim N 1 l N E F ( λ N u ) = 0 ,
F ( u ) = n = 0 1 l n E G ( λ n u ) .
F sing ( u ) = 1 l E F sing ( λ u ) ,
F sing ( u ) = A ( u ) ( u ) E / μ ,
μ = ln λ / ln l
A ( u ) = A ( λ u ) = n A n exp [ 2 π in ( ln u / ln λ ) ] ,
X ( t ) = n = 1 a n [ 1 exp ( i b n ω 0 t ) ] e i ϕ n ,
Δ X ( t , τ ) = X ( t + τ ) X ( t ) = n = b n ( 2 d ) { exp ( i b n t ) exp [ i b n ( t + τ ) ] } e i ϕ n ,
C ( τ ) = | Δ X ( t , τ ) | 2 = n = b n ( 4 2 d ) 2 [ 1 cos ( b n τ ) ] ,
C ( b τ ) = b 2 ( 2 d ) C ( τ ) ,
C ( ρ ) = | Δ X ( r , ρ ) | 2 ,
C ( b ρ ) = b 2 ( 3 d ) C ( ρ ) ,
X ( r ) = ( ln b M ) 1 / 2 m = 1 M A m × n = ( k 0 b n ) ( d 3 ) { 1 exp [ i b n k 0 r cos ( θ α m ) ] } e i ϕ m n .
X ( r , t ) = ( ln b M ) 1 / 2 m = 1 M A m n = b n ( 3 d ) × cos [ k n ( x cos θ m + y sin θ m ) + Φ m n ] .
Φ m n = ϕ m n ω n t ,
ω n g k n = g k 0 b n / 2 ,
X ( r , t ) = [ ln b M k 0 2 ( 3 d ) ] 1 / 2 m = 1 M A m × n = b n ( 3 d ) cos [ k 0 b n ( x cos θ m + y sin θ m ) + ϕ m n ω 0 b n / 2 t ] ,
Ψ ( k ) = g 1 / 2 B u * G ( θ ) k β ,
d 2 k Ψ ( k ) = ζ 2 ,
ζ ( r , t ) = 0 π π cos [ k ( x cos θ + y sin θ ) ω n t + ϕ ( k , θ ) ] × [ Ψ ( k ) k d k d θ ] 1 / 2 ,
[ Ψ ( k ) k d k d θ ] 1 / 2 [ 2 π g 1 / 2 u * B G ( θ m ) k 0 β 2 M ] 1 / 2 b n 2 ( 2 β ) .
ζ ( r , t ) = ( 2 π g 1 / 2 u * B ln b M k 0 β 2 ) 1 / 2 m = 1 M [ G ( θ m ) ] 1 / 2 × n = 0 b ( n / 2 ) ( 2 β ) cos [ k 0 b n ( x cos θ m + y sin θ m ) ω 0 b n / 2 t + ϕ m n ] .
A m = [ 2 π g 1 / 2 u * B G ( θ m ) ] 1 / 2 ,
d = 4 β / 2 .
Ψ ( k ) = G ( θ ) f ( x ) g 2 μ U 4 μ k 4 2 μ ,
β = 4 2 μ
α = 2 + μ .
C ( τ ) = b 2 ( 2 d ) C ( b τ ) .
C ( τ ) = A ( τ ) τ α ,
α = 2 ( 2 d ) ,
S ( ω ) C ( ω ) 1 ω α + 1 ,
S ( f , τ ) = 4 Re 0 C ( t , τ ) exp ( 2 π ift ) d t = 4 τ 1 + ( 2 π f τ ) 2 .
S ( f ) = 0 4 τ ρ ( τ ) d τ 1 + ( 2 π f τ ) 2 .
S ( ω , σ ) = 4 τ ¯ ( 2 π σ 2 ) 1 / 2 0 exp { [ ln ( τ / τ ¯ ) ] 2 / 2 σ 2 } 1 + ω 2 τ 2 / τ ¯ 2 d ( τ τ ¯ ) ,
ω = 2 π f τ ¯ .
S ( ω , σ ) 1 ( 2 π σ 2 ) 1 / 2 0 d z 1 + ω 2 z 2 ( π 8 σ 2 ) 1 / 2 1 ω ,
c ( ξ ) = ( 1 p ) [ C ( ξ ) + p N C ( ξ / N ) + p 2 N 2 C ( ξ / N 2 ) + ] ,
c ( ξ ) = p N c ( ξ / N ) + ( 1 p ) C ( ξ ) ,
c ( t ) = A ( t ) t α 1 ,
S ( f ) 1 / f α .
{ 2 + k 2 [ 1 + n 1 ( x ) ] 2 } υ ( x ) = 0 ,
υ ( x , y , z + Δ z ) = υ 0 ( x , y , z ) exp [ i Ψ ( x ) ] .
Ψ ( x ) = k z Δ z z n 1 ( x , y , z ) d z .
Ψ ( x , t ) = j = 1 N Ψ j ( x , t ) .
Ψ ( N ) = 1 B N j = 1 N Ψ j A N ,
C Ψ ( 2 ) = C 1 Ψ 1 + C 2 Ψ 2 ,
P ( κ ) = d ψ exp ( i κ ψ ) P ( ψ )
P ( ψ 1 , ψ 2 ) = P 1 ( ψ 1 ) P 2 ( ψ 2 ) .
P ( C κ ) = exp ( i C κ Ψ ( 2 ) ) = d ψ 1 d ψ 2 exp [ i κ ( C 1 ψ 1 + C 2 ψ 2 ) P 1 ( ψ 1 ) P 2 ( ψ 2 ) ,
P ( C κ ) = P 1 ( C 1 κ ) P 2 ( C 2 κ ) .
P ( C κ ) = P ( C 1 κ ) P ( C 2 κ ) ,
q ( κ ) = log P ( κ ) .
q ( C κ ) = q ( C 1 κ ) + q ( C 2 κ ) .
m q ( κ ) = q ( a m κ )
q ( κ / a m ) = 1 m q ( κ ) .
M q ( κ ) = q ( a M κ ) ,
M m q ( κ ) = q ( κ a M / a m ) .
μ q ( κ ) = q ( λ κ ) ,
lim μ 1 λ ( μ ) = 1 .
q ( κ ) = A | κ | α
lim | κ | P ( κ ) = 0 ,
α = ln μ / ln λ .
q ( κ ) = γ ( 1 + i β κ | κ | ) | κ | α , γ > 0 .
C α = C 1 α + C 2 α ,
P ( κ ) = exp [ γ ( 1 + i β κ | κ | ) | κ | α ]
q ( κ ) = γ | κ | α [ 1 + i β ω ( α , κ ) κ | κ | ] , 0 < α 2 ,
ω ( α , κ ) = { tan ( π α / 2 ) if α 1 2 π log κ if α = 1 ,
P ( ψ ) = 1 2 π d k exp ( i κ ψ ) exp { γ | κ | α [ 1 + i β ω ( α , κ ) κ | κ | ] }
C α = j = 1 N C j α ,
C = N 1 / α ,
Ψ ( N ) = 1 N 1 / α j = 1 N Ψ j .
P ( κ ) = exp { γ | κ | α [ 1 + i β ω ( α , κ ) κ | κ | ] } , α 1 .
P ( N ) ( B N κ ) = [ P ( κ ) ] N ,
P ( N ) ( κ ) = [ P ( κ / N 1 / α ) ] N = exp { γ | κ | α [ 1 + i β ω ( α , κ ) κ | κ | ] } .
P ( κ , ζ ) = exp ( γ | κ | α ζ ) , ζ 0 ,
P ( ψ , x ) = 1 2 π d κ exp ( i κ ψ ) exp ( γ | κ | α | x | ) .
P ( ψ 2 ψ 2 , ζ ) = d ψ P ( ψ 2 ψ , ζ x ) P ( ψ ψ 1 , x ) ,
P ( κ , ζ ) = P ( κ , ζ x ) P ( κ , x ) ,
P ( Γ 1 / α ψ , Γ ζ ) = Γ 1 / α P ( ψ , ζ )
P ( ψ , ζ ) = μ γ | ζ | Γ ( μ ) sin ( μ π / 2 ) π | ψ | α + 1 as | ψ | .
P ( N ) ( κ ) = T P ( N 1 ) ( κ ) ,
P ( N ) ( κ ) = T N P ( κ ) .
T N P ( κ ) = [ P ( κ / N 1 / α ) ] N = P ( κ ) ,
Ψ ( N ) = 1 N 1 / α j = 1 N Ψ j
Ψ ( N s ) = 1 N s 1 / α Ψ m ( s ) ,
Ψ m ( s ) = j m = 1 s Ψ j m .
Ψ ( N ) = λ s Ψ ( N s ) ( s ) .
T s P ( κ ) = [ P ( λ s κ ) ] s = exp [ s γ | λ s κ | α ( 1 + i β ω ( α , κ ) κ | κ | ) ] ,
λ s = 1 / s 1 / α
T s P ( κ ) = P ( κ ) .
υ 0 ( x ) = exp [ i k Ψ ( x ) ] at z = 0 + ,
( i 2 k z + 2 x 2 + k 2 ) u ( x , z ) = 0 ,
u ( x , z ) = e ikz ( k 2 π i z ) 1 / 2 d x exp [ i k 2 z ( x x ) 2 ] u 0 ( x ) .
u ( x , z ) = e ikz ( k 2 π i z ) 1 / 2 × d x exp [ i k 2 z ( x x ) 2 ] u 0 ( x ) .
u 0 ( x ) = u 0 ( x , 0 + ) = exp [ i k Ψ ( x ) ] .
Ψ ( x ) = 0 x d Ψ ( x ) + Ψ ( 0 ) ,
u 0 ( x ) = exp [ i k Ψ ( 0 ) ] exp [ i k 0 x d Ψ ( x ) ] .
u ( x , z ) 0 = 0 .
u ( x + ζ , z ) u * ( x , z ) = k 2 π z d x 1 d x 2 × exp { i k 2 z [ ( x + ζ x 1 ) 2 ( x x 2 ) 2 } u 0 ( x 1 ) u 0 * ( x 2 ) .
u ( x + ζ , z ) u * ( x , z ) = k 2 π z d x 1 d x 2 × exp { i k 2 z [ ( x + ζ x 1 ) 2 ( x x 2 ) 2 } exp ( γ | k | α | x 1 x 2 | ) .
u ( x + ζ , z ) u * ( x , z ) = exp ( γ k α | ζ | ) .
P u ( K ) = d ζ e i K ζ u ( x + ζ , z ) u * ( x , z ) = 2 γ k α K 2 + γ 2 k 2 α .
P u ( k ) P u ( 0 ) = γ 2 k 2 α K 2 + γ 2 k 2 α 1 ,
K k γ k α 1 Θ ,
k P u ( K / k , Θ ) = 2 Θ ( K / k ) 2 + Θ 2 ,
P u ( K / k , Θ ) = 1 Θ P u ( K / Θ k , 1 ) .
C ( R ) = C 0 ( R ) exp [ ½ D ( | R | ) ] .
C ( R , σ ) = exp ( R 2 / σ 2 ) ,
C ( R ) ¯ = 0 C ( R , σ ) P ( σ ) d σ .
P L ( σ ) = a 1 a n = 0 1 a n δ ( σ σ L / b n ) ,
C L ( R ) ¯ = a 1 a n = 0 1 a n exp ( b 2 n R 2 / σ L 2 ) ,
C L ¯ ( R ) = a 1 a exp ( R 2 / σ L 2 ) + 1 a C L ¯ ( b R ) .
C L ¯ ( R ) = ( R / σ L ) α Q L ( R ) + q L ( R ) ,
Q L ( R ) = a 1 2 a ln b m = Γ ( μ / 2 + π i m / ln b ) × exp [ 2 π i m ln b ln ( R / σ L ) ]
q L ( R ) a 1 a n = 0 ( R 2 σ L 2 ) n 1 n ! 1 1 b 2 n / a ,
F L ( κ ) = d 2 R C L ¯ ( R ) e i κ · R = a 1 a n = 0 π σ L 2 a n b 2 n exp ( σ L 2 κ 2 4 b 2 n ) ,
F L ( κ ) = Q L ( κ ) ( σ L κ ) α + 2 + q ¯ L ( κ ) ,
Q L ( κ ) = ( a 1 ) a π σ L 2 ln b 2 α + 1 m = Γ ( 1 α 2 + i m π ln b ) × exp [ 2 π i m ln b ln ( σ L κ / 2 ) ] ,
q L ( κ ) = a 1 a m = 0 1 m ! ( 1 b 2 m 2 / a ) ( σ L 2 κ 2 / 4 ) m .
P s ( σ ) = a 1 a n = 1 1 a n δ ( σ σ s b m ) ,
C S ¯ ( R ) = a 1 a n = 0 1 a n exp ( R 2 / σ s 2 b 2 n ) ,
F s ( κ ) = a 1 a n = 0 b 2 n a n π σ s 2 exp ( b 2 n σ s 2 κ 2 / 4 ) ,
F s ( κ ) = Q s ( κ ) ( σ s κ ) 2 α + q s ( κ ) , α = ln a / ln b > 0 ,
F ( κ ) = π σ L 2 { f L ( κ ) ( σ s σ L ) 2 f s ( κ ) } .
X ( ω ) = e i ω t X ( t ) d t ,
X ( t ) = 1 2 π e i ω t X ( ω ) d t .
g ( t , ω 0 ) = exp [ ( 2 t / Δ t ) 2 ln 2 ] exp ( i ω 0 t ) ,
X ( t i , ω j ) = d t g * ( t t i , ω j ) X ( t ) .
C g = 1 2 π | g ( ω ) | 2 d ω ω <
g ( t ) d t = δ ( ω ) g ( ω ) d ω = g ( 0 ) = 0 ,
g ( t ) = ( 1 t 2 ) e t 2 / 2 .
g ( ω ) = ω 2 e ω 2 / 2 ,
g ( t ) = e i Ω t ( e t 2 / 2 2 e Ω 2 / 4 e t 2 ) ,
g ( ω ) = exp [ ( ω Ω ) 2 / 2 ] e Ω 2 / 4 exp [ ( ω Ω ) 2 / 4 ]
T g ( a , b ) X = 1 ( a C g ) 1 / 2 g * [ ( t b ) / a ] X ( t ) d t
= ( a C g ) 1 / 2 g * ( a ω ) e i b ω X ( ω ) d ω ,
X ( t ) = 1 C g 1 / 2 d a d b a 5 / 2 g [ ( t b ) / a ] T g ( a , b ) X ,
d t | X ( t ) | 2 <
d t | X ( t ) | 2 = d a d b a 2 | T g ( a , b ) X | 2 ,
X t 0 ( t ) X ( t + t 0 ) X ( t ) .
X t 0 ( λ t ) = λ α ( t 0 ) X t 0 ( t ) .
T g ( λ a , λ b + t 0 ) X = 1 ( λ a C g ) 1 / 2 d t g * [ ( t t 0 λ b ) / λ a ] X ( t ) = 1 ( λ a C g ) 1 / 2 d t g * [ ( t λ b / λ a ) ] [ X ( t + t 0 ) X ( t ) ] ,
T g ( λ a , λ b + t 0 ) X = ( λ a C g ) 1 / 2 d t g * [ ( t b ) / a ] X t 0 ( λ t ) ,
T g ( λ a , λ b + t 0 ) X = λ α ( t 0 ) + 1 / 2 T g ( a , b + t 0 ) X .
X ( t ) = { 0 for t 0 t α for 0 < t < 1 , 1 for t 1
T g ( a , 0 ) X = [ 0 , 1 ] 1 a g ( t / a ) t α d t + [ 1 , ] 1 a g ( t / a ) d t = a α [ 0 , 1 / a ] g ( t ) t α d t + [ 1 / a , ] g ( t ) d t = T g ( l ) X + T g ( g ) X .
T g ( a , b ) P = 1 a n g [ ( t b ) / a ] P ( t ) d t ,
P ( λ t , t 0 ) λ α ( t a ) P ( t , t 0 ) ,
T g ( λ a , λ b + t 0 ) P λ α ( t 0 ) n T g ( a , b + t 0 ) P ,
N α ( l ) l f ( α )
α ( t 0 ) lim t 0 ln P ( t , t 0 ) ln t .
q = d d α f ( α ) ,
D q = 1 q 1 [ q α f ( α ) ] .
α = d d q [ ( q 1 ) D q ] .
cos ( b n ω o t ) = 1 2 π i C i C + i Γ ( s ) cos ( π s / 2 ) ( b n ω 0 t ) s d s , 0 < c = Re ( s ) < 1 .
W ( t ) = 1 2 π i C i C + i d s Γ ( s ) cos ( π s / 2 ) ( ω 0 t ) s n = 0 1 ( a b s ) n = 1 2 π i C i C + i d s Γ ( s ) cos ( π s / 2 ) ( ω 0 t ) s ( 1 1 / a b s ) ,
α = ln a / ln b .
W ( t ) = t α Q ( t ) + m = 0 ( 1 ) m ( ω o t ) 2 m ( 2 m ) ! ( 1 b 2 m / a ) ,
Q ( t ) = ω o ln b n = Γ ( α + 2 π n i ln b ) cos [ ( α + 2 π n i ln b ) π / 2 ] × exp [ 2 π n i ln ( ω o t ) ln b ] .
I = exp [ i k 0 x d Ψ ( x ) ] ,
I = lim N Δ x 0 j = 1 N 1 exp [ i k j Δ x ( j + 1 ) Δ x d Ψ ( x ) ] .
I = lim N Δ x 0 j = 0 N 1 exp { i k Ψ [ ( j + 1 ) Δ x ] Ψ ( j Δ x ) } ,
I = lim N Δ x 0 j = 0 N 1 exp [ i k Δ Ψ j ( Δ x ) ] P ( Δ Ψ j , Δ x ) d Δ Ψ j .
I = lim N Δ x 0 j = 0 N 1 P ( k , Δ x ) ,
x = lim N Δ x 0 N Δ x = const .
I = exp ( γ | k | μ x ) , x 0 ,
u 0 ( x 1 ) u 0 * ( x 2 ) = exp { i k [ 0 x 1 d Ψ ( x ) 0 x 2 d Ψ ( x ) ] } ,
u 0 ( x 1 ) u 0 ( x 2 ) = lim N , M Δ x 0 j = M N 1 exp [ i k j Δ x ( j + 1 ) Δ x d Ψ ( x ) ] ,
u 0 ( x 1 ) u 0 ( x 2 ) = lim N , M Δ x 0 j = M N 1 exp [ i k Δ Ψ j ( Δ x ) ] × P ( Δ Ψ j , Δ x ) d Δ Ψ j = lim N , M Δ x 0 j = M N 1 P ( k , Δ x ) .
x 1 lim n Δ x 0 N Δ x = const .
x 2 lim n Δ x 0 M Δ x = const . ,
u 0 ( x 1 ) u 0 * ( x 2 ) = exp [ γ | k | μ ( x 1 x 2 ) ] , x 1 x 2 .
I n = 1 1 a n exp ( b 2 n R 2 / σ L 2 ) = n = 0 g ( n ) ,
n = 0 g ( n ) , = 1 2 g ( 0 ) + 0 g ( x ) d x + 2 m = 1 0 g ( x ) cos ( 2 π m x ) d x ,
g ( x ) = exp ( x ln a ) exp [ ( σ 2 / σ 4 2 ) exp ( 2 x ln b ) ] .
I = 1 2 exp ( R 2 / σ L 2 ) + 1 2 ln b × m = 1 y 1 μ / 2 + ( π i m / ln b ) exp ( y R 2 / σ L 2 ) d y ,
I = 1 2 exp ( R 2 / σ L 2 ) + 1 2 ln b m = ( σ L 2 R 2 ) μ / 2 + ( i m π / ln b ) Γ ( μ / 2 + i m π / ln b ) 1 2 ln b m = l = 0 1 l ! ( μ / 2 + l + i m π / ln b ) ( R 2 / σ L 2 ) l .
q L ( R ) = a 1 a [ 1 2 exp ( R 2 / σ L 2 ) 1 2 ln b × m = l = 0 1 l ! [ μ / 2 + l + i m π / ln b ] ( R 2 / σ L 2 ) l ] .
m = 1 μ / 2 + l + i m π / ln b = m = 1 2 ( l μ / 2 ) ( l μ / 2 ) 2 + ( i m π / ln b ) 2 + 1 l μ / 2 = ln b coth [ ln b ( l μ / 2 ) ] ,
q L ( R ) = a 1 a { 1 2 exp ( R 2 σ L 2 ) 1 2 l = 0 coth [ ln b ( l μ / 2 ) ] l ! ( R 2 / σ L 2 ) l } .
q L ( R ) = a 1 a l = 0 ( R 2 / σ L 2 ) l 1 l ! ( 1 b 2 l / a ) ,

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