Research Article Tachyon Warm Intermediate and Logamediate...

Research ArticleTachyon Warm Intermediate and Logamediate Inflation in theBrane World Model in the Light of Planck Data

V Kamali1 and M R Setare2

1Department of Physics Faculty of Science Bu-Ali Sina University Hamedan 65178 Iran2Department of Science University of Kurdistan Campus of Bijar Bijar 15175 Iran

Correspondence should be addressed to V Kamali vkamali1362gmailcom

Received 4 April 2016 Revised 2 June 2016 Accepted 10 July 2016

Academic Editor Vladimir Dzhunushaliev

Copyright copy 2016 V Kamali and M R Setare This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

Tachyon inflationary universe model on the brane in the context of warm inflation is studied In slow-roll approximation and inlongitudinal gauge we find the primordial perturbation spectrums for this scenario We also present the general expressions of thetensor-scalar ratio scalar spectral index and its running We develop our model by using exponential potential the characteristicsof this model are calculated in great detail We also study our model in the context of intermediate (where scale factor expandsas 119886 = 119886

0exp(119860119905119891)) and logamediate (where the scale factor expands as 119886 = 119886

0exp(119860[ln 119905]])) models of inflation In these two

sectors dissipative parameter is considered as a constant parameter and a function of tachyon field Our model is compatible withobservational data The parameters of the model are restricted by Planck data

1 Introduction

Inflation as a theoretical framework presents the betterdescription of the early phase of our universe Main problemsof Big Bang model (horizon flatness etc) could be solved inthe context of inflation scenario [1 2] Lagrangian formalismin terms of scalar fields can explain this scenario Quantumfluctuations of the scalar field provide a description ofanisotropy of cosmic microwave background (CMB) andorigin of the distribution of large scale structure (LSS) [3ndash6] Standard model of inflation ldquocold inflationrdquo has tworegimes slow-roll and (p)reheating In the slow-roll limitkinematic energy is small compared to the potential energyterm and the universe expands Interaction between scalarfield (inflation) and other fields (massive and radiation fields)is neglected After this period kinetic energy is comparableto the potential energy in (p)reheating epoch In this erainflation oscillates around the minimum of the potentialwhile losing their energy to other fields (radiation masslessfields) which are presented in the theory After reheatingthe universe is filled by radiation In (p)reheating epochobserved universe attaches to the end of inflationary period

Another view of reheating is based on quantum mechanicalproduction of massive particles in classical backgroundinflation [7 8] Preheating is probably the most efficient andplausible bridge that could connect inflation to a hot radiationdominated universe [9 10]

In warm inflationary scenario radiation productionoccurs during the slow-roll inflation epoch and (p)reheatingis avoided [11 12] In this scenario thermal fluctuations couldplay a dominant role to produce initial fluctuations whichare necessary for LSS formation [13 14] Warm inflationaryperiod ends when the universe stops inflating After thisperiod the universe enters in the radiation phase [11 12] Someextensions of this model are found in [15ndash19]

In warm inflation there has to be continuously particleproduction For this to be possible then the microscopicprocesses that produce these particles must occur at atimescalemuch faster thanHubble expansionThus the decayrates Γ

119894(not to be confused with the dissipative coefficient)

must be bigger than 119867 Also these produced particles mustthermalize Thus the scattering processes amongst theseproduced particles must occur at a rate bigger than119867 Theseadiabatic conditions were outlined since the early warm

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 9682398 18 pageshttpdxdoiorg10115520169682398

2 Advances in High Energy Physics

inflation papers such as [20 21]More recently there has beenconsiderable explicit calculation fromQuantumFieldTheory(QFT) that explicitly computes all these relevant decay andscattering rates in warm inflation models [22 23]

The inflation era in the early evolution of the universecould be described by tachyonic field associated with unsta-ble D-brane because of the tachyon condensation near themaximum of the effective potential [24ndash26] At the late timestachyonic fields may add a nonrelativistic fluid or a new formof cosmological darkmatter to the universe [27]The tachyoninflation is a k-inflation model [28] for scalar field 120601 with apositive potential 119881(120601) Tachyon potentials have two specialproperties firstly a maximum of this potential is obtainedwhere 120601 rarr 0 and second property is the minimum ofthese potentials which is obtained where 120601 rarr infin If thetachyon field starts to roll down the potential then universedominated by a new form of matter will smoothly evolvefrom inflationary universe to an era which is dominatedby a nonrelativistic fluid [27] So we can explain the phaseof acceleration expansion (inflation) in terms of tachyonfield Tachyon fields in the ordinary (cold) tachyon inflationframework after slow-roll epoch evolve towards minimumof the potential without oscillating about it [26] so the(p)reheating mechanism in cold tachyon inflation does notwork Warm tachyon inflation is a picture where there aredissipative effects playing during inflation As a result of thisthe inflation evolves in a thermal radiation bath thereforethe reheating problem of cold tachyon inflation [26] can besolved in the framework of warm tachyon inflation We notethat the cold tachyonic inflation era cannaturally endwith thecollision of the two branes In this situation we do not needwarm inflation If the collision of two branes does not arisenaturally warm inflation is perfectly good scenario that cansolve the problem of end of thachyon inflation

We may live on a brane which is embedded in a higherdimensional universe This realization has significant impli-cations to cosmology [29ndash34] In this scenario which ismotivated by string theory gravity (closed string modes) canpropagate in the bulk while the standard model of particles(matter fields which are related to open string modes) isconfined to the lower-dimensional brane [35ndash37] In termsof Randall-Sundrum suggestion there are two similar butphenomenologically different brane world scenarios [33 34]In this paper we will consider that the brane world modelcorresponds to the Randall-Sundrum II brane world [34]

The brane world picture is described by the followingaction [29ndash34]

119878 =

1

1205812int1198895119909radicminus119892(119877

5+

12

1198972) minus 120582int119889

4119909radicminus119892brane

+ int1198894119909radicminus119892brane119871matter

(1)

In this scenario we have a 3-brane universe which is locatedin the 5D Anti-de Sitter (AdS) space-time where this space-time is effectively compactified with curvature scale 119897 ofAdS space-time 1198775 is the Ricci scalar in five dimensionsand 120581 = 8120587119866

5= 8120587119872

3

5 where 119866

5is the 5D Newtonrsquos

constant and1198725is Planck scale in five dimensions 120582 is the

tension of the brane and if we have no matter on the brane1205812120582 = 6119897 where the brane becomes Minkowski space-

time In the brane world model the gravity could propagatein the 5D space-time and the Newtonian gravity in fourdimensions is reproduced at the scales larger than 119897 on thebrane 4D Einsteinrsquos equation projected onto the brane hasbeen found in [38] Friedmann equation and the equations oflinear perturbation theory [39 40] may be modified by theseprojections Einsteinrsquos equations which are projected onto thebrane with cosmological constant andmatter fields which areconfined to 3-brane tension have the following form [38]

119866120583] = minusΛ 4119892120583] + (

8120587

1198722

4

)119879120583] + (

8120587

1198723

5

)

2

120587120583] minus 119864120583] (2)

where 119864120583] is a projection of 5D weyl tensor 119879

120583] is energy

density tensor on the brane1198724= radic119897119872

3

5is the Planck scales

in 4D and 120587120583] is a tensor quadratic in 119879120583]

120587120583] = minus

1

4

119879120583120572119879120572

] +1

12

119879120572

120572119879120583]

+

1

24

(3119879120572120573119879120572120573minus (119879120572

120572)2) 119892120583]

(3)

Cosmological constant Λ4on the brane in terms of 3-

brane tension 120582 and 5D cosmological constant Λ is given by

Λ4=

4120587

1198723

5

(Λ +

4120587

31198723

5

1205822) (4)

4D Planck scale is determined by 5D Planck scale as

1198724= radic

3

4120587

(

1198722

5

radic120582

)1198725= radic

4120587120582

3

119897 (5)

The natural boundary conditions to specify the perturbationsof this model are imposed where the perturbations do notdiverge at the horizon of the AdS space-time and we assumethat theWeyl curvature may be neglected On the large scalethe behavior of cosmological perturbations on the braneworld models is the same as a closed system on the branewithout the effects of the perturbations along the extradi-mensions in the bulk [41ndash43] On the large scale limit theperturbation parameters of inflation models have a completeset of perturbed equations on the brane which may be solvedin quasi-stable and slow-roll limit [41ndash44] The study of theperturbation evolution of warm inflation in the brane worldmodel on the large scale by using equations solely on thebrane and without solving the bulk perturbations is found in[44] This model has a complete set of perturbed equationson the brane We would like to study the warm tachyoninflation model on the brane using this approach Thereforewewill consider the linear cosmological perturbations theoryfor warm tachyon inflation model on the brane In spatiallyflat FRW model the Friedmann equation by using Einsteinrsquosequation (3) has following form [38]

1198672=

Λ4

3

+ (

8120587

31198722

4

)120588 + (

4120587

31198723

5

)1205882+

120576

1198864 (6)

Advances in High Energy Physics 3

where 119886 is scale factor of the model and 119867 is Hubbleparameter and 120588 is the total energy density on the braneThe last term in the above equation denotes the influence ofthe bulk gravitons on the brane where 120576 is an integrationconstant which arising from Weyl tensor 119864

120583] This termmay be rapidly diluted once inflation begins and we willneglect it Therefore the projected Weyl tensor term in theeffective Einstein equation may be neglected and this termdoes not give the significant contributions to the observableperturbation parameters We will also takeΛ

4to be vanished

at least in the early universe So the Friedmann equationreduces to

1198672=

8120587

31198722

4

120588 (1 +

120588

2120582

) (7)

Thebrane tension120582has been constrained fromnucleosynthe-sis [45] 120582 gt (1MeV)4 and a stronger limit of it results fromcurrent tests for deviation from Newtonrsquos law 120582 ge (10TeV)4[46 47]

In the warm inflationary models the total energy density120588 = 120588

120601+ 120588120574is presented on the brane [48] where 120588

120574is the

energy density of the radiation The Friedmann equation hasthis form

1198672=

8120587

31198722

4

(120588120601+ 120588120574) (1 +

120588120601+ 120588120574

2120582

) (8)

Cosmological perturbations of warm inflation model havebeen studied in [49] Warm tachyon inflationary universemodel has been studied in [50ndash53] also warm inflation onthe brane has been studied in [44] Inflation era is locatedin a period of dynamical evolution of the universe that theeffect of stringM-theory is relevant On the other handstringM-theory is related to higher dimension theories suchas space-like branes [24] Therefore in the present work wewill study warm tachyon inspired inflation in the context ofa higher dimensional theory instead of General Relativitythat is Randall-Sundrum brane world and cosmologicalperturbations of the model by using the above modifiedEinstein and Friedmann equations

Recently there has been a new perspective of warminflation [54] which is considered warm inflationary era asa quasi-de Sitter epoch of universe expansion on the otherhand as we mentioned it is believed that we may live on thebrane therefore we are interested to study warm tachyoninflation on the brane by using quasi-de Sitter solutions ofscale factor

In one sector of the present work we would like toconsider warm tachyon model on the brane in the contextof ldquointermediate inflationrdquo This scenario is one of the exactsolutions of inflationary field equation in the Einstein theorywith scale factor 119886(119905) = 119886

0exp(119860119905119891) (119860 gt 0 0 lt 119891 lt 1)

this solution of the scale factor in the context of a modifiedtensor-scalar theory has been found in [55] The study ofthis model is motivated by stringM-theory [56] If we addthe higher order curvature correction which is proportionalto Gauss-Bonnet (GB) term and to Einstein-Hilbert actionthen we obtain a free-ghost action [57 58] Gauss-Bonnetinteraction is leading order of the ldquo120572rdquo expansion to low energy

string effective action [57 58] (120572 is inverse string tension)This theory may be applied for black hole solutions [59]acceleration of the late time universe [60 61] and initialsingularity problems [62] The GB interaction in 4D withdynamical dilatonic scalar coupling leads to an intermediateform of scale factor [56] Expansion of the universe in theintermediate inflation scenario is slower than standard deSitter inflation with scale factor 119886 = 119886

0exp(119867

0119905) (1198860 1198670gt 0)

which arises as 119891 = 1 but faster than power-low inflationwith scale factor 119886 = 119905

119901 (119901 gt 1) Harrison-Zeldovich[63ndash65] spectrum of density perturbation that is 119899

119904= 1

for intermediate inflation models driven by scalar field ispresented for exact values of parameter 119891 [66]

On the other hand we will also study our model in thecontext of ldquologamediate inflationrdquo with scale factor 119886(119905) =1198860exp(119886[ln 119905]]) (] gt 1 119860 gt 0) [67] This model is converted

to power-law inflation for ] = 1 casesThis scenario is appliedin a number of scalar-tensor theories [68] The study oflogamediate scenario is motivated by imposing weak generalconditions on the cosmological models which have indefiniteexpansion [67] The effective potential of the logamediatemodel has been considered in dark energy models [69] Thisform of potential is also used in supergravity Kaluza-Kleintheories and superstring models [68 70] For logamediatemodels the power spectrum could be either red or bluetilted [71 72] In [67] we can find eight possible asymptoticscale factor solutions for cosmological dynamics Three ofthese solutions are noninflationary scale factor another threesolutions give power-low de Sitter and intermediate scalefactors Finally two cases of these solutions have asymptoticexpansion with logamediate scale factor We will study ourmodel using intermediate and logamediate scenarios

Warm inflation models based on ordinary scalar fieldshave been studied in [15 44 73ndash77] Particular model ofwarm inflation which is driven by tachyon field can befound in [50ndash53] In [78] the consistency of warm tachyoninflation with viscous pressure has been studied and thestability analysis for that model has been done In the presentpaper we will study warm tachyon inflation without viscosityeffect on the brane We also extended our model by usingexact solutions of the scale factor by Barrow [67] that isinter(loga)mediate solution

The paper is organized as follows in the next sectionwe will describe warm tachyon inflationary universe modelin the brane scenario in the background level In Section 3we present the perturbation parameters for our model InSection 4 we study our model using the exponential potentialin high dissipative regime and high energy limit In Section 5we study the model using intermediate scenario In Section 6we develop ourmodel in the context of logamediate inflationFinally in Section 7 we close by some concluding remarks

2 The Model

Tachyon scalar field 120601 is described by relativistic Lagrangian[27] as

119871 = radicminus119892(

119877

16120587119866

minus 119881 (120601)radic1 minus 119892120583]120597120583120601120597]120601) (9)


The stress-energy tensor in a spatially flat Friedmann Robert-son Walker (FRW) space-time is presented by

119879120583

] =120597119871

120597 (120597120583120601)

120597]120601 minus 119892120583

]119871 = diag (minus120588120601 119875120601 119875120601 119875120601) (10)

From the above equation energy density and pressure for aspatially homogeneous field have the following forms

120588120601=

119881 (120601)

radic1 minus120601

2

119875120601= minus119881 (120601)

radic1 minus

120601

2

(11)

where 119881(120601) is a scalar potential associated with the tachyonfield 120601 Important characteristics of this potential are119889119881119889120601 lt 0 and 119881(120601 rarr infin) rarr 0 [79] In this section we willpresent the characteristics of warm tachyon inflation modelon the brane in the background level This model may bedescribed by an effective fluid where the energy-momentumtensor of this fluid was recognized in the above equation

The dynamic of the warm tachyon inflation in spatiallyflat FRWmodel on the brane is described by these equations

1198672=

8120587

31198722

4

[

[

[

119881 (120601)

radic1 minus120601

2

+ 120588120574

]

]

]

sdot[

[

[

1 +

1

2120582

(

119881 (120601)

radic1 minus120601

2

+ 120588120574)]

]

]

(12)

120601+ 3119867(119875

120601+ 120588120601) = minusΓ

120601

2

997904rArr

120601

1 minus120601

2+ 3119867

120601 +

1198811015840

119881

= minus

Γ

119881

radic1 minus

120601

2120601

(13)

120574+ 4119867120588

120574= Γ

120601

2

(14)

where Γ is the dissipative coefficient In the above equationsdots ldquo rdquo mean derivative with respect to cosmic time andprime denotes derivative with respect to scalar field120601 Duringslow-roll inflation era the energy density (11) is the order ofpotential 120588

120601sim 119881 and dominates over the radiation energy

120588120601gt 120588120574 Using the slow-roll limit when

120601 ≪ 1 and 120601 ≪

(3119867 + Γ119881)120601 [11 12] and also when the inflation radiation

production is quasi-stable (120574≪ 4119867Γ

120574≪ Γ

120601

2) thedynamic equations (12) and (13) are reduced to

1198672=

8120587

31198722

4

119881(1 +

119881

2120582

) (15)

3119867 (1 + 119903)120601 = minus

1198811015840

119881

(16)

where 119903 = Γ3119867119881 In canonical warm inflation scenario therelative strength of thermal damping (Γ) should be comparedto expansion damping (119867) We must analyse the warminflation model in background and linear perturbation levelson our expanding over timescales which are shorter thanthe variation of expansion rate but large compared to themicrophysical processes

119881

Γ

≪ 120591 ≪ 119867minus1997904rArr

Γ ≫ 119867119881

(17)

Formore discussion please seeAppendix Particle productionin fact takes place at a constant rate during warm inflationfor canonical scalar field where strength of thermal dampingdominates over the effect of expansion damping (Γ gt 119867) butfor tachyon scalar fields as presented in the above equationΓ gt 119867119881 We will study our model in high dissipative regime(119903 ≫ 1) Using these conditions we have Γ ≫ 119867119881 whichagrees with particle production condition (Γ gt 119881119867)

From (14) (15) and (16) 120588120574could be written as

120588120574=

Γ120601

2

4119867

=

1198722

4119903

32120587 (1 + 119903)2(1 + 1198812120582)

(

1198811015840

119881

)

2

= 1205901198794

119903 (18)

where 119879119903is the temperature of thermal bath and 120590 is Stefan-

Boltzmann constant We introduce the slow-roll parametersfor our model as

120598 = minus

1198672≃

1198722

4

16120587

11988110158402

(1 + 119903) 1198813

1 + 119881120582

(1 + 1198812120582)2 (19)

120578 = minus

119867

≃

1198722

4

8120587

1198811015840

1198812(1 + 119903) [1 + 1198812120582]

[

211988110158401015840

1198811015840minus

1198811015840

119881

minus

1199031015840

(1 + 119903)

+

1198811015840

120582 + 119881

] minus 2120598

(20)

A relation between two energy densities 120588120601and 120588

120574is

obtained from (18) and (19)

120588120574=

119903

2 (1 + 119903)

[1 + 1205881206012120582]

[1 + 120588120601120582]

120588120601120598

≃

119903

2 (1 + 119903)

[1 + 1198812120582]

[1 + 119881120582]

119881120598

(21)

The condition of inflation epoch gt 0 could be obtainedby inequality 120598 lt 1 Therefore from above equation warmtachyon inflation on the brane could take place when

2 (1 + 119903)

119903

120588120574lt

1 + 1205881206012120582

1 + 120588120601120582

120588120601 (22)

Inflation period ends when 120598 ≃ 1 which implies

1198722

4

8120587

[

1198811015840

119891

119881119891

]

2

1 + 119881119891120582

(1 + 1198811198912120582)

2

1

119881119891

≃ 2 (1 + 119903119891) (23)


where the subscript 119891 denotes the end of inflation Thenumber of e-folds is given by

119873 = int

120601119891

120601lowast

119867119889119905 = int

120601119891

120601lowast

119867

120601

119889120601

= minus

8120587

1198722

4

int

120601119891

120601lowast

1198812

1198811015840(1 + 119903) [1 +

119881

2120582

] 119889120601

(24)

where the subscript lowast denotes the epoch when the cosmolog-ical scale exits the horizon

3 Perturbation

In this section we will study inhomogeneous perturbationsof the FRW background As we have mentioned in theintroduction we ignore the influence of the bulk gravitonson the brane arising from Weyl tensor 119864

120583] so we neglectthe back-reaction due to metric perturbations in the fifthdimension These perturbations in the longitudinal gaugemay be described by the perturbed FRWmetric

1198891199042= (1 + 2Φ) 119889119905

2minus 1198862(119905) (1 minus 2Ψ) 120575119894119895

119889119909119894119889119909119895 (25)

where Φ and Ψ are gauge-invariant metric perturbationvariables [80 81] The equation of motion is given by

120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601

+ [minus119886minus2nabla2+ (

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(26)

We expand the small change of field 120575120601 into Fourier compo-nents as

120575120601 (119909)

= int

1198893119896

(2120587)3[119890119894119896119909120575120601 (119896 119905) 119886

119896+ 119890minus119894119896119909120575120601 (119896 119905) 119886

lowast

119896]

(27)

In warm inflation thermal fluctuations of the inflation dom-inate over the quantum ones therefore we have classicalperturbation of scalar field 120575120601 All perturbed quantities havea spatial sector of the form 119890119894119896119909 where 119896 is the wave number

Perturbed Einstein field equations in momentum space haveonly the temporal parts

Φ = Ψ (28)

Φ + 119867Φ =

4120587

1198722

4

[

[

[

minus

4120588120574119886V3119896

+

119881120601

radic1 minus120601

2

120575120601]

]

]

sdot[

[

[

1 +

1

120582

[

[

[

120588120574+

119881

radic1 minus120601

2

]

]

]

]

]

]

(29)

120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601 + [

1198962

1198862+ (

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]

sdot 120575120601 minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(30)

(

120575120588120574) + 4119867120575120588

120574+

4

3

119896119886120588120574V minus 4120588

120574Φ minus

120601

2

Γ1015840120575120601

minus Γ120601

2

[2 (120575120601) minus 3

120601Φ] = 0

(31)

V + 4119867V +119896

119886

[Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601] = 0 (32)

The above equations are obtained for Fourier components119890119894119896119909 where the subscript 119896 is omitted V in the above set ofequations is presented by the decomposition of the velocityfield (120575119906

119895= minus(119894119886119896

119869119896)V119890119894119896119909 119895 = 1 2 3) [80 81]

Note that the effect of the bulk (extradimension) to per-turbed projected Einstein field equations on the branemay befound in (29) We will describe the nondecreasing adiabaticand isocurvature modes of our model on large scale limitIn this limit we have obtained a complete set of perturbationequations on the brane Therefore the perturbation variablesalong the extradimensions in the bulk could not have anycontribution to the perturbation equations on super-horizonscales (see eg [41ndash44])The same approach for nontachyonwarm inflation model on the brane in [44] is presentedWarm inflation model may be considered as a hybrid-likeinflationary model where the inflation field interacts withradiation field [49 82 83] Entropy perturbation may berelated to dissipation term [84] Perturbation of entropy inwarm inflation model is given by [85]

120575119878 = 119890 = minus119881120601119879120575120601 minus 119881

119879119879120575119879 (33)

In this paperwewill study potential of themodel as a functionof scalar field (119881(120601)) therefore the entropy perturbation willbe neglected We will study this important issue (potential asfunction of temperature 119881(120601 119879)) in future works

During inflationary phase with slow-roll approximationfor nondecreasing adiabatic modes on large scale limit 119896 ≪119886119867 we assume that the perturbed quantities could not varystrongly Sowe have119867Φ ≫ Φ ( 120575120601) ≪ (Γ+3119867)( 120575120601) (

120575120588120574) ≪

120575120588120574 and V ≪ 4119867V In the slow-roll limit and by using the


above limitations the set of perturbed equations are reducedto

Φ ≃

4120587

1198671198722

4

[minus

4120588120574119886V3119896

+ 119881120601120575120601] [1 +

119881

120582

] (34)

[3119867 +

Γ

119881

]120575120601 + [(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

≃ [120601

Γ

119881

minus 2(

1198811015840

119881

)]Φ

(35)

120575120588120574

120588120574

≃

Γ1015840

Γ

120575120601 minus 3Φ (36)

V ≃ minus119896

4119886119867

(Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601) (37)

Using (34) (36) and (37) perturbation variable Φ is deter-mined

Φ

=

4120587

1198722

4

(

119881120601

119867

)[1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

) 120575120601

(38)

We can solve the above equations by taking tachyon field120601 as the independent variable in place of cosmic time 119905 Using(16) we find

(3119867 +

Γ

119881

)

119889

119889119905

= (3119867 +

Γ

119881

)120601

119889

119889120601

= minus

1198811015840

119881

119889

119889120601

(39)

From above equation (35) and (38) the expression(120575120601)1015840120575120601 is obtained

(120575120601)1015840

120575120601

=

119881

1198811015840[(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

+

4120587

1198722

4

(minus120601

Γ

119881

+ 2(

1198811015840

119881

)

1015840

)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)]

(40)

We will return to the above relation Following [44 50ndash53 84] we introduce auxiliary function 120594 as

120594 =

119881120575120601

1198811015840exp [int 1

3119867 + Γ119881

(

Γ

119881

)

1015840

119889120601] (41)

From above definition we have

1205941015840

120594

=

(120575120601)1015840

120575120601

minus

119881

1198811015840(

1198811015840

119881

)

1015840

+

(Γ119881)1015840

3119867 + Γ119881

(42)

Using above equation and (40) we find

1205941015840

120594

=

4120587

1198722

4

(minus

119881120601

1198811015840

Γ

119881

+ 2)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)

(43)

We could rewrite this equation using (15) and (16)

1205941015840

120594

= minus

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

(44)

A solution for the above equation is

120594 (120601) = 119862 exp(minusintminus98

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

119889120601) (45)

where 119862 is integration constant From above equation and(42) we find small change of variable 120575120601 as

120575120601 = 119862

1198811015840

119881

exp (I (120601)) (46)

where

I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

)]119889120601

(47)

In the above calculations we have used the perturbationmethods in warm inflation models [44 50ndash53 84] where


the small change of variable 120575120601may be generated by thermalfluctuations instead of quantum fluctuations [86] and theintegration constant119862may be driven by boundary conditionsfor field perturbation Perturbed matter fields of our modelare inflation 120575120601 radiation 120575120588

119903 and velocity 119896minus1(119875 + 120588)V

119894

We can explain the cosmological perturbations in terms ofgauge-invariant variables These variables are important fordevelopment of perturbation after the end of inflation periodThe curvature perturbationR and entropy perturbation 119890 aredefied by [87 88]

R = Φ minus 119896minus1119886119867V

119890 = 120575119875 minus 1198882

119904120575120588

(48)

where 1198882119904= The boundary condition of warm inflation

models is found in very large scale limits that is 119896 ≪ 119886119867

where the curvature perturbationR sim const and the entropyperturbation vanishes [85]

Finally the density perturbation is given by [89 90]

120575119867=

2

5

1198722

4

119881 exp (minusI (120601))1198811015840

120575120601

=

2

15

1198722

4

exp (minusI (120601))119867119903

120601

120575120601

(49)

For high or low energy limit (119881 ≫ 120582 or 119881 ≪ 120582)and by inserting Γ = 0 the above equation reduces to120575119867≃ (119867

120601)120575120601 which agrees with the density perturbation

in cold inflation model [1 2] In the warm inflation modelthe fluctuations of the scalar field in high dissipative regime(119903 ≫ 1) may be generated by thermal fluctuation instead ofquantum fluctuations [86] as

(120575120601)2≃

119896119865119879119903

21205872 (50)

where in this limit freeze-out wave number 119896119865= radicΓ119867119881 =

119867radic3119903 ge 119867 corresponds to the freeze-out scale at the pointwhen dissipation damps out to thermally excited fluctuations(119881101584010158401198811015840 lt Γ119867119881) [86] 120575120601 in (50) can be found in [86]where Fourier transformed to momentum space is used (seeeg Appendix of [86] and Section 4 of [66]) therefore 120575120601 isintroduced in Fourier space andwe can present spectral indexand running in Fourier space With the help of (49) and (50)in high energy (119881 ≫ 120582) and high dissipative regime (119903 ≫ 1)we find

1205752

119867=

2radic3

7512058721198724

4

exp (minus2I (120601))radic119903

119879119903

119867

(51)

or equivalently

1205752

119867=

41198725

412058212

25 (2120587)5212059014119881minus34119903minus12120598minus34 exp (minus2I (120601)) (52)

where

I (120601) = minusint[1

3119867119903

(

Γ

119881

)

1015840

+

9

4

(1 minus

(ln Γ)1015840 1198811015840119881361199031198672

)

1198811015840

119881

]119889120601

(53)

=

1198722

4120582

4120587119903

11988110158402

1198814 (54)

An important perturbation parameter of inflationmodelsis scalar index 119899

119904which in high dissipative regime is presented

by

119899119904= 1 +

119889 ln 1205752119867

119889 ln 119896

asymp 1 minus

3

4

+

3

4

+ (

119881

1198811015840)(2

I1015840

(120601) +

1199031015840

2119903

)

(55)

where

=

1198722

4120582

4120587119903

1198811015840

1198813[

211988110158401015840

1198811015840minus

1199031015840

119903

] minus 2 (56)

In (55) we have used a relation between small change ofthe number of e-folds and interval in wave number (119889119873 =

minus119889 ln 119896) Running of the scalar spectral index may be foundas

120572119904=

119889119899119904

119889 ln 119896= minus

119889119899119904

119889119873

= minus

119889120601

119889119873

119889119899119904

119889120601

=

1198722

4120582

4120587119903

11988110158401198991015840

119904

1198813 (57)

This parameter is one of the interesting cosmologicalperturbation parameters which is approximately minus0038 byusing observational results [3 4] During inflation epochthere are two independent components of gravitationalwaves (ℎ

times+) with action of massless scalar field which are

produced by the generation of tensor perturbations Tensorperturbations do not couple to the thermal backgroundtherefore gravitational waves are only generated by quan-tum fluctuations the same as in standard fluctuations [86]However if the gravitational sector is modified then theexpression for tensor power spectrum changeswith respect toGeneral Relativity In particular the amplitude of the tensorperturbation on the brane is presented as [91 92]

1198602

119892=

16120587

1198724

4

(

119867

2120587

)

2

1198652(119909) =

16

31198722

4120582

11988121198652(119909) (58)

where the temperature 119879 in extra factor coth[1198962119879] denotesthe temperature of the thermal background of gravitationalwave [93] 119909 = [3119867

21198722

44120587120582]

12 and 119865(119909) = radic1 + 1199092minus

1199092sinhminus1(1119909)minus12 (in high energy limit 119881 ≫ 120582 we have119865(119909) = [27119872

2

416120587120582]

1411986712= [3120587120582

31198722

4]1411988112) Spectral

index 119899119892is presented as

119899119892=

119889

119889 ln 119896(ln[

1198602

119892

coth (1198962119879)]) ≃ minus2 (59)


where119860119892prop 119896119899119892 coth[1198962119879] [93]Using (51) and (58)wewrite

the tensor-scalar ratio in high dissipative regime

119877 (119896) =

1198602

119892

119875119877

1003816100381610038161003816100381610038161003816100381610038161003816119896=1198960

=

16252120587114120590141198811341199031212059834

334sdot 119872152

412058294

sdot exp (2I (120601)) coth( 1198962119879

)

(60)

where 1198960is referred to pivot point [93] and 119875

119877= (254)120575

2

119867

An upper bound for this parameter is given by using Planckdata 119877 lt 011 [3 4]

4 Exponential Potential

In this section we consider our model with the tachyoniceffective potential

119881 (120601) = 1198810exp (minus120572120601) (61)

where parameter 120572 gt 0 is related to mass of tachyon field[94]The exponential form of the potential has characteristicsof tachyon field (119889119881119889120601 lt 0 and 119881(120601 rarr 0) rarr 119881max)We develop our model in high dissipative regime that is119903 ≫ 1 and high energy limit that is 119881 ≫ 120582 for a constantdissipation coefficient Γ From (54) slow-roll parameter inthe present case has the form

=

1198722

4120582

8120587

1205722

1199031198812

0119890minus2120572120601

(62)

Also the other slow-roll parameter is obtained from (56)

= minus

1198722

4

4120587

1205722

1199031198812

0119890minus2120572120601

(63)

Dissipation parameter 119903 = Γ3119867119881 in this case is given by

119903 =radicΓ2

01198722

4120582

12120587

1198902120572120601

1198812

0

(64)

We find the evolution of tachyon field with the help of (16)

120601 (119905) =

1

120572

ln[12057221198810

Γ0

119905 + 119890120572120601119894] (65)

where 120601119894= 120601(119905 = 0) Hubble parameter for our model has

this form

119867 = radic

4120587

31198722

4120582

1198810119890minus120572120601 (66)

Using (21) and (62) the energy density of the radiationfield in high dissipative limit becomes

120588120574=

311987241205722

16Γ0

1198812

0

radic3120587120582

119890minus2120572120601 (67)

and in terms of tachyon field energy density 120588120601becomes

120588120574=

31198722

4

16radic3120587120582

(

1205722

Γ0

)1205882

120601 (68)

From (24) the number of e-folds at the end of inflationby using the potential (61) for our inflationmodel is presentedby

119873total = radic4120587120582

31198722

4

Γ0

120572

(120601119891minus 120601119894) (69)

or equivalently

119873total = radic4120587120582

31198722

4

Γ0

1205722ln(

119881119894

119881119891

) (70)

where 119881119894gt 119881119891 Using (51) and (60) we could find the scalar

spectrum and scalar-tensor ratio

1205752

119867= 119860 exp (minus7

2

120572120601) (71)

where119860 = (16radic375120587)(1198813201198722

41205722)(Γ2

01198722

412058212120587)

12(31198722

4120582

4120587)14 and

119877 = 119861 exp (minus120572120601) (72)

where 119861 = (50120587321205722119881031198727

412058232119879119903)(36120587

2Γ2

01198724

41205824)14 In

the above equation we have used (53) where

I (120601) = minus5

4

ln119881 (73)

These parameters may by restricted by Planck observa-tional data [3ndash6]

5 Intermediate Inflation

Intermediate inflation is denoted by the scale factor

119886 (119905) = 1198860exp (119860119905119891) 0 lt 119891 lt 1 (74)

This model of inflation is faster than power-low inflation andslower than de Sitter inflation In this section we will studyour model in the context of intermediate inflation in twocases (1) Γ = Γ

0and (2) Γ = Γ

1119881(120601) which have been

considered in the literature [50ndash53]

51 Γ = Γ0Case In high dissipative (119903 ≫ 1) and high energy

(119881 ≫ 120582) limits the equations of the slow-roll inflation that is(12) and (13) are simplified as

119881 = (

31205821198722

4

4120587

)

12

119867

120601

2

= minus

Γ

(75)


Inflation field may be derived from above equations in thiscase (Γ = Γ

0)

120601 minus 1206010= 1205731199051198912 (76)

where 120573 = (12120582119872241198602(1 minus 119891)

21205871198912Γ2

0) Using above equation

and the scale factor of intermediate inflation tachyonicpotential and Hubble parameter are presented as

119867(120601) = 119891119860(

120601 minus 1206010

120573

)

(2119891minus2)119891

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

120601 minus 1206010

120573

)

(2119891minus2)119891

(77)

Dissipative parameter 119903 is given by using above equation

119903 =

Γ0

3119867119881

=

4120587Γ0

9 (119891119860)21198722

4120582

(

120601 minus 1206010

120573

)

(4minus4119891)119891

(78)

The slow-roll parameters of themodel in the present casemaybe obtained as

120598 = minus

1198672=

1 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

120578 = minus

119867

=

2 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

(79)

We present the number of e-folds as

119873 = int

119905

1199051

119867119889119905 = 119860([

120601 minus 1206010

120573

]

2

minus [

1206011minus 1206010

120573

]

2

) (80)

where 1206011= 1206010+ 120573((1 minus 119891)119891119860)

12 is the scalar field at thebeginning of the inflation From the above equation we canpresent the scalar field in terms of number of e-folds andintermediate parameters

120601 = 120573(

119873

119860

+

1 minus 119891

119891119860

)

12

+ 1206010 (81)

Nowwe could find the perturbation parameters of themodelThe power spectrum is obtained from (51) (53) and (73)

119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

11988174

1199031212059834

= 1198601(

120601 minus 1206010

120573

)

(14119891minus11)2119891

= 1198601(

119873

119860

+

1 minus 119891

119891119860

)

(14119891minus11)4119891

(82)

where 1198601= 252119872314

4(3120582)158(119891119860)72(4120587)31812059014Γ12

0(1 minus

119891)34 We present the spectral index 119899

119904which is one of the

important perturbation parameters from (55) and (73)

119899119904= 1 +

3

4

120578 minus

17

4

120598 = 1 minus

11 minus 14119891

4119891119860

(

120601 minus 1206010

120573

)

minus2

= 1 minus

11 minus 14119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(83)

40 60 80 10020N

f = 57

ns

090

092

094

096

098

100

102

104

Figure 1 In this graph we plot the spectral index 119899119904in terms of the

number of e-folds119873 We can find best range of spectral index ratiowhere119873 gt 60

Harrison-Zeldovich spectrum that is 119899119904= 1 is obtained

for an exact value of parameter 119891 (ie 119891 = 1114) For119891 lt 1114 we found the 119899

119904lt 1 cases which is compatible

with observational dataIn Figure 1 we plot the spectral index in terms of number

of e-folds where 119891 = 57 For119873 gt 60 we can see the spectralindex is confined to 098 lt 119899

119904lt 1 which is compatible with

Planck data [3 4]Tensor-scalar ratio of the model in this case is presented

by using (60) and (74)

119877 = 1198611(

120601 minus 1206010

120573

)

(minus4119891+1)2119891

coth [ 1198962119879

]

= 1198611(

119873

119860

+

1 minus 119891

119891119860

)

(minus4119891+1)4119891

coth [ 1198962119879

]

= 1198611(

4119891119860

11 minus 14119891

(1 minus 119899119904))

(4119891minus1)4119891

(84)

where 1198611

= (232(4120587)238Γ12

012059014(1 minus 119891)

34

3158119872314

4120582158(119891119860)32)(31198911198602120582)

12 In Figure 2 tensor-

scalar ratio in terms of number of e-folds is plotted where119891 = 56 We could see 60 lt 119873 lt 80 lead to 119877 lt 011

[3 5 6] The expression for the perturbation 120575120601 given by(43) is valid when 119879 gt 119867 The choice of the parameters ofthe model has to be consistent with this condition 119879 gt 119867 InFigure 3 we plot 119879119867 in terms of spectral index that showsthe model is compatible with observational data in warminflation limit 119879 gt 119867 We also checked the high dissipativecondition Γ

0gt 3119867119881 in Figure 4 that we can see agreement

with observational data

52 Γ = Γ1119881(120601) Case Dissipative parameter may be

considered as a function of scalar field [50ndash53] We will studyour model in the context of intermediate inflation where


f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R

Figure 2 In this graph we plot the scalar-tensor ratio 119877 in termsof the number of e-folds119873 We can find best range of tensor-scalarratio where 60 lt 119873 lt 80

096 097 098 099095ns

08

09

10

11

12

13

14

15

TH

Figure 3 In this graphwe plot the temperature toHubble parameterratio 119879119867 in terms of the spectral index 119899

119904 We can find best fit of

warm inflation condition (119879 gt 119867) with the Planck data

Γ03HV

06

08

10

12

14

096 097 098 099095ns

Figure 4 In this graph we plot the dissipative to Hubble parameterratio Γ

03119867119881 in terms of the spectral index 119899

119904 We can find best fit

of high dissipative regime Γ0gt 3119867119881 with the Planck data for three

cases of Γ0

Γ = Γ1119881(120601) In this case the scalar field is determined from

(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)

Therefor the Hubble parameter and potential of the model interms of tachyon potential have the following forms

119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)

Dissipative parameter 119903 is presented by using above equation

119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)

Important parameters of the slow-roll inflation in this caseare presented as

120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)

The number of e-folds is given by

119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)

where 1206011is the tachyon field at the beginning of the inflation

period We find this field where the slow-roll parameter 120598 isequal to one

1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)

From above equations we present the scalar field in terms ofnumber of e-folds and intermediate parameters 119891 and 119860

120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)

Spectral index 119899119904is presented using (55)

119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)


We can find the scale invariant spectrum (Harrison-Zeldovich spectrum) that is 119899

119904= 1 where 119891 = 1720 In

Figure 5 we plot the spectral index in terms of number of e-folds where 119891 = 56 For 119873 gt 60 we can see the spectralindex is confined to 098 lt 119899


Planck data [3 4] Power spectrum and scalar-tensor ratio ofthis model may be obtained from (51) and (60) respectively

119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)

In Figure 6 we can see high dissipative condition agreeswith Planck data In Figure 7 tensor-scalar ratio in terms ofnumber of e-folds is plotted where 119891 = 56 We could see60 lt 119873 lead to 119877 lt 011 [3 5 6]

6 Logamediate Inflation

In this section we will study warm tachyon inflationmodel inthe context of logamediate scenario The scale factor of thismodel is given by

119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)

where 119860 is a positive constant and ] gt 1 We consider thismodel in two cases (1) Dissipative parameter Γ is constant(2) Dissipative parameter is proportional to tachyon fieldpotential 119881(120601)

61 Γ = Γ0Case In this case the scalar field is given by using

(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns

Figure 6 In this graph we plot the dissipation to Hubble parameterratio Γ



high dissipative regime Γ0gt 3119867 with the Planck data for three cases

of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R

Figure 7 In this graph we plot the scalar-tensor ratio 119877 in termsof the number of e-folds119873 We can find best range of tensor-scalarratio where 60 lt 119873


where 120596 = (312058211987224]211986022120587Γ2

0)14 Using above equation the

Hubble parameter and tachyon potential have the followingforms

119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)

We derive the slow-roll parameters in logamediate scenario

120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)

The number of e-folds for present model of inflation ispresented as

119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)

1206011= 1206010+ (2120596(] + 1))(119860])(1+])2(1minus]) is the inflation at the

beginning of the inflation era From above equation the scalarfield is presented in terms of number of e-folds

120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)

Dissipative parameter 119903 is given by

119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)

Power spectrum and scalar-tensor ratio of logamediate infla-tion are derived from (51) and (60)

119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)

By using (55) we could find the spectral index 119899119904

119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)

In Figure 8 the dependence of spectral index on the numberof e-folds is shown (for ] = 50 and ] = 5 cases) It isobserved that the small values of the number of e-folds areassured for large values of ] parameter This figure shows thescale invariant spectrum (Harrison-Zeldovich spectrum ie119899119904= 1) could be approximately obtained for (] 119873) = (50 60)

From above equation and (102) a relation between scalar-tensor ratio and spectral index is obtained

119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)

In Figure 9 two trajectories in the 119899119904-119877 plane are shown

There is a range of values of 119877 and 119899119904which is compatible

with the Planck data


62 Γ = Γ1119881(120601) Warm tachyon inflation in the context of

logamediate scenario with dissipation Γ = Γ1119881(120601) will be

studied In this case we can find the scalar field using (75) and(95)

120601 minus 1206010=

2

radicΓ1

11990512 (106)

We also derive the Hubble parameter tachyonic potential anddissipative parameter 119903 from above equation

119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)

The slow-roll parameters 120598 and 120578 are presented respec-tively

120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)

Number of e-folds at the end of inflation is given by

119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)

where 1206011is beginning inflation At the beginning point of

inflation period we have 120598 = 1 therefore the inflation in thispoint has the following form

1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)

Using above equation we could find the scalar field in termsof number of e-folds

1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)

Important perturbation parameters119875119877(power spectrum) and

119877 (scalar-tensor ratio) could be derived in terms of scalar fieldand number of e-folds

119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)

The spectral index 119899119904is derived in this case as

119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)

In Figure 10 the dependence of spectral index on the numberof e-folds is shown (for ] = 50 and ] = 5 cases) It is observedthat the small values of number of e-folds are assured for largevalues of ] parameter This figure shows the scale invariantspectrum (Harrison-Zeldovich spectrum ie 119899

119904= 1) could

be approximately obtained for (] 119873) = (50 60) From aboveequation and (112) we find the tensor-scalar ratio in terms ofspectral index

119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)


There is a range of values of119877 and 119899119904which is compatible with

14 Advances in High Energy Physicsns

00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50

Figure 8 Spectral index in terms of number of e-folds ] = 50 bydashed line and ] = 5 by green line

120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R

Figure 9 Tensor-scalar ratio in terms of spectral index 119899119904 ] = 50

by dashed line and ] = 5 by green line

the Planck data In order to produce our plots we assumesome values for the several parameters (119891119860 ] 120582 Γ

0 Γ1) for

the above cases studied these parameters coincide with 1120590confidence level of Planck data We will use a new methodto constrain the parameters of the model in future worksIn Figure 12 we plot the tachyonic potential in terms of thespectral index 119899

119904in logamediate case We can find the best fit

of high energy limit119881 ≫ 120582with the Planck data that we haveused in this paper

7 Conclusion and Discussion

Tachyon inflation model on the brane with everlasting formof potential 119881(120601) = 119881

0exp(minus120572120601) which agrees with tachyon

potential properties has been studied The main problem ofthe inflation theory is how to attach the universe to the endof the inflation period One of the solutions of this problemis the study of inflation in the context of warm inflation[11 12] In this scenario radiation is produced during inflation

ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100

Figure 12 In this graph we plot the tachyonic potential in terms ofthe spectral index 119899

119904 We can find best fit of high energy limit119881 ≫ 120582



periodwhere its energy density is kept nearly constantThis isphenomenologically fulfilled by introducing the dissipationterm Γ The study of warm inflation model as a mechanismthat gives an end for the tachyon inflation motivated usto consider the warm tachyon inflation model We notethat the I(120601) factor (47) which appears in the perturbationparameters (51) (55) (57) and (60) in high energy limit (119881 ≫120582) for warm tachyon inflation model on the brane has animportant differencewith the same factorwhichwas obtainedfor usual warm tachyon inflation model [50ndash53]

I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)

The density square term in the effective Einstein equationon the brane is responsible for this difference Thereforethe perturbation parameters which may be constrained byPlanck observational data are modified due to the effect ofdensity square term in effective Einstein equation Also theslow-roll parameters (19) and (20) which are derived in thebackground level are modified because of the density squareterm in modified Friedmann equation (15) The slow-rollparameters appeared in the perturbation parameters (51)(55) (57) (59) and (60) As have been shown in [50ndash53] theslow-roll parameters of warm tachyon inflation model havethe forms

120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)

These parameters are obviously different from the slow-rollparameters (19) and (20) Perturbation parameters of warmtachyon inflation model have following from [50ndash53]

120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)

The above parameters are also different from the perturbationparameters of our model on the brane (51) (55) (57) (59)

and (60) because of the density square term in the effectiveEinstein equation on the brane So from above discussionwe know the density square term in the effective Einsteinequation on the brane gives the significant contributionsto the observable parameters 119875

119877 119877 119899

119904 and 120572

119904 Also the

different observable perturbation parameters for the modelsof nontachyon warm inflation and nontachyon warm infla-tion model on the brane are presented in [49] and [44]respectively

In tachyon Randall-Sundrum brane world scenario Ein-steinrsquos equation and therefore the Friedmann equation aremodified Warm tachyon inflation parameters on the branehave important differences with the same parameters whichwere presented for usual warm inflation model [44] becauseof this modification The density square term in the effectiveEinstein equation on the brane is responsible for this differ-ence Therefore the perturbation parameters which may beconstrained by Planck observational data are modified due tothe effect of density square term in effective Einstein equationandmodification of tachyonic scalar field equation of motion(EMO) instead of normal scalar fields EMO In this paper wehave considered warm tachyon inflationary universe modelon the brane In the slow-roll approximation the generalrelation between energy density of radiation and energydensity of tachyon field is presented In the longitudinal gaugeand the slow-roll limit the explicit expressions for the tensor-scalar ratio 119877 scalar spectrum 119875

119877index 119899

119904and its running

120572119904 have been presented We have developed our specific

model by exponential potential with a constant dissipationcoefficient In this case we have found perturbation param-eters and constrained these parameters Planck observationaldata Intermediate and logamediate inflation are consideredfor two cases of dissipative parameters (1) Γ is constantparameter (2) Γ is a function of tachyon field In these twocases we have found that the models are compatible withobservational data Harrison-Zeldovich spectrum that is119899119904= 1 is obtained exactly by one parameter in intermediate

scenario (119891 = 1114 for Γ = Γ0case and 119891 = 1720

for Γ = Γ(120601)) In logamediate scenario we have presentedapproximately scale invariant spectrum that is 119899 ≃ 1 where(119873 ]) = (60 50)

Appendix

In this paper we have studied the model in natural unit(ℎ2120587 = 119888 = 1) therefore we have ([mass] = 119872 [time] = 119879and [length] = 119871 where [119860]means dimension of ldquo119860rdquo)

[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)

where119881 and119875 are potential and pressurewith dimension1198724From (11) we have

[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)

It appears that tachyon scalar field has dimension119872minus1 whichagrees with the tachyonic potential (61) In (13) right-handside and left-hand side have dimension1198724

[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)

In (16) we have used dimensionless parameter 119903 =

(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)

119881Γ has dimension time (119867minus1) therefore in our paper wehave used Γ119881 instead of Γ We note that from abovediscussion that 120594 in (41) has dimension119872minus2 which leads to[119862] = 119872

minus2 in (45) and (49) has correct dimension

[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)

In (47) we have 2119867+Γ119881 where the analysis of dimension isgiven by

[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)

Equation (49) has correct dimension for cold inflation wehave [120575

119867] = ([119867][

120601])[120575120601] = 1 in warm inflation also we

have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)

We note that (50) is in momentum space [66 86] Henceinserting (50) into (49) means that (51) and the followingequations are in momentum space

Competing Interests

The authors declare that they have no competing interests

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[3] P A R Ade N AghanimM Arnaud et al ldquoPlanck 2015 resultsXX Constraints on inflationrdquo httpsarxivorgabs150202114

[4] G Hinshaw D Larson E Komatsu et al ldquoNine-yearWilkinsonmicrowave anisotropy probe (WMAP) observations cosmolog-ical parameter resultsrdquo The Astrophysical Journal SupplementSeries vol 208 no 2 p 19 2013

[5] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XVI Cosmological parametersrdquo Astronomy ampAstrophysics vol 571 article A16 2014

[6] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 2014

[7] J H Traschen and R H Brandenberger ldquoParticle productionduring out-of-equilibrium phase transitionsrdquo Physical ReviewD vol 42 no 8 pp 2491ndash2504 1990

[8] L Kofman A Linde and A A Starobinsky ldquoReheating afterinflationrdquo Physical Review Letters vol 73 no 24 pp 3195ndash31981994

[9] Y Shtanov J Traschen and R Brandenberger ldquoUniversereheating after inflationrdquo Physical Review D vol 51 no 10 pp5438ndash5455 1995

[10] L Kofman A Linde and A A Starobinsky ldquoTowards thetheory of reheating after inflationrdquo Physical Review D vol 56no 6 pp 3258ndash3295 1997

[11] A Berera ldquoWarm inflationrdquo Physical Review Letters vol 75 no18 pp 3218ndash3221 1995

[12] A Berera ldquoInterpolating the stage of exponential expansion inthe early universe possible alternative with no reheatingrdquo Phys-ical ReviewDmdashParticles Fields Gravitation and Cosmology vol55 no 6 pp 3346ndash3357 1997

[13] I G Moss ldquoPrimordial inflation with spontaneous symmetrybreakingrdquo Physics Letters B vol 154 no 2-3 pp 120ndash124 1985

[14] A Berera ldquoWarm inflation in the adiabatic regimemdasha modelan existence proof for inflationary dynamics in quantum fieldtheoryrdquo Nuclear Physics B vol 585 no 3 pp 666ndash714 2000

[15] Y-F Cai J B Dent and D A Easson ldquoWarm dirac-born-infeldinflationrdquo Physical Review D vol 83 no 10 Article ID 1013012011

[16] R Cerezo and J G Rosa ldquoWarm inflectionrdquo Journal of HighEnergy Physics vol 2013 article 24 2013

[17] S Bartrum A Berera and J G Rosa ldquoGravitino cosmologyin supersymmetric warm inflationrdquo Physical Review D vol 86Article ID 123525 2012

[18] M Bastero-Gil A Berera R O Ramos and J G Rosa ldquoWarmbaryogenesisrdquo Physics Letters Section B Nuclear ElementaryParticle and High-Energy Physics vol 712 no 4-5 pp 425ndash4292012


[19] M Bastero-Gil A Berera and J G Rosa ldquoWarming up brane-antibrane inflationrdquo Physical Review D vol 84 no 10 ArticleID 103503 2011

[20] A Berera M Gleiser and R O Ramos ldquoStrong dissipativebehavior in quantum field theoryrdquo Physical Review D vol 58Article ID 123508 1998

[21] A Berera M Gleiser and R O Ramos ldquoA first principleswarm inflation model that solves the cosmological horizon andflatness problemsrdquo Physical Review Letters vol 83 no 2 pp264ndash267 1999

[22] M Bastero-Gil A Berera and R O Ramos ldquoDissipation coef-ficients from scalar and fermion quantum field interactionsrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 9article 033 2011

[23] M Bastero-Gil A Berera R O Ramos and J G Rosa ldquoGeneraldissipation coefficient in low-temperature warm inflationrdquoJournal of Cosmology and Astroparticle Physics vol 2013 no 1article 016 2013

[24] A Sen ldquoRolling Tachyonrdquo Journal of High Energy Physics vol204 p 48 2002

[25] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002

[26] M Sami P Chingangbam andTQureshi ldquoAspects of tachyonicinflation with an exponential potentialrdquo Physical Review D vol66 no 4 Article ID 043530 2002

[27] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002

[28] C Armendariz-Picon T Damour and V Mukhanov ldquok-Inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999

[29] K Akama Gauge Theory and Gravitation vol 176 of LectureNotes in Physics Springer Berlin Germany 1982

[30] V A Rubakov and M E Shaposhnikov ldquoAn exotic class ofKaluza-Klein modelsrdquo Physics Letters B vol 159 no 1 pp 22ndash25 1985

[31] NArkani-Hamed SDimopoulos andGDvali ldquoThehierarchyproblem and new dimensions at a millimeterrdquo Physics LettersSection B Nuclear Elementary Particle andHigh-Energy Physicsvol 429 no 3-4 pp 263ndash272 1998

[32] M Gogberashvili ldquoOur world as an expanding shellrdquo Euro-physics Letters vol 49 no 3 pp 396ndash399 2000

[33] L Randall and R Sundrum ldquoLargemass hierarchy from a smallextra dimensionrdquo Physical Review Letters vol 83 no 17 pp3370ndash3373 1999

[34] L Randall and R Sundrum ldquoAn alternative to compactifica-tionrdquo Physical Review Letters vol 83 no 23 pp 4690ndash46931999

[35] J Polchinski ldquoDirichlet branes and Ramond-Ramond chargesrdquoPhysical Review Letters vol 75 no 26 pp 4724ndash4727 1995

[36] P Horava and E Witten ldquoHeterotic and type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3 pp506ndash524 1996

[37] A Lukas B A Ovrut and D Waldram ldquoCosmological solu-tions of Horava-Witten theoryrdquo Physical Review D vol 60 no8 Article ID 086001 1999

[38] T Shiromizu K-I Maeda and M Sasaki ldquoThe Einsteinequations on the 3-brane worldrdquo Physical Review D vol 62 no2 Article ID 024012 2000

[39] D Langlois RMaartensM Sasaki andDWands ldquoLarge-scalecosmological perturbations on the branerdquo Physical Review Dvol 63 no 8 Article ID 084009 2001

[40] P R Ashcroft C van de Bruck and A-C Davis ldquoSuppressionof entropy perturbations in multifield inflation on the branerdquoPhysical Review D vol 66 no 12 Article ID 121302 5 pages2002

[41] R Maartens ldquoCosmological dynamics on the branerdquo PhysicalReview D vol 62 no 8 Article ID 084023 14 pages 2000

[42] C Gordon and R Maartens ldquoDensity perturbations in thebrane-worldrdquo Physical Review D vol 63 no 4 Article ID044022 2001

[43] D Folini and R Walder ldquoTheoretical predictions forthe cold part of the colliding wind interaction zonerdquohttparxivorgabsastro-ph0012132

[44] M A Cid S del Campo and R Herrera ldquoWarm inflation onthe branerdquo Journal of Cosmology and Astroparticle Physics vol2007 no 10 p 5 2007

[45] J M Cline C Grojean and G Servant ldquoCosmological expan-sion in the presence of an extra dimensionrdquo Physical ReviewLetters vol 83 no 21 pp 4245ndash4248 1999

[46] P Brax and C van de Bruck ldquoCosmology and brane worlds areviewrdquoClassical and QuantumGravity vol 20 no 9 pp R201ndashR232 2003

[47] T Clifton P G Ferreira A Padilla and C Skordis ldquoModifiedgravity and cosmologyrdquo Physics Reports vol 513 no 1ndash3 pp 1ndash189 2012

[48] S del Campo and R Herrera ldquoWarm inflation in the DGPbrane-worldmodelrdquo Physics Letters B vol 653 no 2ndash4 pp 122ndash128 2007

[49] H P de Oliveira ldquoDensity perturbations in warm inflation andCOBEnormalizationrdquo Physics Letters B vol 526 no 1-2 pp 1ndash82002

[50] R Herrera S del Campo and C Campuzano ldquoTachyonwarm inflationary universe modelsrdquo Journal of Cosmology andAstroparticle Physics vol 2006 no 10 p 9 2006

[51] M R Setare and V Kamali ldquoCosmological perturbationsin warm-tachyon inflationary universe model with viscouspressure on the branerdquo Journal of High Energy Physics vol 2013no 3 article 066 2013

[52] M R Setare and V Kamali ldquoTachyon warm-logamediate infla-tionary universe model in a high dissipative regimerdquo PhysicalReview D vol 87 no 8 Article ID 083524 2013

[53] A Deshamukhya and S Panda ldquoWarm tachyonic inflation ina warped backgroundrdquo International Journal of Modern PhysicsD vol 18 no 14 pp 2093ndash2106 2009

[54] T Clifton and J D Barrow ldquoDecay of the cosmic vacuumenergyrdquo httpsarxivorgabs14125465

[55] A Cid G Leon and Y Leyva ldquoIntermediate acceleratedsolutions as generic late-time attractors in a modified Jordan-Brans-Dicke theorrdquo Journal of Cosmology and AstroparticlePhysics vol 2016 no 2 article 027 2016

[56] A K Sanyal ldquoIf Gauss-Bonnet interaction plays the role of darkenergyrdquo Physics Letters B vol 645 no 1 pp 1ndash5 2007

[57] T Koivisto and D F Mota ldquoCosmology and astrophysicalconstraints of Gauss-Bonnet dark energyrdquo Physics Letters B vol644 no 2-3 pp 104ndash108 2007

[58] T Koivisto and D F Mota ldquoGauss-Bonnet quintessencebackground evolution large scale structure and cosmologicalconstraintsrdquo Physical Review D vol 75 Article ID 023518 2007

[59] S Mignemi and N R Stewart ldquoCharged black holes in effectivestring theoryrdquo Physical Review D vol 47 no 12 pp 5259ndash52691993


[60] S Nojiri S D Odintsov and M Sasaki ldquoGauss-Bonnet darkenergyrdquo Physical Review D vol 71 no 12 Article ID 1235092005

[61] G Cognola E Elizalde S Nojiri S D Odintsov and SZerbini ldquoDark energy in modified Gauss-Bonnet gravity late-time acceleration and the hierarchy problemrdquo Physical ReviewD vol 73 no 8 Article ID 084007 2006

[62] I Antoniadis J Rizos and K Tamvakis ldquoSingularity-free cos-mological solutions of the superstring effective actionrdquo NuclearPhysics Section B vol 415 no 2 pp 497ndash514 1994

[63] J D Barrow and A R Liddle ldquoInfluence of induced magneticfields on the static properties of Josephson-junction arraysrdquoPhysical Review D vol 47 pp 5219ndash5229 1993

[64] A Vallinotto E J Copeland E W Kolb A R Liddle andD A Steer ldquoInflationary potentials yielding constant scalarperturbation spectral indicesrdquo Physical Review D vol 69 no10 Article ID 103519 8 pages 2004

[65] A A Starobinsky ldquoInflaton field potential producing an exactlyflat spectrum of adiabatic perturbationsrdquo JETP Letters vol 82no 4 pp 169ndash173 2005

[66] M R Setare and V Kamali ldquoTachyon warm-intermediateinflationary universe model in high dissipative regimerdquo Journalof Cosmology and Astroparticle Physics vol 2012 no 8 article034 16 pages 2012

[67] J D Barrow ldquoVarieties of expanding universerdquo Classical andQuantum Gravity vol 13 no 11 pp 2965ndash2975 1996

[68] J D Barrow ldquoSlow-roll inflation in scalar-tensor theoriesrdquoPhysical Review D vol 51 p 2729 1995

[69] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[70] P G Ferreira and M Joyce ldquoCosmology with a primordialscaling fieldrdquoPhysical ReviewD vol 58 no 2 Article ID0235031998

[71] J D Barrow and N J Nunes ldquoDynamics of lsquologamediatersquoinflationrdquo Physical Review D vol 76 no 4 Article ID 0435012007

[72] J Yokoyama and K Maeda ldquoOn the dynamics of the power lawinflation due to an exponential potentialrdquo Physics Letters B vol207 no 1 pp 31ndash35 1988

[73] R Herrera ldquoWarm inflationary model in loop quantum cos-mologyrdquo Physical Review D vol 81 Article ID 123511 2010

[74] K Xiao and J Y Zhu ldquoA phenomenology analysis of the tachyonwarm inflation in loop quantum cosmologyrdquo Physics Letters Bvol 699 no 4 pp 217ndash223 2011

[75] R Herrera and E SanMartin ldquoWarm-intermediate inflationaryuniverse model in braneworld cosmologiesrdquo The EuropeanPhysical Journal C vol 71 article 1701 2011

[76] R Herrera and M Olivares ldquoWarm-logamediate inflationaryuniverse modelrdquo International Journal of Modern Physics D vol21 no 5 Article ID 1250047 13 pages 2012


[78] A Cid ldquoOn the consistency of tachyon warm inflation withviscous pressurerdquo Physics Letters B vol 743 pp 127ndash133 2015

[79] A Sen ldquoTachyon condensation on the brane antibrane systemrdquoJournal of High Energy Physics vol 1998 no 08 1998

[80] J M Bardeen ldquoGauge-invariant cosmological perturbationsrdquoPhysical Review D Particles and Fields Third Series vol 22 no8 pp 1882ndash1905 1980

[81] V F Mukhanov H A Feldman and R H BrandenbergerldquoTheory of cosmological perturbationsrdquo Physics Reports AReview Section of Physics Letters vol 215 no 5-6 pp 203ndash3331992

[82] A A Starobinsky and J Yokoyama ldquoDensity fluctuations inBrans-Dicke inflationrdquo in Proceedings of the 4th Workshop onGeneral Relativity and Gravitation (JGRG rsquo94) p 381 1994httpsinspirehepnetsearchp=find+eprint+GR-QC9502002

[83] A A Starobinsky S Tsujikawa and J Yokoyama ldquoCosmo-logical perturbations from multi-field inflation in generalizedEinstein theoriesrdquo Nuclear Physics B vol 610 no 1-2 pp 383ndash410 2001

[84] H P de Oliveira and S E Joras ldquoPerturbations in warminflationrdquo Physical Review D vol 64 Article ID 063513 2001

[85] L M H Hall I G Moss and A Berera ldquoScalar perturbationspectra from warm inflationrdquo Physical Review D vol 69 no 8Article ID 083525 2004

[86] A N Taylor and A Berera ldquoPerturbation spectra in the warminflationary scenariordquo Physical Review D vol 62 no 8 ArticleID 083517 2000

[87] V N Lukash ldquoProduction of phonons in an isotropic universerdquoSoviet PhysicsmdashJETP vol 52 pp 807ndash814 1980

[88] H Kodama andM Sasaki ldquoCosmological perturbation theoryrdquoProgress of Theoretical Physics Supplement vol 78 pp 1ndash1661984

[89] J E Lidsey A R Liddle E W Kolb E J Copeland T Barreiroand M Abney ldquoReconstructing the inflaton potentialmdashanoverviewrdquoReviews ofModern Physics vol 69 no 2 pp 373ndash4101997

[90] B A Bassett S Tsujikawa and D Wands ldquoInflation dynamicsand reheatingrdquo Reviews of Modern Physics vol 78 no 2 pp537ndash589 2006

[91] D Langlois R Maartens and D Wands ldquoGravitational wavesfrom inflation on the branerdquo Physics Letters B vol 489 no 3-4pp 259ndash267 2000

[92] R Herrera N Videla and M Olivares ldquoWarm intermediateinflation in the RandallndashSundrum II model in the light ofPlanck 2015 and BICEP2 results a general dissipative coeffi-cientrdquoTheEuropean Physical Journal C vol 75 article 205 2015

[93] K Bhattacharya S Mohanty and A Nautiyal ldquoEnhancedpolarization of the cosmic microwave background radiationfrom thermal gravitational wavesrdquo Physical Review Letters vol97 no 25 Article ID 251301 2006

[94] M Fairbairn andMH Tytgat ldquoInflation from a tachyon fluidrdquoPhysics Letters B vol 546 no 1-2 pp 1ndash7 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

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Journal of



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OpticsInternational Journal of



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Journal of



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Biophysics


ThermodynamicsJournal of


inflation papers such as [20 21]More recently there has beenconsiderable explicit calculation fromQuantumFieldTheory(QFT) that explicitly computes all these relevant decay andscattering rates in warm inflation models [22 23]

The inflation era in the early evolution of the universecould be described by tachyonic field associated with unsta-ble D-brane because of the tachyon condensation near themaximum of the effective potential [24ndash26] At the late timestachyonic fields may add a nonrelativistic fluid or a new formof cosmological darkmatter to the universe [27]The tachyoninflation is a k-inflation model [28] for scalar field 120601 with apositive potential 119881(120601) Tachyon potentials have two specialproperties firstly a maximum of this potential is obtainedwhere 120601 rarr 0 and second property is the minimum ofthese potentials which is obtained where 120601 rarr infin If thetachyon field starts to roll down the potential then universedominated by a new form of matter will smoothly evolvefrom inflationary universe to an era which is dominatedby a nonrelativistic fluid [27] So we can explain the phaseof acceleration expansion (inflation) in terms of tachyonfield Tachyon fields in the ordinary (cold) tachyon inflationframework after slow-roll epoch evolve towards minimumof the potential without oscillating about it [26] so the(p)reheating mechanism in cold tachyon inflation does notwork Warm tachyon inflation is a picture where there aredissipative effects playing during inflation As a result of thisthe inflation evolves in a thermal radiation bath thereforethe reheating problem of cold tachyon inflation [26] can besolved in the framework of warm tachyon inflation We notethat the cold tachyonic inflation era cannaturally endwith thecollision of the two branes In this situation we do not needwarm inflation If the collision of two branes does not arisenaturally warm inflation is perfectly good scenario that cansolve the problem of end of thachyon inflation

We may live on a brane which is embedded in a higherdimensional universe This realization has significant impli-cations to cosmology [29ndash34] In this scenario which ismotivated by string theory gravity (closed string modes) canpropagate in the bulk while the standard model of particles(matter fields which are related to open string modes) isconfined to the lower-dimensional brane [35ndash37] In termsof Randall-Sundrum suggestion there are two similar butphenomenologically different brane world scenarios [33 34]In this paper we will consider that the brane world modelcorresponds to the Randall-Sundrum II brane world [34]

The brane world picture is described by the followingaction [29ndash34]

119878 =

1

1205812int1198895119909radicminus119892(119877

5+

12

1198972) minus 120582int119889

4119909radicminus119892brane

+ int1198894119909radicminus119892brane119871matter

(1)

In this scenario we have a 3-brane universe which is locatedin the 5D Anti-de Sitter (AdS) space-time where this space-time is effectively compactified with curvature scale 119897 ofAdS space-time 1198775 is the Ricci scalar in five dimensionsand 120581 = 8120587119866

5= 8120587119872

3

5 where 119866

5is the 5D Newtonrsquos

constant and1198725is Planck scale in five dimensions 120582 is the

tension of the brane and if we have no matter on the brane1205812120582 = 6119897 where the brane becomes Minkowski space-

time In the brane world model the gravity could propagatein the 5D space-time and the Newtonian gravity in fourdimensions is reproduced at the scales larger than 119897 on thebrane 4D Einsteinrsquos equation projected onto the brane hasbeen found in [38] Friedmann equation and the equations oflinear perturbation theory [39 40] may be modified by theseprojections Einsteinrsquos equations which are projected onto thebrane with cosmological constant andmatter fields which areconfined to 3-brane tension have the following form [38]

119866120583] = minusΛ 4119892120583] + (

8120587

1198722

4

)119879120583] + (

8120587

1198723

5

)

2

120587120583] minus 119864120583] (2)

where 119864120583] is a projection of 5D weyl tensor 119879

120583] is energy

density tensor on the brane1198724= radic119897119872

3

5is the Planck scales

in 4D and 120587120583] is a tensor quadratic in 119879120583]

120587120583] = minus

1

4

119879120583120572119879120572

] +1

12

119879120572

120572119879120583]

+

1

24

(3119879120572120573119879120572120573minus (119879120572

120572)2) 119892120583]

(3)

Cosmological constant Λ4on the brane in terms of 3-

brane tension 120582 and 5D cosmological constant Λ is given by

Λ4=

4120587

1198723

5

(Λ +

4120587

31198723

5

1205822) (4)

4D Planck scale is determined by 5D Planck scale as

1198724= radic

3

4120587

(

1198722

5

radic120582

)1198725= radic

4120587120582

3

119897 (5)

The natural boundary conditions to specify the perturbationsof this model are imposed where the perturbations do notdiverge at the horizon of the AdS space-time and we assumethat theWeyl curvature may be neglected On the large scalethe behavior of cosmological perturbations on the braneworld models is the same as a closed system on the branewithout the effects of the perturbations along the extradi-mensions in the bulk [41ndash43] On the large scale limit theperturbation parameters of inflation models have a completeset of perturbed equations on the brane which may be solvedin quasi-stable and slow-roll limit [41ndash44] The study of theperturbation evolution of warm inflation in the brane worldmodel on the large scale by using equations solely on thebrane and without solving the bulk perturbations is found in[44] This model has a complete set of perturbed equationson the brane We would like to study the warm tachyoninflation model on the brane using this approach Thereforewewill consider the linear cosmological perturbations theoryfor warm tachyon inflation model on the brane In spatiallyflat FRW model the Friedmann equation by using Einsteinrsquosequation (3) has following form [38]

1198672=

Λ4

3

+ (

8120587

31198722

4

)120588 + (

4120587

31198723

5

)1205882+

120576

1198864 (6)




4to be vanished


1198672=

8120587

31198722

4

120588 (1 +

120588

2120582

) (7)




120574is the


1198672=

8120587

31198722

4

(120588120601+ 120588120574) (1 +

120588120601+ 120588120574

2120582

) (8)




0exp(119860119905119891) (119860 gt 0 0 lt 119891 lt 1)



0exp(119867

0119905) (1198860 1198670gt 0)



119904= 1






2 The Model


119871 = radicminus119892(

119877

16120587119866




119879120583

] =120597119871

120597 (120597120583120601)

120597]120601 minus 119892120583

]119871 = diag (minus120588120601 119875120601 119875120601 119875120601) (10)


120588120601=

119881 (120601)

radic1 minus120601

2

119875120601= minus119881 (120601)

radic1 minus

120601

2

(11)



1198672=

8120587

31198722

4

[

[

[

119881 (120601)

radic1 minus120601

2

+ 120588120574

]

]

]

sdot[

[

[

1 +

1

2120582

(

119881 (120601)

radic1 minus120601

2

+ 120588120574)]

]

]

(12)

120601+ 3119867(119875

120601+ 120588120601) = minusΓ

120601

2

997904rArr

120601

1 minus120601

2+ 3119867

120601 +

1198811015840

119881

= minus

Γ

119881

radic1 minus

120601

2120601

(13)

120574+ 4119867120588

120574= Γ

120601

2

(14)




120601 ≪ 1 and 120601 ≪



120574≪ Γ

120601


1198672=

8120587

31198722

4

119881(1 +

119881

2120582

) (15)

3119867 (1 + 119903)120601 = minus

1198811015840

119881

(16)


119881

Γ

≪ 120591 ≪ 119867minus1997904rArr

Γ ≫ 119867119881

(17)



120588120574=

Γ120601

2

4119867

=

1198722

4119903

32120587 (1 + 119903)2(1 + 1198812120582)

(

1198811015840

119881

)

2

= 1205901198794

119903 (18)



120598 = minus

1198672≃

1198722

4

16120587

11988110158402

(1 + 119903) 1198813

1 + 119881120582

(1 + 1198812120582)2 (19)

120578 = minus

119867

≃

1198722

4

8120587

1198811015840

1198812(1 + 119903) [1 + 1198812120582]

[

211988110158401015840

1198811015840minus

1198811015840

119881

minus

1199031015840

(1 + 119903)

+

1198811015840

120582 + 119881

] minus 2120598

(20)


120574is


120588120574=

119903

2 (1 + 119903)

[1 + 1205881206012120582]

[1 + 120588120601120582]

120588120601120598

≃

119903

2 (1 + 119903)

[1 + 1198812120582]

[1 + 119881120582]

119881120598

(21)


2 (1 + 119903)

119903

120588120574lt

1 + 1205881206012120582

1 + 120588120601120582

120588120601 (22)


1198722

4

8120587

[

1198811015840

119891

119881119891

]

2

1 + 119881119891120582

(1 + 1198811198912120582)

2

1

119881119891

≃ 2 (1 + 119903119891) (23)



119873 = int

120601119891

120601lowast

119867119889119905 = int

120601119891

120601lowast

119867

120601

119889120601

= minus

8120587

1198722

4

int

120601119891

120601lowast

1198812

1198811015840(1 + 119903) [1 +

119881

2120582

] 119889120601

(24)


3 Perturbation



1198891199042= (1 + 2Φ) 119889119905

2minus 1198862(119905) (1 minus 2Ψ) 120575119894119895

119889119909119894119889119909119895 (25)


120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601


1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(26)


120575120601 (119909)

= int

1198893119896

(2120587)3[119890119894119896119909120575120601 (119896 119905) 119886

119896+ 119890minus119894119896119909120575120601 (119896 119905) 119886

lowast

119896]

(27)



Φ = Ψ (28)

Φ + 119867Φ =

4120587

1198722

4

[

[

[

minus

4120588120574119886V3119896

+

119881120601

radic1 minus120601

2

120575120601]

]

]

sdot[

[

[

1 +

1

120582

[

[

[

120588120574+

119881

radic1 minus120601

2

]

]

]

]

]

]

(29)

120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601 + [

1198962

1198862+ (

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]

sdot 120575120601 minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(30)

(

120575120588120574) + 4119867120575120588

120574+

4

3

119896119886120588120574V minus 4120588

120574Φ minus

120601

2

Γ1015840120575120601

minus Γ120601

2

[2 (120575120601) minus 3

120601Φ] = 0

(31)

V + 4119867V +119896

119886

[Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601] = 0 (32)


119895= minus(119894119886119896

119869119896)V119890119894119896119909 119895 = 1 2 3) [80 81]


120575119878 = 119890 = minus119881120601119879120575120601 minus 119881

119879119879120575119879 (33)



120575120588120574) ≪




Φ ≃

4120587

1198671198722

4

[minus

4120588120574119886V3119896

+ 119881120601120575120601] [1 +

119881

120582

] (34)

[3119867 +

Γ

119881

]120575120601 + [(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

≃ [120601

Γ

119881

minus 2(

1198811015840

119881

)]Φ

(35)

120575120588120574

120588120574

≃

Γ1015840

Γ

120575120601 minus 3Φ (36)

V ≃ minus119896

4119886119867

(Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601) (37)


Φ

=

4120587

1198722

4

(

119881120601

119867

)[1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

) 120575120601

(38)


(3119867 +

Γ

119881

)

119889

119889119905

= (3119867 +

Γ

119881

)120601

119889

119889120601

= minus

1198811015840

119881

119889

119889120601

(39)


(120575120601)1015840

120575120601

=

119881

1198811015840[(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

+

4120587

1198722

4

(minus120601

Γ

119881

+ 2(

1198811015840

119881

)

1015840

)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)]

(40)


120594 =

119881120575120601

1198811015840exp [int 1

3119867 + Γ119881

(

Γ

119881

)

1015840

119889120601] (41)


1205941015840

120594

=

(120575120601)1015840

120575120601

minus

119881

1198811015840(

1198811015840

119881

)

1015840

+

(Γ119881)1015840

3119867 + Γ119881

(42)


1205941015840

120594

=

4120587

1198722

4

(minus

119881120601

1198811015840

Γ

119881

+ 2)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)

(43)


1205941015840

120594

= minus

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

(44)



2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

119889120601) (45)


120575120601 = 119862

1198811015840

119881

exp (I (120601)) (46)

where

I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

)]119889120601

(47)





119894


R = Φ minus 119896minus1119886119867V

119890 = 120575119875 minus 1198882

119904120575120588

(48)





120575119867=

2

5

1198722

4

119881 exp (minusI (120601))1198811015840

120575120601

=

2

15

1198722

4

exp (minusI (120601))119867119903

120601

120575120601

(49)




(120575120601)2≃

119896119865119879119903

21205872 (50)



1205752

119867=

2radic3

7512058721198724

4


119879119903

119867

(51)

or equivalently

1205752

119867=

41198725

412058212


where

I (120601) = minusint[1

3119867119903

(

Γ

119881

)

1015840

+

9

4

(1 minus

(ln Γ)1015840 1198811015840119881361199031198672

)

1198811015840

119881

]119889120601

(53)

=

1198722

4120582

4120587119903

11988110158402

1198814 (54)



by

119899119904= 1 +

119889 ln 1205752119867

119889 ln 119896

asymp 1 minus

3

4

+

3

4

+ (

119881

1198811015840)(2

I1015840

(120601) +

1199031015840

2119903

)

(55)

where

=

1198722

4120582

4120587119903

1198811015840

1198813[

211988110158401015840

1198811015840minus

1199031015840

119903

] minus 2 (56)



120572119904=

119889119899119904

119889 ln 119896= minus

119889119899119904

119889119873

= minus

119889120601

119889119873

119889119899119904

119889120601

=

1198722

4120582

4120587119903

11988110158401198991015840

119904

1198813 (57)




1198602

119892=

16120587

1198724

4

(

119867

2120587

)

2

1198652(119909) =

16

31198722

4120582

11988121198652(119909) (58)


21198722

44120587120582]

12 and 119865(119909) = radic1 + 1199092minus


2

416120587120582]

1411986712= [3120587120582

31198722

4]1411988112) Spectral


119899119892=

119889

119889 ln 119896(ln[

1198602

119892

coth (1198962119879)]) ≃ minus2 (59)




119877 (119896) =

1198602

119892

119875119877

1003816100381610038161003816100381610038161003816100381610038161003816119896=1198960

=

16252120587114120590141198811341199031212059834

334sdot 119872152

412058294

sdot exp (2I (120601)) coth( 1198962119879

)

(60)


119877= (254)120575

2

119867




119881 (120601) = 1198810exp (minus120572120601) (61)


=

1198722

4120582

8120587

1205722

1199031198812

0119890minus2120572120601

(62)


= minus

1198722

4

4120587

1205722

1199031198812

0119890minus2120572120601

(63)


119903 =radicΓ2

01198722

4120582

12120587

1198902120572120601

1198812

0

(64)


120601 (119905) =

1

120572

ln[12057221198810

Γ0

119905 + 119890120572120601119894] (65)


this form

119867 = radic

4120587

31198722

4120582

1198810119890minus120572120601 (66)


120588120574=

311987241205722

16Γ0

1198812

0

radic3120587120582

119890minus2120572120601 (67)


120588120574=

31198722

4

16radic3120587120582

(

1205722

Γ0

)1205882

120601 (68)


119873total = radic4120587120582

31198722

4

Γ0

120572

(120601119891minus 120601119894) (69)

or equivalently

119873total = radic4120587120582

31198722

4

Γ0

1205722ln(

119881119894

119881119891

) (70)



1205752

119867= 119860 exp (minus7

2

120572120601) (71)

where119860 = (16radic375120587)(1198813201198722

41205722)(Γ2

01198722

412058212120587)

12(31198722

4120582

4120587)14 and

119877 = 119861 exp (minus120572120601) (72)

where 119861 = (50120587321205722119881031198727

412058232119879119903)(36120587

2Γ2

01198724

41205824)14 In


I (120601) = minus5

4

ln119881 (73)




119886 (119905) = 1198860exp (119860119905119891) 0 lt 119891 lt 1 (74)


0and (2) Γ = Γ





119881 = (

31205821198722

4

4120587

)

12

119867

120601

2

= minus

Γ

(75)



0)

120601 minus 1206010= 1205731199051198912 (76)

where 120573 = (12120582119872241198602(1 minus 119891)

21205871198912Γ2



119867(120601) = 119891119860(

120601 minus 1206010

120573

)

(2119891minus2)119891

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

120601 minus 1206010

120573

)

(2119891minus2)119891

(77)


119903 =

Γ0

3119867119881

=

4120587Γ0

9 (119891119860)21198722

4120582

(

120601 minus 1206010

120573

)

(4minus4119891)119891

(78)


120598 = minus

1198672=

1 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

120578 = minus

119867

=

2 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

(79)


119873 = int

119905

1199051

119867119889119905 = 119860([

120601 minus 1206010

120573

]

2

minus [

1206011minus 1206010

120573

]

2

) (80)

where 1206011= 1206010+ 120573((1 minus 119891)119891119860)


120601 = 120573(

119873

119860

+

1 minus 119891

119891119860

)

12

+ 1206010 (81)


119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

11988174

1199031212059834

= 1198601(

120601 minus 1206010

120573

)

(14119891minus11)2119891

= 1198601(

119873

119860

+

1 minus 119891

119891119860

)

(14119891minus11)4119891

(82)

where 1198601= 252119872314

4(3120582)158(119891119860)72(4120587)31812059014Γ12

0(1 minus




119899119904= 1 +

3

4

120578 minus

17

4

120598 = 1 minus

11 minus 14119891

4119891119860

(

120601 minus 1206010

120573

)

minus2

= 1 minus

11 minus 14119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(83)

40 60 80 10020N

f = 57

ns

090

092

094

096

098

100

102

104











119877 = 1198611(

120601 minus 1206010

120573

)

(minus4119891+1)2119891

coth [ 1198962119879

]

= 1198611(

119873

119860

+

1 minus 119891

119891119860

)

(minus4119891+1)4119891

coth [ 1198962119879

]

= 1198611(

4119891119860

11 minus 14119891

(1 minus 119899119904))

(4119891minus1)4119891

(84)

where 1198611

= (232(4120587)238Γ12

012059014(1 minus 119891)

34

3158119872314

4120582158(119891119860)32)(31198911198602120582)









f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R


096 097 098 099095ns

08

09

10

11

12

13

14

15

TH




Γ03HV

06

08

10

12

14

096 097 098 099095ns





cases of Γ0


(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)


119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)


119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)


120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)


119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)



1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)


120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)


119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)



119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






4to be vanished


1198672=

8120587

31198722

4

120588 (1 +

120588

2120582

) (7)




120574is the


1198672=

8120587

31198722

4

(120588120601+ 120588120574) (1 +

120588120601+ 120588120574

2120582

) (8)




0exp(119860119905119891) (119860 gt 0 0 lt 119891 lt 1)



0exp(119867

0119905) (1198860 1198670gt 0)



119904= 1






2 The Model


119871 = radicminus119892(

119877

16120587119866




119879120583

] =120597119871

120597 (120597120583120601)

120597]120601 minus 119892120583

]119871 = diag (minus120588120601 119875120601 119875120601 119875120601) (10)


120588120601=

119881 (120601)

radic1 minus120601

2

119875120601= minus119881 (120601)

radic1 minus

120601

2

(11)



1198672=

8120587

31198722

4

[

[

[

119881 (120601)

radic1 minus120601

2

+ 120588120574

]

]

]

sdot[

[

[

1 +

1

2120582

(

119881 (120601)

radic1 minus120601

2

+ 120588120574)]

]

]

(12)

120601+ 3119867(119875

120601+ 120588120601) = minusΓ

120601

2

997904rArr

120601

1 minus120601

2+ 3119867

120601 +

1198811015840

119881

= minus

Γ

119881

radic1 minus

120601

2120601

(13)

120574+ 4119867120588

120574= Γ

120601

2

(14)




120601 ≪ 1 and 120601 ≪



120574≪ Γ

120601


1198672=

8120587

31198722

4

119881(1 +

119881

2120582

) (15)

3119867 (1 + 119903)120601 = minus

1198811015840

119881

(16)


119881

Γ

≪ 120591 ≪ 119867minus1997904rArr

Γ ≫ 119867119881

(17)



120588120574=

Γ120601

2

4119867

=

1198722

4119903

32120587 (1 + 119903)2(1 + 1198812120582)

(

1198811015840

119881

)

2

= 1205901198794

119903 (18)



120598 = minus

1198672≃

1198722

4

16120587

11988110158402

(1 + 119903) 1198813

1 + 119881120582

(1 + 1198812120582)2 (19)

120578 = minus

119867

≃

1198722

4

8120587

1198811015840

1198812(1 + 119903) [1 + 1198812120582]

[

211988110158401015840

1198811015840minus

1198811015840

119881

minus

1199031015840

(1 + 119903)

+

1198811015840

120582 + 119881

] minus 2120598

(20)


120574is


120588120574=

119903

2 (1 + 119903)

[1 + 1205881206012120582]

[1 + 120588120601120582]

120588120601120598

≃

119903

2 (1 + 119903)

[1 + 1198812120582]

[1 + 119881120582]

119881120598

(21)


2 (1 + 119903)

119903

120588120574lt

1 + 1205881206012120582

1 + 120588120601120582

120588120601 (22)


1198722

4

8120587

[

1198811015840

119891

119881119891

]

2

1 + 119881119891120582

(1 + 1198811198912120582)

2

1

119881119891

≃ 2 (1 + 119903119891) (23)



119873 = int

120601119891

120601lowast

119867119889119905 = int

120601119891

120601lowast

119867

120601

119889120601

= minus

8120587

1198722

4

int

120601119891

120601lowast

1198812

1198811015840(1 + 119903) [1 +

119881

2120582

] 119889120601

(24)


3 Perturbation



1198891199042= (1 + 2Φ) 119889119905

2minus 1198862(119905) (1 minus 2Ψ) 120575119894119895

119889119909119894119889119909119895 (25)


120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601


1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(26)


120575120601 (119909)

= int

1198893119896

(2120587)3[119890119894119896119909120575120601 (119896 119905) 119886

119896+ 119890minus119894119896119909120575120601 (119896 119905) 119886

lowast

119896]

(27)



Φ = Ψ (28)

Φ + 119867Φ =

4120587

1198722

4

[

[

[

minus

4120588120574119886V3119896

+

119881120601

radic1 minus120601

2

120575120601]

]

]

sdot[

[

[

1 +

1

120582

[

[

[

120588120574+

119881

radic1 minus120601

2

]

]

]

]

]

]

(29)

120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601 + [

1198962

1198862+ (

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]

sdot 120575120601 minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(30)

(

120575120588120574) + 4119867120575120588

120574+

4

3

119896119886120588120574V minus 4120588

120574Φ minus

120601

2

Γ1015840120575120601

minus Γ120601

2

[2 (120575120601) minus 3

120601Φ] = 0

(31)

V + 4119867V +119896

119886

[Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601] = 0 (32)


119895= minus(119894119886119896

119869119896)V119890119894119896119909 119895 = 1 2 3) [80 81]


120575119878 = 119890 = minus119881120601119879120575120601 minus 119881

119879119879120575119879 (33)



120575120588120574) ≪




Φ ≃

4120587

1198671198722

4

[minus

4120588120574119886V3119896

+ 119881120601120575120601] [1 +

119881

120582

] (34)

[3119867 +

Γ

119881

]120575120601 + [(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

≃ [120601

Γ

119881

minus 2(

1198811015840

119881

)]Φ

(35)

120575120588120574

120588120574

≃

Γ1015840

Γ

120575120601 minus 3Φ (36)

V ≃ minus119896

4119886119867

(Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601) (37)


Φ

=

4120587

1198722

4

(

119881120601

119867

)[1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

) 120575120601

(38)


(3119867 +

Γ

119881

)

119889

119889119905

= (3119867 +

Γ

119881

)120601

119889

119889120601

= minus

1198811015840

119881

119889

119889120601

(39)


(120575120601)1015840

120575120601

=

119881

1198811015840[(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

+

4120587

1198722

4

(minus120601

Γ

119881

+ 2(

1198811015840

119881

)

1015840

)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)]

(40)


120594 =

119881120575120601

1198811015840exp [int 1

3119867 + Γ119881

(

Γ

119881

)

1015840

119889120601] (41)


1205941015840

120594

=

(120575120601)1015840

120575120601

minus

119881

1198811015840(

1198811015840

119881

)

1015840

+

(Γ119881)1015840

3119867 + Γ119881

(42)


1205941015840

120594

=

4120587

1198722

4

(minus

119881120601

1198811015840

Γ

119881

+ 2)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)

(43)


1205941015840

120594

= minus

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

(44)



2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

119889120601) (45)


120575120601 = 119862

1198811015840

119881

exp (I (120601)) (46)

where

I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

)]119889120601

(47)





119894


R = Φ minus 119896minus1119886119867V

119890 = 120575119875 minus 1198882

119904120575120588

(48)





120575119867=

2

5

1198722

4

119881 exp (minusI (120601))1198811015840

120575120601

=

2

15

1198722

4

exp (minusI (120601))119867119903

120601

120575120601

(49)




(120575120601)2≃

119896119865119879119903

21205872 (50)



1205752

119867=

2radic3

7512058721198724

4


119879119903

119867

(51)

or equivalently

1205752

119867=

41198725

412058212


where

I (120601) = minusint[1

3119867119903

(

Γ

119881

)

1015840

+

9

4

(1 minus

(ln Γ)1015840 1198811015840119881361199031198672

)

1198811015840

119881

]119889120601

(53)

=

1198722

4120582

4120587119903

11988110158402

1198814 (54)



by

119899119904= 1 +

119889 ln 1205752119867

119889 ln 119896

asymp 1 minus

3

4

+

3

4

+ (

119881

1198811015840)(2

I1015840

(120601) +

1199031015840

2119903

)

(55)

where

=

1198722

4120582

4120587119903

1198811015840

1198813[

211988110158401015840

1198811015840minus

1199031015840

119903

] minus 2 (56)



120572119904=

119889119899119904

119889 ln 119896= minus

119889119899119904

119889119873

= minus

119889120601

119889119873

119889119899119904

119889120601

=

1198722

4120582

4120587119903

11988110158401198991015840

119904

1198813 (57)




1198602

119892=

16120587

1198724

4

(

119867

2120587

)

2

1198652(119909) =

16

31198722

4120582

11988121198652(119909) (58)


21198722

44120587120582]

12 and 119865(119909) = radic1 + 1199092minus


2

416120587120582]

1411986712= [3120587120582

31198722

4]1411988112) Spectral


119899119892=

119889

119889 ln 119896(ln[

1198602

119892

coth (1198962119879)]) ≃ minus2 (59)




119877 (119896) =

1198602

119892

119875119877

1003816100381610038161003816100381610038161003816100381610038161003816119896=1198960

=

16252120587114120590141198811341199031212059834

334sdot 119872152

412058294

sdot exp (2I (120601)) coth( 1198962119879

)

(60)


119877= (254)120575

2

119867




119881 (120601) = 1198810exp (minus120572120601) (61)


=

1198722

4120582

8120587

1205722

1199031198812

0119890minus2120572120601

(62)


= minus

1198722

4

4120587

1205722

1199031198812

0119890minus2120572120601

(63)


119903 =radicΓ2

01198722

4120582

12120587

1198902120572120601

1198812

0

(64)


120601 (119905) =

1

120572

ln[12057221198810

Γ0

119905 + 119890120572120601119894] (65)


this form

119867 = radic

4120587

31198722

4120582

1198810119890minus120572120601 (66)


120588120574=

311987241205722

16Γ0

1198812

0

radic3120587120582

119890minus2120572120601 (67)


120588120574=

31198722

4

16radic3120587120582

(

1205722

Γ0

)1205882

120601 (68)


119873total = radic4120587120582

31198722

4

Γ0

120572

(120601119891minus 120601119894) (69)

or equivalently

119873total = radic4120587120582

31198722

4

Γ0

1205722ln(

119881119894

119881119891

) (70)



1205752

119867= 119860 exp (minus7

2

120572120601) (71)

where119860 = (16radic375120587)(1198813201198722

41205722)(Γ2

01198722

412058212120587)

12(31198722

4120582

4120587)14 and

119877 = 119861 exp (minus120572120601) (72)

where 119861 = (50120587321205722119881031198727

412058232119879119903)(36120587

2Γ2

01198724

41205824)14 In


I (120601) = minus5

4

ln119881 (73)




119886 (119905) = 1198860exp (119860119905119891) 0 lt 119891 lt 1 (74)


0and (2) Γ = Γ





119881 = (

31205821198722

4

4120587

)

12

119867

120601

2

= minus

Γ

(75)



0)

120601 minus 1206010= 1205731199051198912 (76)

where 120573 = (12120582119872241198602(1 minus 119891)

21205871198912Γ2



119867(120601) = 119891119860(

120601 minus 1206010

120573

)

(2119891minus2)119891

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

120601 minus 1206010

120573

)

(2119891minus2)119891

(77)


119903 =

Γ0

3119867119881

=

4120587Γ0

9 (119891119860)21198722

4120582

(

120601 minus 1206010

120573

)

(4minus4119891)119891

(78)


120598 = minus

1198672=

1 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

120578 = minus

119867

=

2 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

(79)


119873 = int

119905

1199051

119867119889119905 = 119860([

120601 minus 1206010

120573

]

2

minus [

1206011minus 1206010

120573

]

2

) (80)

where 1206011= 1206010+ 120573((1 minus 119891)119891119860)


120601 = 120573(

119873

119860

+

1 minus 119891

119891119860

)

12

+ 1206010 (81)


119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

11988174

1199031212059834

= 1198601(

120601 minus 1206010

120573

)

(14119891minus11)2119891

= 1198601(

119873

119860

+

1 minus 119891

119891119860

)

(14119891minus11)4119891

(82)

where 1198601= 252119872314

4(3120582)158(119891119860)72(4120587)31812059014Γ12

0(1 minus




119899119904= 1 +

3

4

120578 minus

17

4

120598 = 1 minus

11 minus 14119891

4119891119860

(

120601 minus 1206010

120573

)

minus2

= 1 minus

11 minus 14119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(83)

40 60 80 10020N

f = 57

ns

090

092

094

096

098

100

102

104











119877 = 1198611(

120601 minus 1206010

120573

)

(minus4119891+1)2119891

coth [ 1198962119879

]

= 1198611(

119873

119860

+

1 minus 119891

119891119860

)

(minus4119891+1)4119891

coth [ 1198962119879

]

= 1198611(

4119891119860

11 minus 14119891

(1 minus 119899119904))

(4119891minus1)4119891

(84)

where 1198611

= (232(4120587)238Γ12

012059014(1 minus 119891)

34

3158119872314

4120582158(119891119860)32)(31198911198602120582)









f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R


096 097 098 099095ns

08

09

10

11

12

13

14

15

TH




Γ03HV

06

08

10

12

14

096 097 098 099095ns





cases of Γ0


(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)


119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)


119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)


120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)


119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)



1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)


120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)


119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)



119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119879120583

] =120597119871

120597 (120597120583120601)

120597]120601 minus 119892120583

]119871 = diag (minus120588120601 119875120601 119875120601 119875120601) (10)


120588120601=

119881 (120601)

radic1 minus120601

2

119875120601= minus119881 (120601)

radic1 minus

120601

2

(11)



1198672=

8120587

31198722

4

[

[

[

119881 (120601)

radic1 minus120601

2

+ 120588120574

]

]

]

sdot[

[

[

1 +

1

2120582

(

119881 (120601)

radic1 minus120601

2

+ 120588120574)]

]

]

(12)

120601+ 3119867(119875

120601+ 120588120601) = minusΓ

120601

2

997904rArr

120601

1 minus120601

2+ 3119867

120601 +

1198811015840

119881

= minus

Γ

119881

radic1 minus

120601

2120601

(13)

120574+ 4119867120588

120574= Γ

120601

2

(14)




120601 ≪ 1 and 120601 ≪



120574≪ Γ

120601


1198672=

8120587

31198722

4

119881(1 +

119881

2120582

) (15)

3119867 (1 + 119903)120601 = minus

1198811015840

119881

(16)


119881

Γ

≪ 120591 ≪ 119867minus1997904rArr

Γ ≫ 119867119881

(17)



120588120574=

Γ120601

2

4119867

=

1198722

4119903

32120587 (1 + 119903)2(1 + 1198812120582)

(

1198811015840

119881

)

2

= 1205901198794

119903 (18)



120598 = minus

1198672≃

1198722

4

16120587

11988110158402

(1 + 119903) 1198813

1 + 119881120582

(1 + 1198812120582)2 (19)

120578 = minus

119867

≃

1198722

4

8120587

1198811015840

1198812(1 + 119903) [1 + 1198812120582]

[

211988110158401015840

1198811015840minus

1198811015840

119881

minus

1199031015840

(1 + 119903)

+

1198811015840

120582 + 119881

] minus 2120598

(20)


120574is


120588120574=

119903

2 (1 + 119903)

[1 + 1205881206012120582]

[1 + 120588120601120582]

120588120601120598

≃

119903

2 (1 + 119903)

[1 + 1198812120582]

[1 + 119881120582]

119881120598

(21)


2 (1 + 119903)

119903

120588120574lt

1 + 1205881206012120582

1 + 120588120601120582

120588120601 (22)


1198722

4

8120587

[

1198811015840

119891

119881119891

]

2

1 + 119881119891120582

(1 + 1198811198912120582)

2

1

119881119891

≃ 2 (1 + 119903119891) (23)



119873 = int

120601119891

120601lowast

119867119889119905 = int

120601119891

120601lowast

119867

120601

119889120601

= minus

8120587

1198722

4

int

120601119891

120601lowast

1198812

1198811015840(1 + 119903) [1 +

119881

2120582

] 119889120601

(24)


3 Perturbation



1198891199042= (1 + 2Φ) 119889119905

2minus 1198862(119905) (1 minus 2Ψ) 120575119894119895

119889119909119894119889119909119895 (25)


120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601


1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(26)


120575120601 (119909)

= int

1198893119896

(2120587)3[119890119894119896119909120575120601 (119896 119905) 119886

119896+ 119890minus119894119896119909120575120601 (119896 119905) 119886

lowast

119896]

(27)



Φ = Ψ (28)

Φ + 119867Φ =

4120587

1198722

4

[

[

[

minus

4120588120574119886V3119896

+

119881120601

radic1 minus120601

2

120575120601]

]

]

sdot[

[

[

1 +

1

120582

[

[

[

120588120574+

119881

radic1 minus120601

2

]

]

]

]

]

]

(29)

120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601 + [

1198962

1198862+ (

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]

sdot 120575120601 minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(30)

(

120575120588120574) + 4119867120575120588

120574+

4

3

119896119886120588120574V minus 4120588

120574Φ minus

120601

2

Γ1015840120575120601

minus Γ120601

2

[2 (120575120601) minus 3

120601Φ] = 0

(31)

V + 4119867V +119896

119886

[Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601] = 0 (32)


119895= minus(119894119886119896

119869119896)V119890119894119896119909 119895 = 1 2 3) [80 81]


120575119878 = 119890 = minus119881120601119879120575120601 minus 119881

119879119879120575119879 (33)



120575120588120574) ≪




Φ ≃

4120587

1198671198722

4

[minus

4120588120574119886V3119896

+ 119881120601120575120601] [1 +

119881

120582

] (34)

[3119867 +

Γ

119881

]120575120601 + [(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

≃ [120601

Γ

119881

minus 2(

1198811015840

119881

)]Φ

(35)

120575120588120574

120588120574

≃

Γ1015840

Γ

120575120601 minus 3Φ (36)

V ≃ minus119896

4119886119867

(Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601) (37)


Φ

=

4120587

1198722

4

(

119881120601

119867

)[1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

) 120575120601

(38)


(3119867 +

Γ

119881

)

119889

119889119905

= (3119867 +

Γ

119881

)120601

119889

119889120601

= minus

1198811015840

119881

119889

119889120601

(39)


(120575120601)1015840

120575120601

=

119881

1198811015840[(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

+

4120587

1198722

4

(minus120601

Γ

119881

+ 2(

1198811015840

119881

)

1015840

)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)]

(40)


120594 =

119881120575120601

1198811015840exp [int 1

3119867 + Γ119881

(

Γ

119881

)

1015840

119889120601] (41)


1205941015840

120594

=

(120575120601)1015840

120575120601

minus

119881

1198811015840(

1198811015840

119881

)

1015840

+

(Γ119881)1015840

3119867 + Γ119881

(42)


1205941015840

120594

=

4120587

1198722

4

(minus

119881120601

1198811015840

Γ

119881

+ 2)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)

(43)


1205941015840

120594

= minus

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

(44)



2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

119889120601) (45)


120575120601 = 119862

1198811015840

119881

exp (I (120601)) (46)

where

I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

)]119889120601

(47)





119894


R = Φ minus 119896minus1119886119867V

119890 = 120575119875 minus 1198882

119904120575120588

(48)





120575119867=

2

5

1198722

4

119881 exp (minusI (120601))1198811015840

120575120601

=

2

15

1198722

4

exp (minusI (120601))119867119903

120601

120575120601

(49)




(120575120601)2≃

119896119865119879119903

21205872 (50)



1205752

119867=

2radic3

7512058721198724

4


119879119903

119867

(51)

or equivalently

1205752

119867=

41198725

412058212


where

I (120601) = minusint[1

3119867119903

(

Γ

119881

)

1015840

+

9

4

(1 minus

(ln Γ)1015840 1198811015840119881361199031198672

)

1198811015840

119881

]119889120601

(53)

=

1198722

4120582

4120587119903

11988110158402

1198814 (54)



by

119899119904= 1 +

119889 ln 1205752119867

119889 ln 119896

asymp 1 minus

3

4

+

3

4

+ (

119881

1198811015840)(2

I1015840

(120601) +

1199031015840

2119903

)

(55)

where

=

1198722

4120582

4120587119903

1198811015840

1198813[

211988110158401015840

1198811015840minus

1199031015840

119903

] minus 2 (56)



120572119904=

119889119899119904

119889 ln 119896= minus

119889119899119904

119889119873

= minus

119889120601

119889119873

119889119899119904

119889120601

=

1198722

4120582

4120587119903

11988110158401198991015840

119904

1198813 (57)




1198602

119892=

16120587

1198724

4

(

119867

2120587

)

2

1198652(119909) =

16

31198722

4120582

11988121198652(119909) (58)


21198722

44120587120582]

12 and 119865(119909) = radic1 + 1199092minus


2

416120587120582]

1411986712= [3120587120582

31198722

4]1411988112) Spectral


119899119892=

119889

119889 ln 119896(ln[

1198602

119892

coth (1198962119879)]) ≃ minus2 (59)




119877 (119896) =

1198602

119892

119875119877

1003816100381610038161003816100381610038161003816100381610038161003816119896=1198960

=

16252120587114120590141198811341199031212059834

334sdot 119872152

412058294

sdot exp (2I (120601)) coth( 1198962119879

)

(60)


119877= (254)120575

2

119867




119881 (120601) = 1198810exp (minus120572120601) (61)


=

1198722

4120582

8120587

1205722

1199031198812

0119890minus2120572120601

(62)


= minus

1198722

4

4120587

1205722

1199031198812

0119890minus2120572120601

(63)


119903 =radicΓ2

01198722

4120582

12120587

1198902120572120601

1198812

0

(64)


120601 (119905) =

1

120572

ln[12057221198810

Γ0

119905 + 119890120572120601119894] (65)


this form

119867 = radic

4120587

31198722

4120582

1198810119890minus120572120601 (66)


120588120574=

311987241205722

16Γ0

1198812

0

radic3120587120582

119890minus2120572120601 (67)


120588120574=

31198722

4

16radic3120587120582

(

1205722

Γ0

)1205882

120601 (68)


119873total = radic4120587120582

31198722

4

Γ0

120572

(120601119891minus 120601119894) (69)

or equivalently

119873total = radic4120587120582

31198722

4

Γ0

1205722ln(

119881119894

119881119891

) (70)



1205752

119867= 119860 exp (minus7

2

120572120601) (71)

where119860 = (16radic375120587)(1198813201198722

41205722)(Γ2

01198722

412058212120587)

12(31198722

4120582

4120587)14 and

119877 = 119861 exp (minus120572120601) (72)

where 119861 = (50120587321205722119881031198727

412058232119879119903)(36120587

2Γ2

01198724

41205824)14 In


I (120601) = minus5

4

ln119881 (73)




119886 (119905) = 1198860exp (119860119905119891) 0 lt 119891 lt 1 (74)


0and (2) Γ = Γ





119881 = (

31205821198722

4

4120587

)

12

119867

120601

2

= minus

Γ

(75)



0)

120601 minus 1206010= 1205731199051198912 (76)

where 120573 = (12120582119872241198602(1 minus 119891)

21205871198912Γ2



119867(120601) = 119891119860(

120601 minus 1206010

120573

)

(2119891minus2)119891

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

120601 minus 1206010

120573

)

(2119891minus2)119891

(77)


119903 =

Γ0

3119867119881

=

4120587Γ0

9 (119891119860)21198722

4120582

(

120601 minus 1206010

120573

)

(4minus4119891)119891

(78)


120598 = minus

1198672=

1 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

120578 = minus

119867

=

2 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

(79)


119873 = int

119905

1199051

119867119889119905 = 119860([

120601 minus 1206010

120573

]

2

minus [

1206011minus 1206010

120573

]

2

) (80)

where 1206011= 1206010+ 120573((1 minus 119891)119891119860)


120601 = 120573(

119873

119860

+

1 minus 119891

119891119860

)

12

+ 1206010 (81)


119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

11988174

1199031212059834

= 1198601(

120601 minus 1206010

120573

)

(14119891minus11)2119891

= 1198601(

119873

119860

+

1 minus 119891

119891119860

)

(14119891minus11)4119891

(82)

where 1198601= 252119872314

4(3120582)158(119891119860)72(4120587)31812059014Γ12

0(1 minus




119899119904= 1 +

3

4

120578 minus

17

4

120598 = 1 minus

11 minus 14119891

4119891119860

(

120601 minus 1206010

120573

)

minus2

= 1 minus

11 minus 14119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(83)

40 60 80 10020N

f = 57

ns

090

092

094

096

098

100

102

104











119877 = 1198611(

120601 minus 1206010

120573

)

(minus4119891+1)2119891

coth [ 1198962119879

]

= 1198611(

119873

119860

+

1 minus 119891

119891119860

)

(minus4119891+1)4119891

coth [ 1198962119879

]

= 1198611(

4119891119860

11 minus 14119891

(1 minus 119899119904))

(4119891minus1)4119891

(84)

where 1198611

= (232(4120587)238Γ12

012059014(1 minus 119891)

34

3158119872314

4120582158(119891119860)32)(31198911198602120582)









f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R


096 097 098 099095ns

08

09

10

11

12

13

14

15

TH




Γ03HV

06

08

10

12

14

096 097 098 099095ns





cases of Γ0


(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)


119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)


119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)


120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)


119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)



1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)


120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)


119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)



119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119873 = int

120601119891

120601lowast

119867119889119905 = int

120601119891

120601lowast

119867

120601

119889120601

= minus

8120587

1198722

4

int

120601119891

120601lowast

1198812

1198811015840(1 + 119903) [1 +

119881

2120582

] 119889120601

(24)


3 Perturbation



1198891199042= (1 + 2Φ) 119889119905

2minus 1198862(119905) (1 minus 2Ψ) 120575119894119895

119889119909119894119889119909119895 (25)


120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601


1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(26)


120575120601 (119909)

= int

1198893119896

(2120587)3[119890119894119896119909120575120601 (119896 119905) 119886

119896+ 119890minus119894119896119909120575120601 (119896 119905) 119886

lowast

119896]

(27)



Φ = Ψ (28)

Φ + 119867Φ =

4120587

1198722

4

[

[

[

minus

4120588120574119886V3119896

+

119881120601

radic1 minus120601

2

120575120601]

]

]

sdot[

[

[

1 +

1

120582

[

[

[

120588120574+

119881

radic1 minus120601

2

]

]

]

]

]

]

(29)

120575120601

1 minus120601

2+ [3119867 +

Γ

119881

]120575120601 + [

1198962

1198862+ (

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]

sdot 120575120601 minus [

1

1 minus120601

2+ 3]

120601Φ minus [

120601

Γ

119881

minus 2

1198811015840

119881

]Φ = 0

(30)

(

120575120588120574) + 4119867120575120588

120574+

4

3

119896119886120588120574V minus 4120588

120574Φ minus

120601

2

Γ1015840120575120601

minus Γ120601

2

[2 (120575120601) minus 3

120601Φ] = 0

(31)

V + 4119867V +119896

119886

[Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601] = 0 (32)


119895= minus(119894119886119896

119869119896)V119890119894119896119909 119895 = 1 2 3) [80 81]


120575119878 = 119890 = minus119881120601119879120575120601 minus 119881

119879119879120575119879 (33)



120575120588120574) ≪




Φ ≃

4120587

1198671198722

4

[minus

4120588120574119886V3119896

+ 119881120601120575120601] [1 +

119881

120582

] (34)

[3119867 +

Γ

119881

]120575120601 + [(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

≃ [120601

Γ

119881

minus 2(

1198811015840

119881

)]Φ

(35)

120575120588120574

120588120574

≃

Γ1015840

Γ

120575120601 minus 3Φ (36)

V ≃ minus119896

4119886119867

(Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601) (37)


Φ

=

4120587

1198722

4

(

119881120601

119867

)[1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

) 120575120601

(38)


(3119867 +

Γ

119881

)

119889

119889119905

= (3119867 +

Γ

119881

)120601

119889

119889120601

= minus

1198811015840

119881

119889

119889120601

(39)


(120575120601)1015840

120575120601

=

119881

1198811015840[(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

+

4120587

1198722

4

(minus120601

Γ

119881

+ 2(

1198811015840

119881

)

1015840

)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)]

(40)


120594 =

119881120575120601

1198811015840exp [int 1

3119867 + Γ119881

(

Γ

119881

)

1015840

119889120601] (41)


1205941015840

120594

=

(120575120601)1015840

120575120601

minus

119881

1198811015840(

1198811015840

119881

)

1015840

+

(Γ119881)1015840

3119867 + Γ119881

(42)


1205941015840

120594

=

4120587

1198722

4

(minus

119881120601

1198811015840

Γ

119881

+ 2)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)

(43)


1205941015840

120594

= minus

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

(44)



2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

119889120601) (45)


120575120601 = 119862

1198811015840

119881

exp (I (120601)) (46)

where

I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

)]119889120601

(47)





119894


R = Φ minus 119896minus1119886119867V

119890 = 120575119875 minus 1198882

119904120575120588

(48)





120575119867=

2

5

1198722

4

119881 exp (minusI (120601))1198811015840

120575120601

=

2

15

1198722

4

exp (minusI (120601))119867119903

120601

120575120601

(49)




(120575120601)2≃

119896119865119879119903

21205872 (50)



1205752

119867=

2radic3

7512058721198724

4


119879119903

119867

(51)

or equivalently

1205752

119867=

41198725

412058212


where

I (120601) = minusint[1

3119867119903

(

Γ

119881

)

1015840

+

9

4

(1 minus

(ln Γ)1015840 1198811015840119881361199031198672

)

1198811015840

119881

]119889120601

(53)

=

1198722

4120582

4120587119903

11988110158402

1198814 (54)



by

119899119904= 1 +

119889 ln 1205752119867

119889 ln 119896

asymp 1 minus

3

4

+

3

4

+ (

119881

1198811015840)(2

I1015840

(120601) +

1199031015840

2119903

)

(55)

where

=

1198722

4120582

4120587119903

1198811015840

1198813[

211988110158401015840

1198811015840minus

1199031015840

119903

] minus 2 (56)



120572119904=

119889119899119904

119889 ln 119896= minus

119889119899119904

119889119873

= minus

119889120601

119889119873

119889119899119904

119889120601

=

1198722

4120582

4120587119903

11988110158401198991015840

119904

1198813 (57)




1198602

119892=

16120587

1198724

4

(

119867

2120587

)

2

1198652(119909) =

16

31198722

4120582

11988121198652(119909) (58)


21198722

44120587120582]

12 and 119865(119909) = radic1 + 1199092minus


2

416120587120582]

1411986712= [3120587120582

31198722

4]1411988112) Spectral


119899119892=

119889

119889 ln 119896(ln[

1198602

119892

coth (1198962119879)]) ≃ minus2 (59)




119877 (119896) =

1198602

119892

119875119877

1003816100381610038161003816100381610038161003816100381610038161003816119896=1198960

=

16252120587114120590141198811341199031212059834

334sdot 119872152

412058294

sdot exp (2I (120601)) coth( 1198962119879

)

(60)


119877= (254)120575

2

119867




119881 (120601) = 1198810exp (minus120572120601) (61)


=

1198722

4120582

8120587

1205722

1199031198812

0119890minus2120572120601

(62)


= minus

1198722

4

4120587

1205722

1199031198812

0119890minus2120572120601

(63)


119903 =radicΓ2

01198722

4120582

12120587

1198902120572120601

1198812

0

(64)


120601 (119905) =

1

120572

ln[12057221198810

Γ0

119905 + 119890120572120601119894] (65)


this form

119867 = radic

4120587

31198722

4120582

1198810119890minus120572120601 (66)


120588120574=

311987241205722

16Γ0

1198812

0

radic3120587120582

119890minus2120572120601 (67)


120588120574=

31198722

4

16radic3120587120582

(

1205722

Γ0

)1205882

120601 (68)


119873total = radic4120587120582

31198722

4

Γ0

120572

(120601119891minus 120601119894) (69)

or equivalently

119873total = radic4120587120582

31198722

4

Γ0

1205722ln(

119881119894

119881119891

) (70)



1205752

119867= 119860 exp (minus7

2

120572120601) (71)

where119860 = (16radic375120587)(1198813201198722

41205722)(Γ2

01198722

412058212120587)

12(31198722

4120582

4120587)14 and

119877 = 119861 exp (minus120572120601) (72)

where 119861 = (50120587321205722119881031198727

412058232119879119903)(36120587

2Γ2

01198724

41205824)14 In


I (120601) = minus5

4

ln119881 (73)




119886 (119905) = 1198860exp (119860119905119891) 0 lt 119891 lt 1 (74)


0and (2) Γ = Γ





119881 = (

31205821198722

4

4120587

)

12

119867

120601

2

= minus

Γ

(75)



0)

120601 minus 1206010= 1205731199051198912 (76)

where 120573 = (12120582119872241198602(1 minus 119891)

21205871198912Γ2



119867(120601) = 119891119860(

120601 minus 1206010

120573

)

(2119891minus2)119891

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

120601 minus 1206010

120573

)

(2119891minus2)119891

(77)


119903 =

Γ0

3119867119881

=

4120587Γ0

9 (119891119860)21198722

4120582

(

120601 minus 1206010

120573

)

(4minus4119891)119891

(78)


120598 = minus

1198672=

1 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

120578 = minus

119867

=

2 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

(79)


119873 = int

119905

1199051

119867119889119905 = 119860([

120601 minus 1206010

120573

]

2

minus [

1206011minus 1206010

120573

]

2

) (80)

where 1206011= 1206010+ 120573((1 minus 119891)119891119860)


120601 = 120573(

119873

119860

+

1 minus 119891

119891119860

)

12

+ 1206010 (81)


119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

11988174

1199031212059834

= 1198601(

120601 minus 1206010

120573

)

(14119891minus11)2119891

= 1198601(

119873

119860

+

1 minus 119891

119891119860

)

(14119891minus11)4119891

(82)

where 1198601= 252119872314

4(3120582)158(119891119860)72(4120587)31812059014Γ12

0(1 minus




119899119904= 1 +

3

4

120578 minus

17

4

120598 = 1 minus

11 minus 14119891

4119891119860

(

120601 minus 1206010

120573

)

minus2

= 1 minus

11 minus 14119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(83)

40 60 80 10020N

f = 57

ns

090

092

094

096

098

100

102

104











119877 = 1198611(

120601 minus 1206010

120573

)

(minus4119891+1)2119891

coth [ 1198962119879

]

= 1198611(

119873

119860

+

1 minus 119891

119891119860

)

(minus4119891+1)4119891

coth [ 1198962119879

]

= 1198611(

4119891119860

11 minus 14119891

(1 minus 119899119904))

(4119891minus1)4119891

(84)

where 1198611

= (232(4120587)238Γ12

012059014(1 minus 119891)

34

3158119872314

4120582158(119891119860)32)(31198911198602120582)









f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R


096 097 098 099095ns

08

09

10

11

12

13

14

15

TH




Γ03HV

06

08

10

12

14

096 097 098 099095ns





cases of Γ0


(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)


119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)


119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)


120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)


119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)



1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)


120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)


119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)



119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Φ ≃

4120587

1198671198722

4

[minus

4120588120574119886V3119896

+ 119881120601120575120601] [1 +

119881

120582

] (34)

[3119867 +

Γ

119881

]120575120601 + [(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

]120575120601

≃ [120601

Γ

119881

minus 2(

1198811015840

119881

)]Φ

(35)

120575120588120574

120588120574

≃

Γ1015840

Γ

120575120601 minus 3Φ (36)

V ≃ minus119896

4119886119867

(Φ +

120575120588120574

4120588120574

+

3Γ120601

4120588120574

120575120601) (37)


Φ

=

4120587

1198722

4

(

119881120601

119867

)[1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

) 120575120601

(38)


(3119867 +

Γ

119881

)

119889

119889119905

= (3119867 +

Γ

119881

)120601

119889

119889120601

= minus

1198811015840

119881

119889

119889120601

(39)


(120575120601)1015840

120575120601

=

119881

1198811015840[(

1198811015840

119881

)

1015840

+120601 (

Γ

119881

)

1015840

+

4120587

1198722

4

(minus120601

Γ

119881

+ 2(

1198811015840

119881

)

1015840

)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)]

(40)


120594 =

119881120575120601

1198811015840exp [int 1

3119867 + Γ119881

(

Γ

119881

)

1015840

119889120601] (41)


1205941015840

120594

=

(120575120601)1015840

120575120601

minus

119881

1198811015840(

1198811015840

119881

)

1015840

+

(Γ119881)1015840

3119867 + Γ119881

(42)


1205941015840

120594

=

4120587

1198722

4

(minus

119881120601

1198811015840

Γ

119881

+ 2)(

119881120601

119867

)

sdot [1 +

Γ

4119867119881

+

Γ1015840 120601

481198672119881

](1 +

119881

120582

)

(43)


1205941015840

120594

= minus

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

(44)



2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

119889120601) (45)


120575120601 = 119862

1198811015840

119881

exp (I (120601)) (46)

where

I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ10158401198811015840119881

12119867 (3119867 + Γ119881)

)

sdot

1198811015840

1198812

[1 + 119881120582]

1 + 1198812120582

)]119889120601

(47)





119894


R = Φ minus 119896minus1119886119867V

119890 = 120575119875 minus 1198882

119904120575120588

(48)





120575119867=

2

5

1198722

4

119881 exp (minusI (120601))1198811015840

120575120601

=

2

15

1198722

4

exp (minusI (120601))119867119903

120601

120575120601

(49)




(120575120601)2≃

119896119865119879119903

21205872 (50)



1205752

119867=

2radic3

7512058721198724

4


119879119903

119867

(51)

or equivalently

1205752

119867=

41198725

412058212


where

I (120601) = minusint[1

3119867119903

(

Γ

119881

)

1015840

+

9

4

(1 minus

(ln Γ)1015840 1198811015840119881361199031198672

)

1198811015840

119881

]119889120601

(53)

=

1198722

4120582

4120587119903

11988110158402

1198814 (54)



by

119899119904= 1 +

119889 ln 1205752119867

119889 ln 119896

asymp 1 minus

3

4

+

3

4

+ (

119881

1198811015840)(2

I1015840

(120601) +

1199031015840

2119903

)

(55)

where

=

1198722

4120582

4120587119903

1198811015840

1198813[

211988110158401015840

1198811015840minus

1199031015840

119903

] minus 2 (56)



120572119904=

119889119899119904

119889 ln 119896= minus

119889119899119904

119889119873

= minus

119889120601

119889119873

119889119899119904

119889120601

=

1198722

4120582

4120587119903

11988110158401198991015840

119904

1198813 (57)




1198602

119892=

16120587

1198724

4

(

119867

2120587

)

2

1198652(119909) =

16

31198722

4120582

11988121198652(119909) (58)


21198722

44120587120582]

12 and 119865(119909) = radic1 + 1199092minus


2

416120587120582]

1411986712= [3120587120582

31198722

4]1411988112) Spectral


119899119892=

119889

119889 ln 119896(ln[

1198602

119892

coth (1198962119879)]) ≃ minus2 (59)




119877 (119896) =

1198602

119892

119875119877

1003816100381610038161003816100381610038161003816100381610038161003816119896=1198960

=

16252120587114120590141198811341199031212059834

334sdot 119872152

412058294

sdot exp (2I (120601)) coth( 1198962119879

)

(60)


119877= (254)120575

2

119867




119881 (120601) = 1198810exp (minus120572120601) (61)


=

1198722

4120582

8120587

1205722

1199031198812

0119890minus2120572120601

(62)


= minus

1198722

4

4120587

1205722

1199031198812

0119890minus2120572120601

(63)


119903 =radicΓ2

01198722

4120582

12120587

1198902120572120601

1198812

0

(64)


120601 (119905) =

1

120572

ln[12057221198810

Γ0

119905 + 119890120572120601119894] (65)


this form

119867 = radic

4120587

31198722

4120582

1198810119890minus120572120601 (66)


120588120574=

311987241205722

16Γ0

1198812

0

radic3120587120582

119890minus2120572120601 (67)


120588120574=

31198722

4

16radic3120587120582

(

1205722

Γ0

)1205882

120601 (68)


119873total = radic4120587120582

31198722

4

Γ0

120572

(120601119891minus 120601119894) (69)

or equivalently

119873total = radic4120587120582

31198722

4

Γ0

1205722ln(

119881119894

119881119891

) (70)



1205752

119867= 119860 exp (minus7

2

120572120601) (71)

where119860 = (16radic375120587)(1198813201198722

41205722)(Γ2

01198722

412058212120587)

12(31198722

4120582

4120587)14 and

119877 = 119861 exp (minus120572120601) (72)

where 119861 = (50120587321205722119881031198727

412058232119879119903)(36120587

2Γ2

01198724

41205824)14 In


I (120601) = minus5

4

ln119881 (73)




119886 (119905) = 1198860exp (119860119905119891) 0 lt 119891 lt 1 (74)


0and (2) Γ = Γ





119881 = (

31205821198722

4

4120587

)

12

119867

120601

2

= minus

Γ

(75)



0)

120601 minus 1206010= 1205731199051198912 (76)

where 120573 = (12120582119872241198602(1 minus 119891)

21205871198912Γ2



119867(120601) = 119891119860(

120601 minus 1206010

120573

)

(2119891minus2)119891

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

120601 minus 1206010

120573

)

(2119891minus2)119891

(77)


119903 =

Γ0

3119867119881

=

4120587Γ0

9 (119891119860)21198722

4120582

(

120601 minus 1206010

120573

)

(4minus4119891)119891

(78)


120598 = minus

1198672=

1 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

120578 = minus

119867

=

2 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

(79)


119873 = int

119905

1199051

119867119889119905 = 119860([

120601 minus 1206010

120573

]

2

minus [

1206011minus 1206010

120573

]

2

) (80)

where 1206011= 1206010+ 120573((1 minus 119891)119891119860)


120601 = 120573(

119873

119860

+

1 minus 119891

119891119860

)

12

+ 1206010 (81)


119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

11988174

1199031212059834

= 1198601(

120601 minus 1206010

120573

)

(14119891minus11)2119891

= 1198601(

119873

119860

+

1 minus 119891

119891119860

)

(14119891minus11)4119891

(82)

where 1198601= 252119872314

4(3120582)158(119891119860)72(4120587)31812059014Γ12

0(1 minus




119899119904= 1 +

3

4

120578 minus

17

4

120598 = 1 minus

11 minus 14119891

4119891119860

(

120601 minus 1206010

120573

)

minus2

= 1 minus

11 minus 14119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(83)

40 60 80 10020N

f = 57

ns

090

092

094

096

098

100

102

104











119877 = 1198611(

120601 minus 1206010

120573

)

(minus4119891+1)2119891

coth [ 1198962119879

]

= 1198611(

119873

119860

+

1 minus 119891

119891119860

)

(minus4119891+1)4119891

coth [ 1198962119879

]

= 1198611(

4119891119860

11 minus 14119891

(1 minus 119899119904))

(4119891minus1)4119891

(84)

where 1198611

= (232(4120587)238Γ12

012059014(1 minus 119891)

34

3158119872314

4120582158(119891119860)32)(31198911198602120582)









f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R


096 097 098 099095ns

08

09

10

11

12

13

14

15

TH




Γ03HV

06

08

10

12

14

096 097 098 099095ns





cases of Γ0


(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)


119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)


119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)


120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)


119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)



1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)


120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)


119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)



119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119894


R = Φ minus 119896minus1119886119867V

119890 = 120575119875 minus 1198882

119904120575120588

(48)





120575119867=

2

5

1198722

4

119881 exp (minusI (120601))1198811015840

120575120601

=

2

15

1198722

4

exp (minusI (120601))119867119903

120601

120575120601

(49)




(120575120601)2≃

119896119865119879119903

21205872 (50)



1205752

119867=

2radic3

7512058721198724

4


119879119903

119867

(51)

or equivalently

1205752

119867=

41198725

412058212


where

I (120601) = minusint[1

3119867119903

(

Γ

119881

)

1015840

+

9

4

(1 minus

(ln Γ)1015840 1198811015840119881361199031198672

)

1198811015840

119881

]119889120601

(53)

=

1198722

4120582

4120587119903

11988110158402

1198814 (54)



by

119899119904= 1 +

119889 ln 1205752119867

119889 ln 119896

asymp 1 minus

3

4

+

3

4

+ (

119881

1198811015840)(2

I1015840

(120601) +

1199031015840

2119903

)

(55)

where

=

1198722

4120582

4120587119903

1198811015840

1198813[

211988110158401015840

1198811015840minus

1199031015840

119903

] minus 2 (56)



120572119904=

119889119899119904

119889 ln 119896= minus

119889119899119904

119889119873

= minus

119889120601

119889119873

119889119899119904

119889120601

=

1198722

4120582

4120587119903

11988110158401198991015840

119904

1198813 (57)




1198602

119892=

16120587

1198724

4

(

119867

2120587

)

2

1198652(119909) =

16

31198722

4120582

11988121198652(119909) (58)


21198722

44120587120582]

12 and 119865(119909) = radic1 + 1199092minus


2

416120587120582]

1411986712= [3120587120582

31198722

4]1411988112) Spectral


119899119892=

119889

119889 ln 119896(ln[

1198602

119892

coth (1198962119879)]) ≃ minus2 (59)




119877 (119896) =

1198602

119892

119875119877

1003816100381610038161003816100381610038161003816100381610038161003816119896=1198960

=

16252120587114120590141198811341199031212059834

334sdot 119872152

412058294

sdot exp (2I (120601)) coth( 1198962119879

)

(60)


119877= (254)120575

2

119867




119881 (120601) = 1198810exp (minus120572120601) (61)


=

1198722

4120582

8120587

1205722

1199031198812

0119890minus2120572120601

(62)


= minus

1198722

4

4120587

1205722

1199031198812

0119890minus2120572120601

(63)


119903 =radicΓ2

01198722

4120582

12120587

1198902120572120601

1198812

0

(64)


120601 (119905) =

1

120572

ln[12057221198810

Γ0

119905 + 119890120572120601119894] (65)


this form

119867 = radic

4120587

31198722

4120582

1198810119890minus120572120601 (66)


120588120574=

311987241205722

16Γ0

1198812

0

radic3120587120582

119890minus2120572120601 (67)


120588120574=

31198722

4

16radic3120587120582

(

1205722

Γ0

)1205882

120601 (68)


119873total = radic4120587120582

31198722

4

Γ0

120572

(120601119891minus 120601119894) (69)

or equivalently

119873total = radic4120587120582

31198722

4

Γ0

1205722ln(

119881119894

119881119891

) (70)



1205752

119867= 119860 exp (minus7

2

120572120601) (71)

where119860 = (16radic375120587)(1198813201198722

41205722)(Γ2

01198722

412058212120587)

12(31198722

4120582

4120587)14 and

119877 = 119861 exp (minus120572120601) (72)

where 119861 = (50120587321205722119881031198727

412058232119879119903)(36120587

2Γ2

01198724

41205824)14 In


I (120601) = minus5

4

ln119881 (73)




119886 (119905) = 1198860exp (119860119905119891) 0 lt 119891 lt 1 (74)


0and (2) Γ = Γ





119881 = (

31205821198722

4

4120587

)

12

119867

120601

2

= minus

Γ

(75)



0)

120601 minus 1206010= 1205731199051198912 (76)

where 120573 = (12120582119872241198602(1 minus 119891)

21205871198912Γ2



119867(120601) = 119891119860(

120601 minus 1206010

120573

)

(2119891minus2)119891

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

120601 minus 1206010

120573

)

(2119891minus2)119891

(77)


119903 =

Γ0

3119867119881

=

4120587Γ0

9 (119891119860)21198722

4120582

(

120601 minus 1206010

120573

)

(4minus4119891)119891

(78)


120598 = minus

1198672=

1 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

120578 = minus

119867

=

2 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

(79)


119873 = int

119905

1199051

119867119889119905 = 119860([

120601 minus 1206010

120573

]

2

minus [

1206011minus 1206010

120573

]

2

) (80)

where 1206011= 1206010+ 120573((1 minus 119891)119891119860)


120601 = 120573(

119873

119860

+

1 minus 119891

119891119860

)

12

+ 1206010 (81)


119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

11988174

1199031212059834

= 1198601(

120601 minus 1206010

120573

)

(14119891minus11)2119891

= 1198601(

119873

119860

+

1 minus 119891

119891119860

)

(14119891minus11)4119891

(82)

where 1198601= 252119872314

4(3120582)158(119891119860)72(4120587)31812059014Γ12

0(1 minus




119899119904= 1 +

3

4

120578 minus

17

4

120598 = 1 minus

11 minus 14119891

4119891119860

(

120601 minus 1206010

120573

)

minus2

= 1 minus

11 minus 14119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(83)

40 60 80 10020N

f = 57

ns

090

092

094

096

098

100

102

104











119877 = 1198611(

120601 minus 1206010

120573

)

(minus4119891+1)2119891

coth [ 1198962119879

]

= 1198611(

119873

119860

+

1 minus 119891

119891119860

)

(minus4119891+1)4119891

coth [ 1198962119879

]

= 1198611(

4119891119860

11 minus 14119891

(1 minus 119899119904))

(4119891minus1)4119891

(84)

where 1198611

= (232(4120587)238Γ12

012059014(1 minus 119891)

34

3158119872314

4120582158(119891119860)32)(31198911198602120582)









f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R


096 097 098 099095ns

08

09

10

11

12

13

14

15

TH




Γ03HV

06

08

10

12

14

096 097 098 099095ns





cases of Γ0


(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)


119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)


119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)


120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)


119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)



1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)


120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)


119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)



119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119877 (119896) =

1198602

119892

119875119877

1003816100381610038161003816100381610038161003816100381610038161003816119896=1198960

=

16252120587114120590141198811341199031212059834

334sdot 119872152

412058294

sdot exp (2I (120601)) coth( 1198962119879

)

(60)


119877= (254)120575

2

119867




119881 (120601) = 1198810exp (minus120572120601) (61)


=

1198722

4120582

8120587

1205722

1199031198812

0119890minus2120572120601

(62)


= minus

1198722

4

4120587

1205722

1199031198812

0119890minus2120572120601

(63)


119903 =radicΓ2

01198722

4120582

12120587

1198902120572120601

1198812

0

(64)


120601 (119905) =

1

120572

ln[12057221198810

Γ0

119905 + 119890120572120601119894] (65)


this form

119867 = radic

4120587

31198722

4120582

1198810119890minus120572120601 (66)


120588120574=

311987241205722

16Γ0

1198812

0

radic3120587120582

119890minus2120572120601 (67)


120588120574=

31198722

4

16radic3120587120582

(

1205722

Γ0

)1205882

120601 (68)


119873total = radic4120587120582

31198722

4

Γ0

120572

(120601119891minus 120601119894) (69)

or equivalently

119873total = radic4120587120582

31198722

4

Γ0

1205722ln(

119881119894

119881119891

) (70)



1205752

119867= 119860 exp (minus7

2

120572120601) (71)

where119860 = (16radic375120587)(1198813201198722

41205722)(Γ2

01198722

412058212120587)

12(31198722

4120582

4120587)14 and

119877 = 119861 exp (minus120572120601) (72)

where 119861 = (50120587321205722119881031198727

412058232119879119903)(36120587

2Γ2

01198724

41205824)14 In


I (120601) = minus5

4

ln119881 (73)




119886 (119905) = 1198860exp (119860119905119891) 0 lt 119891 lt 1 (74)


0and (2) Γ = Γ





119881 = (

31205821198722

4

4120587

)

12

119867

120601

2

= minus

Γ

(75)



0)

120601 minus 1206010= 1205731199051198912 (76)

where 120573 = (12120582119872241198602(1 minus 119891)

21205871198912Γ2



119867(120601) = 119891119860(

120601 minus 1206010

120573

)

(2119891minus2)119891

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

120601 minus 1206010

120573

)

(2119891minus2)119891

(77)


119903 =

Γ0

3119867119881

=

4120587Γ0

9 (119891119860)21198722

4120582

(

120601 minus 1206010

120573

)

(4minus4119891)119891

(78)


120598 = minus

1198672=

1 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

120578 = minus

119867

=

2 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

(79)


119873 = int

119905

1199051

119867119889119905 = 119860([

120601 minus 1206010

120573

]

2

minus [

1206011minus 1206010

120573

]

2

) (80)

where 1206011= 1206010+ 120573((1 minus 119891)119891119860)


120601 = 120573(

119873

119860

+

1 minus 119891

119891119860

)

12

+ 1206010 (81)


119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

11988174

1199031212059834

= 1198601(

120601 minus 1206010

120573

)

(14119891minus11)2119891

= 1198601(

119873

119860

+

1 minus 119891

119891119860

)

(14119891minus11)4119891

(82)

where 1198601= 252119872314

4(3120582)158(119891119860)72(4120587)31812059014Γ12

0(1 minus




119899119904= 1 +

3

4

120578 minus

17

4

120598 = 1 minus

11 minus 14119891

4119891119860

(

120601 minus 1206010

120573

)

minus2

= 1 minus

11 minus 14119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(83)

40 60 80 10020N

f = 57

ns

090

092

094

096

098

100

102

104











119877 = 1198611(

120601 minus 1206010

120573

)

(minus4119891+1)2119891

coth [ 1198962119879

]

= 1198611(

119873

119860

+

1 minus 119891

119891119860

)

(minus4119891+1)4119891

coth [ 1198962119879

]

= 1198611(

4119891119860

11 minus 14119891

(1 minus 119899119904))

(4119891minus1)4119891

(84)

where 1198611

= (232(4120587)238Γ12

012059014(1 minus 119891)

34

3158119872314

4120582158(119891119860)32)(31198911198602120582)









f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R


096 097 098 099095ns

08

09

10

11

12

13

14

15

TH




Γ03HV

06

08

10

12

14

096 097 098 099095ns





cases of Γ0


(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)


119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)


119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)


120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)


119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)



1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)


120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)


119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)



119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





0)

120601 minus 1206010= 1205731199051198912 (76)

where 120573 = (12120582119872241198602(1 minus 119891)

21205871198912Γ2



119867(120601) = 119891119860(

120601 minus 1206010

120573

)

(2119891minus2)119891

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

120601 minus 1206010

120573

)

(2119891minus2)119891

(77)


119903 =

Γ0

3119867119881

=

4120587Γ0

9 (119891119860)21198722

4120582

(

120601 minus 1206010

120573

)

(4minus4119891)119891

(78)


120598 = minus

1198672=

1 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

120578 = minus

119867

=

2 minus 119891

119891119860

(

120601 minus 1206010

120573

)

minus2

(79)


119873 = int

119905

1199051

119867119889119905 = 119860([

120601 minus 1206010

120573

]

2

minus [

1206011minus 1206010

120573

]

2

) (80)

where 1206011= 1206010+ 120573((1 minus 119891)119891119860)


120601 = 120573(

119873

119860

+

1 minus 119891

119891119860

)

12

+ 1206010 (81)


119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

11988174

1199031212059834

= 1198601(

120601 minus 1206010

120573

)

(14119891minus11)2119891

= 1198601(

119873

119860

+

1 minus 119891

119891119860

)

(14119891minus11)4119891

(82)

where 1198601= 252119872314

4(3120582)158(119891119860)72(4120587)31812059014Γ12

0(1 minus




119899119904= 1 +

3

4

120578 minus

17

4

120598 = 1 minus

11 minus 14119891

4119891119860

(

120601 minus 1206010

120573

)

minus2

= 1 minus

11 minus 14119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(83)

40 60 80 10020N

f = 57

ns

090

092

094

096

098

100

102

104











119877 = 1198611(

120601 minus 1206010

120573

)

(minus4119891+1)2119891

coth [ 1198962119879

]

= 1198611(

119873

119860

+

1 minus 119891

119891119860

)

(minus4119891+1)4119891

coth [ 1198962119879

]

= 1198611(

4119891119860

11 minus 14119891

(1 minus 119899119904))

(4119891minus1)4119891

(84)

where 1198611

= (232(4120587)238Γ12

012059014(1 minus 119891)

34

3158119872314

4120582158(119891119860)32)(31198911198602120582)









f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R


096 097 098 099095ns

08

09

10

11

12

13

14

15

TH




Γ03HV

06

08

10

12

14

096 097 098 099095ns





cases of Γ0


(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)


119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)


119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)


120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)


119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)



1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)


120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)


119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)



119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




f = 57

40 60 80 10020N

002

004

006

008

010

012

014

016

R


096 097 098 099095ns

08

09

10

11

12

13

14

15

TH




Γ03HV

06

08

10

12

14

096 097 098 099095ns





cases of Γ0


(74) and (75)

120601 minus 1206010= (

4 (1 minus 119891)

Γ1

119905)

12

(85)


119867(120601) = 119891119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

119881 (120601) = (

31205821198722

411989121198602

4120587

)

12

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891minus1

(86)


119903 =

Γ1119881 (120601)

3119867119881

=

Γ1

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

1minus119891

(87)


120598 =

1 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

120578 =

2 minus 119891

119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

(88)


119873(120601) = 119860(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

119891

minus 119860(

Γ1(1206011minus 1206010)2

4 (1 minus 119891)

)

119891

(89)



1206011= 1206010+ [

4 (1 minus 119891)

Γ1

(

1 minus 119891

119891119860

)

1119891

]

12

(90)


120601 = 1206010+ [

4 (1 minus 119891)

Γ1

(

119873

119860

+

1 minus 119891

119891119860

)

1119891

]

12

(91)


119899119904= 1 +

3

4

120578 minus

23

4

120598

= 1 minus

17 minus 20119891

4119891119860

(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

minus119891

= 1 minus

17 minus 20119891

4119891119860

(

119873

119860

+

1 minus 119891

119891119860

)

minus1

(92)



119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119904= 1 where 119891 = 1720 In




119875119877=

25

4

1205752

119867=

1198725

412058212

(2120587)5212059014

119881154

1199031212059834

= 1198602(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(20119891minus17)4119891

= 1198602(

119873

119860

+

1 minus 119891

119891119860

)

(20119891minus17)4119891

119877 = 1198612(

Γ1(120601 minus 120601

0)2

4 (1 minus 119891)

)

(minus10119891+7)4119891

coth [ 1198962119879

]

= 1198612(

119873

119860

+

1 minus 119891

119891119860

)

(minus10119891+7)4119891

= 1198612(

119891119860 (1 minus 119899119904)

20119891 minus 17

)

(10119891minus7)4119891

(93)

where

1198602=

119872254

4120582174

(119891119860)53158

12059014Γ12

1(1 minus 119891)

341205873582254

1198612= (

3119891119860

2120582

)

122398

(2120587)138

12059014Γ12

1(1 minus 119891)

34

3158120582198119872358

4(119891119860)3

I (120601) = minus9

4

ln (119881)

(94)




119886 (119905) = 1198860exp (119860 [ln 119905]]) (95)



(75) and (95)

120601 minus 1206010=

2120596

] + 1(ln 119905)(]+1)2 (96)

f = 56

20 40 60 80 1000N

ns

090

092

094

096

098

100

102

104



Γ13HV

08

09

10

11

12

13

14

15

096 097 098 099095ns





of Γ1

f = 56

100 200 300 4000N

00

05

10

15

R



where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where 120596 = (312058211987224]211986022120587Γ2



119867 =

119860] [(] + 1) (120601 minus 1206010) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

119881 =

Γ01205962[(] + 1) (120601 minus 120601

0) 2120596]

2(]minus1)(]+1)

exp ([(] + 1) (120601 minus 1206010) 2120596]

2(]+1))

(97)


120598 =

1

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

120578 =

2

119860][

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

(98)


119873 = 119860([ln 119905]] minus [ln 1199051]])

= 119860([

(] + 1) (120601 minus 1206010)

2120596

]

2](]+1)

minus [

(] + 1) (1206011 minus 1206010)2120596

]

2](]+1)

)

(99)



120601 = 1206010+

2120596

] + 1(

119873

119860

+ (]119860)](1minus]))(]+1)2]

(100)


119903 =

Γ0

3119867119881

=

1

3 (]119860120596)2exp (2 [(] + 1) (120601 minus 120601

0) 2120596]

2(]+1))

[(] + 1) (120601 minus 1206010) 2120596]

4(]minus1)(]+1)

(101)


119875119877= 1198603exp(minus11

4

[

(] + 1) (120601 minus 1206010)

2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)

2120596

]

7(]minus1)(]+1)

= 1198603

sdot exp(minus114

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]7(]minus1)2]

119877 = 1198613exp(1

4

[

(] + 1) (120601 minus 1206010)2120596

]

2(]+1)

)

sdot [

(] + 1) (120601 minus 1206010)2120596

]

4(1minus])(]+1)

= 1198613

sdot exp(14

(

119873

119860

+ (119860])](1minus]))1])

sdot [

119873

119860

+ (119860])](1minus])]4(1minus])2]

(102)

where

1198603=

1198725

412058212Γ74

012059692

(2120587)5212059012(]119860)minus74

1198613= (

3120587Γ0120596

12058231198722

4

)

1416 (2120587)

5212059014Γ14

0

3 (3120596)12(119860])74

(103)


119899119904= 1 minus

11

4]119860[

(] + 1) (120601 minus 1206010)

2120596

]

2(1minus])(]+1)

= 1 minus

11

4]119860[

119873

119860

+ (119860])](1minus])](1minus])]

(104)



119877 = 1198613exp(1

4

(

4]11986011

[1 minus 119899119904])

1(1minus]))

sdot [

4]11986011

(1 minus 119899119904)]

2

(105)








120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







120601 minus 1206010=

2

radicΓ1

11990512 (106)


119867(120601) =

4119860] (ln (Γ1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119881 (120601)

= (

121205821198722

41198602]2

120587

)

12(ln (Γ

1((120601 minus 120601

0)24)))

]minus1

(120601 minus 1206010)2

119903 =

Γ1

12119860](120601 minus 120601

0)2

(ln (Γ1((120601 minus 120601

0)24)))

]minus1

(107)


120598 =

(ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

120578 =

2 (ln (Γ1((120601 minus 120601

0)24)))

1minus]

119860]

(108)


119873 = 119860[(ln(Γ1

(120601 minus 1206010)2

4

))

]

minus (ln(Γ1

(1206011minus 1206010)2

4

))

]

]

(109)



1206011= 1206010+

2

radicΓ1

exp(12

(119860])](1minus])) (110)


1206011= 1206010+

2

radicΓ1

exp(12

[(119860])](1minus]) +119873

119860

]

1]) (111)



119875119877= 1198604(120601 minus 120601

0)minus172

[ln(Γ1

(120601 minus 1206010)2

4

)]

(20]minus9)4

= 1198604(

radicΓ1

2

)

172

sdot exp(minus174

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](20]minus9)4]

119877 = 1198614(120601 minus 120601

0)72[ln(Γ

1

(120601 minus 1206010)2

4

)]

(minus5]+5)2

= 1198614(

2

radicΓ1

)

72

exp(74

[

119873

119860

+ (119860])](1minus])]1])

sdot [

119873

119860

+ (119860])](1minus])](minus5]+5)2]

(112)

where

1198604=

3198498119872354

4(119860])5

12058735812059014120582minus198

1198614=

41412058727812059014(119860])minus3

3198119872354

4Γ12

1120582198

(

361198602]2

1205871205822)

14

(113)


119899119904= 1 minus

17

4119860](ln[

Γ1(120601 minus 120601

0)2

4

])

= 1 minus

17

8119860][

119873

119860

+ (]119860)](1minus])]1]

(114)


119904= 1) could


119877 (119899119904) = 1198614(

4

Γ1

)

74

exp(74

[

4119860]17

(1 minus 119899119904)]

1(1minus]))

sdot [

4119860]17

(1 minus 119899119904)]

52

(115)




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00

02

04

06

08

10

12

20 40 60 800N

120582 = 5 120582 = 50


120582 = 5 120582 = 50

05 10 15 2000ns

00

02

04

06

08

10

R




0 Γ1) for








ns

120582 = 5 120582 = 50

00

02

04

06

08

10

12

50 100 150 2000N


120582 = 5 120582 = 50

05 10 15 2000ns

0

2

4

6

8

10R



V120582

096 097 098 099095ns

0

20

40

60

80

100






I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





I (120601) = minusint[(Γ119881)

1015840

3119867 + Γ119881

+ (

9

8

sdot

2119867 + Γ119881

(3119867 + Γ119881)2(Γ + 4119867119881 minus

Γ1015840(ln119881)1015840

12119867 (3119867 + Γ119881)

)

sdot

(ln119881)1015840

119881

)]119889120601

(116)


120598 =

1198722

4

16120587

1

1 + 119903

[

1198811015840

119881

]

2

1

119881

120578 =

1198722

4

8120587 (1 + 119903)119881

[

11988110158401015840

119881

minus

1

2

(

1198811015840

119881

)

2

]

(117)


120575119867=

radic3

751205872

exp (minus2I (120601))11990312

119899119904= 1 minus [

3

2

+ (

21198811015840

119881

[2I1015840

(120601) minus

1199031015840

4119903

] minus

5

2

)]

120572119904=

2119881

11988110158401198991015840

119904

119899119892= minus2120598

119877 (1198960)

=

240radic3

251198982

119901

[

119903121198673

119879119903

exp (2I (120601)) coth [ 1198962119879

]]

100381610038161003816100381610038161003816100381610038161003816119896=1198960

(118)



119877 119877 119899

119904 and 120572

119904 Also the



119877index 119899






Appendix


[119888] = 119871119879minus1= 1

[ℎ] = 1198721198712119879minus1

dArr

119879 = 119871 = 119872minus1

(A1)


Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


References






































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Using (7) we have

[1198672] = [

8120587

1198722

4

120588119879(1 +

120588119879

2120582

)] 997904rArr

[1198862]

11988621198792=

[120588119879]

[1198722

4]

997904rArr

[120588119879] = [119879

120583]] = [119881] = [119875] = 1198724

(A2)


[120601] = 1 997904rArr

[120601] = 119872minus1

(A3)


[] + [3119867120588] + [3119867119875] = [Γ120601

2

] 997904rArr

[120588]

119879

+

[120588]

119879

+

[119875]

119879

= [Γ] 997904rArr

[Γ] = 1198725

(A4)


(Γ119881)(13119867)

[119903] =

[Γ]

[119867] [119881]

=

1198725

1198721198724= 1 (A5)



[120575120601] = [119862]

[1198811015840]

[119881]

119872minus1= 119872minus2 1

119872minus1

(A6)


[2119867] +

[Γ]

119881

= 119872 +

1198725

1198724 (A7)


119867] = ([119867][


have from (49)

120575119867= [119872

2

4]

[119881] [120575120601]

[1198811015840]

= 1198722119872minus1119872minus1= 1 (A8)


Competing Interests


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 Tachyon Warm Intermediate and Logamediate...

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