research article gurzadyan s problem 5 and improvement of...

Research ArticleGurzadyanrsquos Problem 5 and Improvement of Softeningsfor Cosmological Simulations Using the PP Method

Maxim Eingorn

CREST and NASA Research Centers North Carolina Central University Fayetteville Street 1801 Durham NC 27707 USA

Correspondence should be addressed to Maxim Eingorn maximeingorngmailcom

Received 3 October 2014 Revised 30 November 2014 Accepted 4 December 2014 Published 22 December 2014

Academic Editor Elias C Vagenas

Copyright copy 2014 Maxim Eingorn 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 Thepublication of this article was funded by SCOAP3

This paper is devoted to different modifications of two standard softenings of the gravitational attraction (namely the Plummer andHernquist softenings) which are commonly used in cosmological simulations based on the particle-particle (PP)method and theircomparison It is demonstrated that some of the proposed alternatives lead to almost the same accuracy as in the case of the pureNewtonian interaction even despite the fact that the force resolution is allowed to equal half the minimum interparticle distanceThe revealed way of precision improvement gives an opportunity to succeed in solving Gurzadyanrsquos Problem 5 and bring moderncomputer codes up to a higher standard

1 Introduction

Among Gurzadyanrsquos ldquo10 key problems in stellar dynamicsrdquo[1] any successful step towards solving Problem 5 will exertpowerful influence on the state of affairs in stellar and galacticdynamics The statement of this problem consists in creationof a computer code describing the 119873-body system with thephase trajectory being close to that of the real physical systemfor long enough time scales In view of primary importanceof cosmological simulations in analyzing the large scalestructure formation and comparing results with predictionsof different theories of the Universe evolution the goal ofProblem 5 appears particularly important

In this paper one such step is proposed Obviouslythe higher precision can be achieved in 119873-body com-puter simulations the more rigorous constraints can beimposed on parameters of a concrete cosmological modelThe well known PP method computing forces accordingto the Newton law of gravitation represents an accurate119873-body technique (see eg [2 3]) At the same time itsuffers from an evident shortcoming Since the Newtoniangravitational potential is singular at the particlesrsquo positionssoftening is required in order to avoid divergences of forceswhen the interparticle separation distances are very small

Introduction of softening ensures collisionless behavior ofthe system and simplifies numerical integration of its equa-tions of motion essentially However a high price to payis noticeable reduction of precision Below an attempt ismade to modify two generally accepted softenings in sucha way that the accuracy of computer simulations becomesimproved dramatically without any unjustified complicationof the equations of motion or the integration technique

The paper is organized as follows In Section 2 theequations of motion are written down and two standardsoftenings namely the Plummer and Hernquist ones areintroduced In Section 3 various modifications are proposedand their efficiency is compared with respect to the sameillustrative example Main results are discussed briefly inSection 4

2 Plummer and Hernquist Softenings

According to the mechanical approachnonrelativistic dis-crete cosmology developed recently in a series of papers [4ndash6] in the framework of the conventional ΛCDM (Λ ColdDark Matter) model the scalar cosmological perturbationsin the Universe with flat spatial topology can be described by

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 903642 4 pageshttpdxdoiorg1011552014903642

2 Advances in High Energy Physics

the following perturbed FLRWmetric in both nonrelativisticmatter dominated and dark energy dominated eras

1198891199042asymp 1198862[(1 + 2Φ) 119889120578

2minus (1 minus 2Φ) 120575

120572120573119889119909120572119889119909120573]

120572 120573 = 1 2 3

(1)

where 119886(120578) is the scale factor depending on the conformaltime 120578

Φ(120578 r) = 120593 (r)1198882119886 (120578)

Δ120593 = 4120587119866 (120588 minus 120588) (2)

Here r is the comoving radius-vectorΔ = 1205751205721205731205972(120597119909120572120597119909120573)

stands for the Laplace operator 119866 is the gravitational con-stant and 120588 represents the rest mass density in the comovingcoordinates 119909

1equiv 119909 119909

2equiv 119910 and 119909

3equiv 119911 This quantity

is time-independent within the adopted accuracy (both thenonrelativistic and weak field limits are applied) and 120588

denotes its constant average value Really 120588 changes with thelapse of time in view of peculiar motion of cosmic bodieshowever the corresponding velocities are small enough atthe considered nonrelativistic matter dominated and darkenergy dominated stages of the Universe evolution Conse-quently this temporal change of 120588 may be disregarded whendetermining the gravitational potential from (2) In otherwords it is determined by the positions of cosmic bodiesbut not by their peculiar velocities as it certainly should bein the nonrelativistic and weak field limits Naturally theintroduced function Φ (2) satisfies the linearized Einsteinequations for the metric (1) within the adopted accuracy (see[4 6])

It is worth noting that in the considered flat spatialtopology case the scale factor 119886 may be dimensionless thenthe comoving coordinates 119909120572 have a dimension of length andvice versa In order to be specific let us choose the first optionthen the rest mass density 120588 is measured in its standard units(namely masslength3)

In complete agreement with [7ndash9] the following equa-tions of motion which describe dynamics of the119873-body sys-tem experiencing both the gravitational attraction betweenits constituents and the global cosmological expansion of theUniverse can be immediately derived from (1) and (2) (thecutoff of the gravitational potential in the special relativityspirit introduced in [10] in order to resolve the problem ofnonzero average values of first-order scalar perturbations isnot considered here being irrelevant for the given investi-gation Really this cutoff is supposed to take place at greatdistances of the order of the particle horizon and certainlydoes not affect dynamics on smaller scales described by thewritten down standard equations of motion)

R119894minus

119886

119886R119894= minus119866sum

119895 =119894

119898119895(R119894minus R119895)

10038161003816100381610038161003816R119894minus R119895

10038161003816100381610038161003816

3 (3)

where R119894

= 119886r119894stands for the physical radius-vector of

the 119894th particle and 119898119894represents its mass These equations

are also commonly used for simulations at astrophysical

(ie noncosmological) scales when the second term in theleft hand side of (3) is irrelevant and may be neglected

Apparently the right hand side of (3) is a superpositionof forces which originate from Newtonian gravitationalpotentials of single point-like particles If such a particlepossessing the mass 119898 is located at the point R = 0then its rest mass density in the physical coordinates 120588ph =

119898120575(R) and the potential of the produced gravitational field120601 = minus119866119898119877 is singular at the location point In orderto suppress this singularity for reasons enumerated brieflyin Section 1 (namely avoiding divergences of interparti-cle forces ensuring collisionless dynamics and simplifyingnumerical integration of equations ofmotion) let us considertwo different models dealing with non-point-like gravitatingmasses The density and the potential of a single body in thePlummer model [11ndash15]

120588(P)ph =

3119898

41205871205763(1 +

1198772

1205762)

minus52

120601(P)

= minus119866119898

radic1198772 + 1205762 (4)

The same quantities in the Hernquist model [13ndash16]

120588(H)ph =

119898120576

2120587119877

1

(119877 + 120576)3 120601

(H)= minus

119866119898

119877 + 120576 (5)

where 120576 is the softening lengthparameter (called the forceresolution as well) typically amounting to a small percentof the mean interparticle distance While still dealing withpoint-like particles one usually takes into account the grav-itational interaction by means of 120601

(P) or 120601(H) in modern

computer simulations based on the PP method Thus theforce resolution 120576 should not be attributed to any real physicalsense representing just a mathematical trick eliminatingsingularity Both Plummer 120601(P) andHernquist 120601(H) potentialsconverge to the Newtonian one when 120576 rarr 0 (in thislimit as one can also easily verify both densities in (4) and(5) tend to the same expression 119898120575(R) corresponding to apoint-like massive particle as expected) However for anynonzero value of 120576 the attraction between every two bodiesis changed with respect to the Newton law of gravitation atall separations In particular both analyzed potentials (4) and(5) tend to zero as minus119866119898119877 when 119877 rarr +infin (119877 ≫ 120576) but inthe opposite limit 119877 rarr 0 (119877 ≪ 120576) they behave as a constantminus119866119898120576 so Newtonian singularity is absent The next sectionis entirely devoted to controllable elimination of this defect ofthe above-mentioned softenings

3 Illustration of Accuracy Improvement

For illustration purposes let us consider a hyperbolic trajec-tory of a test particle with the mass 119898 in the gravitationalfield of the mass 119872 resting in the origin of coordinates Thistrajectory is given by the following functions [17] (119883 119884 and

Advances in High Energy Physics 3

119905 denote Cartesian coordinates on the plane of motion andtime resp)

119883 = 119860 (120598 minus cosh 120585)

119884 = 119860radic1205982 minus 1 sinh 120585

119905 = radic1198603

119866119872(120598 sinh 120585 minus 120585)

(6)

where 120598 gt 1 stands for the eccentricity 120585 represents thevarying parameter and 119860 is the so-called semiaxis of ahyperbola being interrelated with the distance to perihelion119877min 119860(120598 minus 1) = 119877min In what follows the values 120598 = 11 and120585 isin [0 015] are used

The functions (6) satisfy the equations of motion

1198892119883

1198891199052= minus119866119872

119883

1198773

1198892119884

1198891199052= minus119866119872

119884

1198773 119877 = radic1198832 + 1198842

(7)

Introducing the normalized quantities

119883 =119883

119860= 120598 minus cosh 120585

=119884

119860= radic1205982 minus 1 sinh 120585

= 119905radic119866119872

1198603= 120598 sinh 120585 minus 120585

(8)

one can rewrite (7) in the form beingmore convenient for thesubsequent numerical integration

1198892119883

1198892= minus

119883

3

1198892

1198892= minus

3 = radic1198832 + 2 (9)

According to (8) if 120585 = 0 then = 0 119883 = 120598 minus 1 and = 0 besides 119889119883119889 = 0 119889119889 = radic1205982 minus 1(120598 minus 1) Theenumerated values will serve as initial conditions hereinafter

The exact solution (8) is depicted on Figure 1 (the blackcurve) together with the numerical solution of (9) (redpoints) The leapfrog ldquodrift-kick-driftrdquo numerical integrationscheme is applied here and below with the fixed time stepΔ = 00025

Orange points correspond to the modified equations ofmotion

1198892119883

1198892= minus

119883

( + 120576)2

1198892

1198892= minus

( + 120576)2

(10)

that is to the ldquoNewton-Hernquistrdquo conversion

1

997888rarr1

+ 120576(11)

in the expression for the gravitational potential The nor-malized softening length 120576 everywhere equals 005 (that is120576 amounts to 50 of min 120576 = min2 meaning quite close

HernquistModified HernquistPlummer

Modified PlummerNewton

092 094 096 098 100

001

002

003

004

005

006

007

Y

10

X

Figure 1 Trajectories for different potentials

approaching) Obviously the orange points lie rather far fromthe red ones Consequently the error is significant

Further one obtains purple points modifying the Hern-quist potential

1

+ 120576

997888rarr2

+ 120576

minus1

+ 2120576

(12)

The idea underlying this modification is simple at eachiteration one can make calculations using both 120576 and 2120576

softenings and then interpolate results to the zero softeningparameter In other words the expression in the right handside of (12) is constructed purposely in such a way that for ≫ 120576 its derivative with respect to being proportionalto the gravitation force behaves as minus12 + 6

2sdot (120576)

2 upto the second order of smallness concerning the ratio 120576The term of the first order is missing therefore the actualsuperposition of twoHernquist potentials with different soft-enings reduces the simulation error in comparison with theprevious case Really the purple points are noticeably closerto the red ones than the orange points However precisionis still low Of course one can increase a number of termsin the superposition and apply a higher order interpolationbut introduction of each additional term requires morecomputational time and consequently is not reasonable

Green points correspond to the modified equations ofmotion

1198892119883

1198892= minus

119883

(2 + 1205762)32

1198892

1198892= minus

(2 + 1205762)32

(13)

that is to the ldquoNewton-Plummerrdquo conversion

1

997888rarr1

radic2 + 1205762(14)

in the expression for the gravitational potential Precision ishigher than in the previous case because the derivative of theexpression in the right hand side of (14) with respect to for


≫ 120576 behaves as minus12 + 152sdot (120576)

2 so the deviationfrom the pure Newtonian behavior minus12 is now four timessmaller

Finally one gets blue points modifying the Plummerpotential [18]

1

radic2 + 1205762997888rarr

1

(4 + 1205764)14

(15)

Now for ≫ 120576 the deviation from the pure Newtonianbehavior represents a quantity of the fourth order of small-ness concerning the ratio 120576 Consequently precision isreally high even despite the fact that the condition 120576 = min2holds true as before

4 Conclusion

In this paper a promising opportunity of increasing theaccuracy of computer 119873-body simulations based on the PPmethod is addressed Namely the inevitable error arisingfrom gravitational softening is reduced considerably bymodifying the commonly used Plummer sim1radic1198772 + 1205762 andHernquist sim1(119877 + 120576) potentials In particular the proposedsim1(119877119899 + 120576

119899)1119899 potential with 119899 gt 2 gives better approxi-

mation since for 119877 gt 120576 the corresponding gravitation forcediffers from the standard Newtonian one in a small quantitysim(120576119877)119899 This is demonstrated explicitly for 119899 = 4 withthe help of the concrete illustrative example of one particlemoving along the hyperbolic trajectory in the softenedgravitational field of another one The force resolution 120576 istaken amounting to half the minimum separation distancebut despite this fact the suggested alternative softening ischaracterized by much higher precision being much closer tothe pure Newtonian picture than the standard ones

Apparently while improving numerical integration in theregion 119877 gt 120576 (where owing to this important inequalitythe expansion into series with respect to the ratio 120576119877 lt

1 is allowed) the developed scheme still misrepresents thepicture for 119877 ⩽ 120576 (where the above-mentioned expansionis forbidden) However if such close approachings seldomhappen thismisrepresentation is not significant for thewhole119873-body system behavior description Thus this scheme canreally play an important role in astrophysicalcosmologicalmodeling and solving Problem 5 In other words the pro-posed modifications reducing simulation errors caused bysoftening can help to bring the phase trajectory of the119873-bodysystem in a corresponding computer codemuch closer to thatof the real physical one

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas supported byNSFCREST awardHRD-1345219and NASA Grant NNX09AV07A The author would like to

thank S J Aarseth for valuable comments and the refereefor critical remarks which have considerably improved thepresentation of the obtained results

References

[1] V G Gurzadyan ldquo10 key problems in stellar dynamics inretrospectrdquo httparxivorgabs14070398

[2] S J Aarseth Gravitational N-Body Simulations CambridgeMonographs on Mathematical Physics Cambridge UniversityPress Cambridge UK 2003

[3] J Makino T Fukushige M Koga and K Namura ldquoGRAPE-6massively-parallel special-purpose computer for astrophysicalparticle simulationsrdquo Publications of the Astronomical Society ofJapan vol 55 no 6 pp 1163ndash1187 2003

[4] M Eingorn and A Zhuk ldquoHubble flows and gravitationalpotentials in observable Universerdquo Journal of Cosmology andAstroparticle Physics vol 9 article 026 2012

[5] M Eingorn A Kudinova and A Zhuk ldquoDynamics of astro-physical objects against the cosmological backgroundrdquo Journalof Cosmology and Astroparticle Physics vol 4 article 010 2013

[6] M Eingorn and A Zhuk ldquoRemarks on mechanical approachto observable Universerdquo Journal of Cosmology and AstroparticlePhysics vol 5 article 024 2014

[7] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[8] V Springel ldquoThe cosmological simulation code GADGET-2rdquoMonthly Notices of the Royal Astronomical Society vol 364 no4 pp 1105ndash1134 2005

[9] L D Landau and E M Lifshitz The Classical Theory of Fieldsvol 2 of Course of Theoretical Physics Series Oxford PergamonPress Oxford UK 4th edition 2000

[10] M Eingorn M Brilenkov and B Vlahovic ldquoZero average val-ues of cosmological perturbations as an indispensable conditionfor the theory and simulationsrdquo httparxivorgabs14073244

[11] H C Plummer ldquoOn the problem of distribution in globular starclustersrdquoMonthly Notices of the Royal Astronomical Society vol71 pp 460ndash470 1911

[12] K Dolag S Borgani S Schindler A Diaferio and AM BykovldquoSimulation techniques for cosmological simulationsrdquo SpaceScience Reviews vol 134 no 1ndash4 pp 229ndash268 2008

[13] F Iannuzzi and K Dolag ldquoAdaptive gravitational softening inGADGETrdquo Monthly Notices of the Royal Astronomical Societyvol 417 no 4 pp 2846ndash2859 2011

[14] J E Barnes ldquoGravitational softening as a smoothing operationrdquoMonthly Notices of the Royal Astronomical Society vol 425 no2 pp 1104ndash1120 2012

[15] B Rottgers T Naab and L Oser ldquoStellar orbits in cosmologicalgalaxy simulations the connection to formation history andline-of-sight kinematicsrdquo Monthly Notices of the Royal Astro-nomical Society vol 445 no 2 pp 1065ndash1083 2014

[16] L Hernquist ldquoAn analytical model for spherical galaxies andbulgesrdquoAstrophysical Journal vol 356 no 2 pp 359ndash364 1990

[17] L D Landau and E M Lifshitz Mechanics vol 1 of Course ofTheoretical Physics Series Oxford Pergamon Press Oxford UK3rd edition 2000

[18] K S Oh D N C Lin and S J Aarseth ldquoOn the tidaldisruption of dwarf spheroidal galaxies around the galaxyrdquoTheAstrophysical Journal vol 442 no 1 pp 142ndash158 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Statistical MechanicsInternational Journal of


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ThermodynamicsJournal of


the following perturbed FLRWmetric in both nonrelativisticmatter dominated and dark energy dominated eras

1198891199042asymp 1198862[(1 + 2Φ) 119889120578

2minus (1 minus 2Φ) 120575

120572120573119889119909120572119889119909120573]

120572 120573 = 1 2 3

(1)

where 119886(120578) is the scale factor depending on the conformaltime 120578

Φ(120578 r) = 120593 (r)1198882119886 (120578)

Δ120593 = 4120587119866 (120588 minus 120588) (2)

Here r is the comoving radius-vectorΔ = 1205751205721205731205972(120597119909120572120597119909120573)

stands for the Laplace operator 119866 is the gravitational con-stant and 120588 represents the rest mass density in the comovingcoordinates 119909

1equiv 119909 119909

2equiv 119910 and 119909

3equiv 119911 This quantity

is time-independent within the adopted accuracy (both thenonrelativistic and weak field limits are applied) and 120588

denotes its constant average value Really 120588 changes with thelapse of time in view of peculiar motion of cosmic bodieshowever the corresponding velocities are small enough atthe considered nonrelativistic matter dominated and darkenergy dominated stages of the Universe evolution Conse-quently this temporal change of 120588 may be disregarded whendetermining the gravitational potential from (2) In otherwords it is determined by the positions of cosmic bodiesbut not by their peculiar velocities as it certainly should bein the nonrelativistic and weak field limits Naturally theintroduced function Φ (2) satisfies the linearized Einsteinequations for the metric (1) within the adopted accuracy (see[4 6])

It is worth noting that in the considered flat spatialtopology case the scale factor 119886 may be dimensionless thenthe comoving coordinates 119909120572 have a dimension of length andvice versa In order to be specific let us choose the first optionthen the rest mass density 120588 is measured in its standard units(namely masslength3)

In complete agreement with [7ndash9] the following equa-tions of motion which describe dynamics of the119873-body sys-tem experiencing both the gravitational attraction betweenits constituents and the global cosmological expansion of theUniverse can be immediately derived from (1) and (2) (thecutoff of the gravitational potential in the special relativityspirit introduced in [10] in order to resolve the problem ofnonzero average values of first-order scalar perturbations isnot considered here being irrelevant for the given investi-gation Really this cutoff is supposed to take place at greatdistances of the order of the particle horizon and certainlydoes not affect dynamics on smaller scales described by thewritten down standard equations of motion)

R119894minus

119886

119886R119894= minus119866sum

119895 =119894

119898119895(R119894minus R119895)

10038161003816100381610038161003816R119894minus R119895

10038161003816100381610038161003816

3 (3)

where R119894

= 119886r119894stands for the physical radius-vector of

the 119894th particle and 119898119894represents its mass These equations

are also commonly used for simulations at astrophysical

(ie noncosmological) scales when the second term in theleft hand side of (3) is irrelevant and may be neglected

Apparently the right hand side of (3) is a superpositionof forces which originate from Newtonian gravitationalpotentials of single point-like particles If such a particlepossessing the mass 119898 is located at the point R = 0then its rest mass density in the physical coordinates 120588ph =

119898120575(R) and the potential of the produced gravitational field120601 = minus119866119898119877 is singular at the location point In orderto suppress this singularity for reasons enumerated brieflyin Section 1 (namely avoiding divergences of interparti-cle forces ensuring collisionless dynamics and simplifyingnumerical integration of equations ofmotion) let us considertwo different models dealing with non-point-like gravitatingmasses The density and the potential of a single body in thePlummer model [11ndash15]

120588(P)ph =

3119898

41205871205763(1 +

1198772

1205762)

minus52

120601(P)

= minus119866119898

radic1198772 + 1205762 (4)

The same quantities in the Hernquist model [13ndash16]

120588(H)ph =

119898120576

2120587119877

1

(119877 + 120576)3 120601

(H)= minus

119866119898

119877 + 120576 (5)

where 120576 is the softening lengthparameter (called the forceresolution as well) typically amounting to a small percentof the mean interparticle distance While still dealing withpoint-like particles one usually takes into account the grav-itational interaction by means of 120601

(P) or 120601(H) in modern

computer simulations based on the PP method Thus theforce resolution 120576 should not be attributed to any real physicalsense representing just a mathematical trick eliminatingsingularity Both Plummer 120601(P) andHernquist 120601(H) potentialsconverge to the Newtonian one when 120576 rarr 0 (in thislimit as one can also easily verify both densities in (4) and(5) tend to the same expression 119898120575(R) corresponding to apoint-like massive particle as expected) However for anynonzero value of 120576 the attraction between every two bodiesis changed with respect to the Newton law of gravitation atall separations In particular both analyzed potentials (4) and(5) tend to zero as minus119866119898119877 when 119877 rarr +infin (119877 ≫ 120576) but inthe opposite limit 119877 rarr 0 (119877 ≪ 120576) they behave as a constantminus119866119898120576 so Newtonian singularity is absent The next sectionis entirely devoted to controllable elimination of this defect ofthe above-mentioned softenings

3 Illustration of Accuracy Improvement

For illustration purposes let us consider a hyperbolic trajec-tory of a test particle with the mass 119898 in the gravitationalfield of the mass 119872 resting in the origin of coordinates Thistrajectory is given by the following functions [17] (119883 119884 and



119883 = 119860 (120598 minus cosh 120585)

119884 = 119860radic1205982 minus 1 sinh 120585

119905 = radic1198603

119866119872(120598 sinh 120585 minus 120585)

(6)



1198892119883

1198891199052= minus119866119872

119883

1198773

1198892119884

1198891199052= minus119866119872

119884

1198773 119877 = radic1198832 + 1198842

(7)


119883 =119883

119860= 120598 minus cosh 120585

=119884

119860= radic1205982 minus 1 sinh 120585

= 119905radic119866119872

1198603= 120598 sinh 120585 minus 120585

(8)


1198892119883

1198892= minus

119883

3

1198892

1198892= minus

3 = radic1198832 + 2 (9)




1198892119883

1198892= minus

119883

( + 120576)2

1198892

1198892= minus

( + 120576)2

(10)


1

997888rarr1

+ 120576(11)




092 094 096 098 100

001

002

003

004

005

006

007

Y

10

X




1

+ 120576

997888rarr2

+ 120576

minus1

+ 2120576

(12)



2sdot (120576)



1198892119883

1198892= minus

119883

(2 + 1205762)32

1198892

1198892= minus

(2 + 1205762)32

(13)


1

997888rarr1

radic2 + 1205762(14)






1

radic2 + 1205762997888rarr

1

(4 + 1205764)14

(15)


4 Conclusion








Acknowledgments



References
























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119883 = 119860 (120598 minus cosh 120585)

119884 = 119860radic1205982 minus 1 sinh 120585

119905 = radic1198603

119866119872(120598 sinh 120585 minus 120585)

(6)



1198892119883

1198891199052= minus119866119872

119883

1198773

1198892119884

1198891199052= minus119866119872

119884

1198773 119877 = radic1198832 + 1198842

(7)


119883 =119883

119860= 120598 minus cosh 120585

=119884

119860= radic1205982 minus 1 sinh 120585

= 119905radic119866119872

1198603= 120598 sinh 120585 minus 120585

(8)


1198892119883

1198892= minus

119883

3

1198892

1198892= minus

3 = radic1198832 + 2 (9)




1198892119883

1198892= minus

119883

( + 120576)2

1198892

1198892= minus

( + 120576)2

(10)


1

997888rarr1

+ 120576(11)




092 094 096 098 100

001

002

003

004

005

006

007

Y

10

X




1

+ 120576

997888rarr2

+ 120576

minus1

+ 2120576

(12)



2sdot (120576)



1198892119883

1198892= minus

119883

(2 + 1205762)32

1198892

1198892= minus

(2 + 1205762)32

(13)


1

997888rarr1

radic2 + 1205762(14)






1

radic2 + 1205762997888rarr

1

(4 + 1205764)14

(15)


4 Conclusion








Acknowledgments



References
























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







1

radic2 + 1205762997888rarr

1

(4 + 1205764)14

(15)


4 Conclusion








Acknowledgments



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 gurzadyan s problem 5 and improvement of...

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