research article essay on kolmogorov law of minus 5 over 3...

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 680678 3 pageshttpdxdoiorg1011552013680678

Research ArticleEssay on Kolmogorov Law of Minus 5 over 3Viewed with Golden Ratio

Ming Li1 and Wei Zhao2

1 School of Information Science amp Technology East China Normal University No 500 Dong-Chuan Road Shanghai 200241 China2Department of Computer and Information Science University of Macau Avenue Padre Tomas Pereira Taipa 1356 Macau

Correspondence should be addressed to Ming Li ming lihkyahoocom

Received 14 July 2013 Accepted 7 August 2013

Academic Editor Carlo Cattani

Copyright copy 2013 M Li and W Zhao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The golden ratio is an astonishing number in high-energy physics neutrino physics and cosmology The Kolmogorov minus53 lawplays a role in describing energy transfer of random data or random functions The contributions of this essay are in twofold Oneis to express the Kolmogorov minus53 law by using the golden ratio The other is to represent the fractal dimension of random datafollowing the Kolmogorov minus53 law with the golden ratio It is our hope that this essay may be helpful to provide a new outlook ofthe Kolmogorov minus53 law from the point of view of the golden ratio

1 Instruction

Let 120593 be the golden ratio It equals to (1 + radic5)2 Its inverse1120593 = (

radic

5minus1)2 is called the goldenmean Both are irrationalnumbers Approximately they are

120593 asymp 1618

1

120593

asymp 0618 (1)

The number 120593 has wide applications to various fieldsranging from physics (Wurm and Martini [1] King [2] Dinget al [3] Feruglio and Paris [4]) to cosmology (Livio [5]Boeyens [6])

In addition to the golden ratio fractal is a mathematicalmodel attracting interests of scientists and physicists in thefield of high-energy physics (Ghosh et al [7])While studyingenergy transfer the Kolmogorov minus53 law introduced byKolmogorov [8] plays a role in the field (Qian [9] Brun etal [10 Chapter 7] Hillebrandt and Kupka [11 Chapter 4]Gomes-Fernandes et al [12 page 309] Lumley and Yaglom[13] Warhaft [14] Geipel et al [15]) Motivated by thosethis essay aims at exhibiting the Kolmogorov minus53 law andthe self-similarity which is an important fractal property(Mandelbrot [16 17] Cattani et al [18 19]) from the pointof the golden ratio It is our expectation that this essay maybe helpful to describe the nature of random data that followsthe Kolmogorov minus53 law

The rest of paper is organized as follows The preliminar-ies are briefed in Section 2 The results that the Kolmogorovminus53 law and the self-similarity are explained from a view ofthe golden ratio are given in Section 3 which is followed byconclusions

2 Preliminaries

21 Golden Ratio Consider a straight line in Figure 1 Arra-nge three points A B and C there such that the ratio below

ACCB=

ABAC

(2)

equals to120593Thenwe say that the straight line is cut in extremeand mean ratio ([5] Ackermann [20] Kaygn et al [21])

There are various ways to synthesize the number 120593 Acco-rding to the definition of extreme and mean ratio one has

119909 =

119909 + 1

119909

(3)

Multiplying 119909 on the both sides of the above yields

119909

2minus 119909 minus 1 = 0

(4)

2 Advances in High Energy Physics

A BC

Figure 1 Illustration of a straight line for the golden ratio

Solving (4) produces

119909

1=

1 +

radic

5

2

119909

2=

1 minus

radic

5

2

(5)

Conventionally 1199091is denoted by 120593 and 119909

2is denoted by

minus1120593

22 Fractal Dimension Let the autocorrelation function(ACF) of a random function 119909(119905) be 119903

119909119909(120591) where 119903

119909119909(120591) =

119864[119909(119905)119909(119905 + 120591)] Then 119903119909119909(120591) for 120591 rarr 0 represents the small-

scaling phenomenon of 119909(119905) Following Davies and Hall [22]if 119903119909119909(120591) is sufficiently smooth on (0infin) and if

119903

119909119909(0) minus 119903

119909119909(120591) sim 119888

1|120591|

120572 for |120591| 997888rarr 0 (6)

where 1198881is a constant and 0 lt 120572 le 2 is the fractal index of

119909(119905) the fractal dimension which is denoted by D of x(t) isin the form

119863 = 2 minus

120572

2

(7)

Note that fractal dimension 1 le 119863 lt 2 is a measure tocharacterize the local self-similarity or local roughness of119909(119905)(Mandelbrot [17] Gneiting and Schlather [23])

Denote the power spectrum density (PSD) function of119909(119905) by 119878

119909119909(120596) Denote by F the operator of Fourier transform

Then

119878

119909119909(120596) = F [119903

119909119909(119905)] = int

infin

minusinfin

119903

119909119909(119905) 119890

minus119895120596119905119889119905 119895 =

radic

minus1 (8)

In the domain of generalized functions we have (Kanwal[24] Gelfand and Vilenkin [25] Li and Lim [26])

F (|120591|120572) = minus2 sin(1205871205722

) Γ (1 + 120572) |120596|

minus(120572+1) (9)

where Γ is the Gamma function Therefore we have an ana-logy of (6) in the frequency domain in the form

F (|120591|120572) sim 1198882|120596|minus(120572+1) for 120596 997888rarr infin (10)

where 1198882is a constant (Chan et al [27])

3 Results

The Kolmogorov minus53 law implies that the PSD of a randomfunction has the asymptotic behavior expressed by

119878

119909119909(120596) sim 119888|120596|

minus53 for 120596 997888rarr infin (11)

where 119888 is a constant (Monin and Yaglom [28 29])The abovewell characterizes the local irregularity of turbulence function119909(119905) (Tropea et al [30])

In order to connect the Kolmogorov minus53 law with thegolden ratio we write

5

3

+ 119890 = 120593 (12)

From the previous discussions we have

119890 = 120593 minus

5

3

=

1 +

radic

5

2

minus

5

3

=

radic

5 minus 1

2

minus

2

3

=

1

120593

minus

2

3

(13)

Thus the Kolmogorov minus53 law expressed by (11) may berewritten by

119878

119909119909(120596) sim 119888|120596|

minus(120593minus119890) for 120596 997888rarr infin (14)

From (10) we see that the fractal index may be expressedby using the golden ratio as

120572 = 120593 minus 119890 minus 1 (15)

Thus we attain the fractal dimension from the point of viewof the golden ratio in the form

119863 = 2 minus

120593 minus 119890 minus 1

2

(16)

From the previous discussions we have the following theo-rems

Theorem 1 Let 119909(119905) be the random function that obeys theKolmogorov minus53 law expressed by (11) Then its PSD may beexpressed using the golden ratio in (14)

Theorem 2 Let 119909(119905) be the random function that obeysthe Kolmogorov minus53 law expressed by (11) Then its fractaldimension may be represented by using the golden ratio as thatin (16) The concrete value of119863 in (16) is 53

Finally we note that the significance of the present resultslies in expressing theKolmogorovminus53 law which plays a rolein energy transfer by using the golden ratio which attractsinterests of researchers and scientists in high-energy physicsneutrino physics cosmology and many others

4 Conclusions

We have explained our results in expressing the Kolmogorovminus53 lawwith the golden ratio In addition we have expressedthe fractal dimension of random data obeying the Kol-mogorov minus53 law based on the golden ratio

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under the Project Grant nos61272402 61070214 and 60873264 and by the 973 plan underthe Project Grant no 2011CB302800

Advances in High Energy Physics 3

References

[1] A Wurm and K M Martini ldquoBreakup of inverse golden meanshearless tori in the two-frequency standard nontwist maprdquoPhysics Letters A vol 377 no 8 pp 622ndash627 2013

[2] S F King ldquoTri-bimaximal-Cabibbo mixingrdquo Physics Letters Bvol 718 no 1 pp 136ndash142 2012

[3] G J Ding L L Everett and A J Stuart ldquoGolden ratio neutrinomixing and A5 flavor symmetryrdquoNuclear Physics B vol 857 no3 pp 219ndash253 2012

[4] F Feruglio and A Paris ldquoThe golden ratio prediction for thesolar angle from a natural model with A5 flavour symmetryrdquoJournal of High Energy Physics vol 2011 no 3 article 101 2011

[5] M LivioThe Golden Ration Random House 2003[6] J C A Boeyens Chemical Cosmology Springer 2010[7] D Ghosh A Deb S Pal et al ldquoEvidence of fractal behavior of

pions and protons in high energy interactionsmdashan experimen-tal investigationrdquo Fractals vol 13 no 4 pp 325ndash339 2005

[8] A N Kolmogorov ldquoLocal structure of turbulence in an incom-pressible viscous fluid at very high Reynolds numbersrdquo SovietPhysics Uspekhi vol 10 no 6 pp 734ndash736 1968

[9] J Qian ldquoGeneralization of the Kolmogorov minus53 law of turbu-lencerdquo Physical Review E vol 50 no 1 pp 611ndash613 1994

[10] C BrunD JuveMManhart andCDMunzNumerical Simu-lation of Turbulent Flows and Noise Generation Springer 2009

[11] W Hillebrandt and F Kupka Interdisciplinary Aspects of Turbu-lence Springer Berlin Germany 2009

[12] R Gomes-Fernandes B Ganapathisubramani and J C Vas-silicos ldquoParticle image velocimetry study of fractal-generatedturbulencerdquo Journal of Fluid Mechanics vol 711 pp 306ndash33362012

[13] J L Lumley andAM Yaglom ldquoACentury of Turbulencerdquo FlowTurbulence and Combustion vol 66 no 3 pp 241ndash286 2001

[14] Z Warhaft ldquoTurbulence in nature and in the laboratoryrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 1 pp 2481ndash2486 2002

[15] P Geipel K H H Goh and R P Lindstedt ldquoFractal-generatedturbulence in opposed jet flowsrdquo Flow Turbulence and Combus-tion vol 85 no 3-4 pp 397ndash419 2010

[16] B B Mandelbrot Gaussian Self-Affinity and Fractals SpringerNew York NY USA 2002

[17] B B Mandelbrot The Fractal Geometry of Nature W HFreeman New York NY USA 1982

[18] C Cattani G Pierro and G Altieri ldquoEntropy and multifrac-tality for the myeloma multiple TET 2 generdquo MathematicalProblems in Engineering vol 2012 Article ID 193761 14 pages2012

[19] C Cattani ldquoFractional calculus and Shannon waveletrdquo Math-ematical Problems in Engineering Article ID 502812 26 pages2012

[20] E C Ackermann ldquoThe Golden SectionrdquoThe American Mathe-matical Monthly vol 2 no 9-10 pp 260ndash264 1895

[21] B Kaygn B Balcin C Yildiz and S Arslan ldquoThe effect of teach-ing the subject of Fibonacci numbers and golden ratio thro-ugh the history of mathematicsrdquo ProcediamdashSocial and Behav-ioral Sciences vol 15 pp 961ndash965 2011

[22] S Davies and P Hall ldquoFractal analysis of surface roughness byusing spatial datardquo Journal of the Royal Statistical Society B vol61 no 1 pp 3ndash37 1999

[23] T Gneiting and M Schlather ldquoStochastic models that separatefractal dimension and the Hurst effectrdquo SIAM Review vol 46no 2 pp 269ndash282 2004

[24] R P Kanwal Generalized Functions Theory and ApplicationsBirkhauser 3rd edition 2004

[25] I M Gelfand and K Vilenkin Generalized Functions vol 1Academic Press New York NY USA 1964

[26] M Li and S C Lim ldquoA rigorous derivation of power spectrumof fractional Gaussian noiserdquo Fluctuation and Noise Letters vol6 no 4 pp C33ndashC36 2006

[27] G Chan P Hall and D S Poskitt ldquoPeriodogram-based estima-tors of fractal propertiesrdquoThe Annals of Statistics vol 23 no 5pp 1684ndash1711 1995

[28] A S Monin and A M Yaglom Statistical Fluid MechanicsMechanics of Turbulence vol 1 The MIT Press Cambridge Mass USA 1971


[30] C Tropea A L Yarin and F John Foss Springer Handbook ofExperimental Fluid Mechanics Springer 2007

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Journal of



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Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of

2 Advances in High Energy Physics

A BC

Figure 1 Illustration of a straight line for the golden ratio

Solving (4) produces

119909

1=

1 +

radic

5

2

119909

2=

1 minus

radic

5

2

(5)

Conventionally 1199091is denoted by 120593 and 119909

2is denoted by

minus1120593

22 Fractal Dimension Let the autocorrelation function(ACF) of a random function 119909(119905) be 119903

119909119909(120591) where 119903

119909119909(120591) =

119864[119909(119905)119909(119905 + 120591)] Then 119903119909119909(120591) for 120591 rarr 0 represents the small-

scaling phenomenon of 119909(119905) Following Davies and Hall [22]if 119903119909119909(120591) is sufficiently smooth on (0infin) and if

119903

119909119909(0) minus 119903

119909119909(120591) sim 119888

1|120591|

120572 for |120591| 997888rarr 0 (6)

where 1198881is a constant and 0 lt 120572 le 2 is the fractal index of

119909(119905) the fractal dimension which is denoted by D of x(t) isin the form

119863 = 2 minus

120572

2

(7)

Note that fractal dimension 1 le 119863 lt 2 is a measure tocharacterize the local self-similarity or local roughness of119909(119905)(Mandelbrot [17] Gneiting and Schlather [23])

Denote the power spectrum density (PSD) function of119909(119905) by 119878

119909119909(120596) Denote by F the operator of Fourier transform

Then

119878

119909119909(120596) = F [119903

119909119909(119905)] = int

infin

minusinfin

119903

119909119909(119905) 119890

minus119895120596119905119889119905 119895 =

radic

minus1 (8)

In the domain of generalized functions we have (Kanwal[24] Gelfand and Vilenkin [25] Li and Lim [26])

F (|120591|120572) = minus2 sin(1205871205722

) Γ (1 + 120572) |120596|

minus(120572+1) (9)

where Γ is the Gamma function Therefore we have an ana-logy of (6) in the frequency domain in the form

F (|120591|120572) sim 1198882|120596|minus(120572+1) for 120596 997888rarr infin (10)

where 1198882is a constant (Chan et al [27])

3 Results

The Kolmogorov minus53 law implies that the PSD of a randomfunction has the asymptotic behavior expressed by

119878

119909119909(120596) sim 119888|120596|

minus53 for 120596 997888rarr infin (11)

where 119888 is a constant (Monin and Yaglom [28 29])The abovewell characterizes the local irregularity of turbulence function119909(119905) (Tropea et al [30])

In order to connect the Kolmogorov minus53 law with thegolden ratio we write

5

3

+ 119890 = 120593 (12)

From the previous discussions we have

119890 = 120593 minus

5

3

=

1 +

radic

5

2

minus

5

3

=

radic

5 minus 1

2

minus

2

3

=

1

120593

minus

2

3

(13)

Thus the Kolmogorov minus53 law expressed by (11) may berewritten by

119878

119909119909(120596) sim 119888|120596|

minus(120593minus119890) for 120596 997888rarr infin (14)

From (10) we see that the fractal index may be expressedby using the golden ratio as

120572 = 120593 minus 119890 minus 1 (15)

Thus we attain the fractal dimension from the point of viewof the golden ratio in the form

119863 = 2 minus

120593 minus 119890 minus 1

2

(16)

From the previous discussions we have the following theo-rems

Theorem 1 Let 119909(119905) be the random function that obeys theKolmogorov minus53 law expressed by (11) Then its PSD may beexpressed using the golden ratio in (14)

Theorem 2 Let 119909(119905) be the random function that obeysthe Kolmogorov minus53 law expressed by (11) Then its fractaldimension may be represented by using the golden ratio as thatin (16) The concrete value of119863 in (16) is 53

Finally we note that the significance of the present resultslies in expressing theKolmogorovminus53 law which plays a rolein energy transfer by using the golden ratio which attractsinterests of researchers and scientists in high-energy physicsneutrino physics cosmology and many others

4 Conclusions

We have explained our results in expressing the Kolmogorovminus53 lawwith the golden ratio In addition we have expressedthe fractal dimension of random data obeying the Kol-mogorov minus53 law based on the golden ratio

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under the Project Grant nos61272402 61070214 and 60873264 and by the 973 plan underthe Project Grant no 2011CB302800


References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article essay on kolmogorov law of minus 5 over 3...

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