Black Holes: Insights and Enigmasdownloads.hindawi.com/journals/specialissues/519381.pdf · Black Holes: Insights and Enigmas Lead Guest Editor: Izzet Sakalli Guest Editors: Eduardo

Advances in High Energy Physics

Black Holes Insights and Enigmas

Lead Guest Editor Izzet SakalliGuest Editors Eduardo Guendelman Douglas Singleton and Habib Mazharimousavi




Lead Guest Editor Izzet SakalliGuest Editors Eduardo Guendelman Douglas Singletonand Habib Mazharimousavi

Copyright copy 2018 Hindawi All rights reserved

This is a special issue published in ldquoAdvances in High Energy Physicsrdquo All articles are open access articles distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the originalwork is properly cited

Editorial Board

Antonio J Accioly BrazilGiovanni Amelino-Camelia ItalyLuis A Anchordoqui USAMichele Arzano ItalyT Asselmeyer-Maluga GermanyAlessandro Baldini ItalyMarco Battaglia SwitzerlandLorenzo Bianchini SwitzerlandRoelof Bijker MexicoBurak Bilki USAAdrian Buzatu UKRong-Gen Cai ChinaAnna Cimmino BelgiumOsvaldo Civitarese ArgentinaAndrea Coccaro ItalyShi-Hai Dong MexicoLorenzo Fortunato ItalyMariana Frank Canada

Ricardo G Felipe PortugalChao-Qiang Geng TaiwanPhilippe Gras FranceXiaochun He USALuis Herrera SpainFilipe R Joaquim PortugalAurelio Juste SpainTheocharis Kosmas GreeceMing Liu USAEnrico Lunghi USASalvatore Mignemi ItalyOmar G Miranda MexicoBogdan Mitrica RomaniaGreacutegory Moreau FrancePiero Nicolini GermanyCarlos Pajares SpainSergio Palomares-Ruiz SpainGiovanni Pauletta Italy

Yvonne Peters UKAnastasios Petkou GreeceAlexey A Petrov USAThomas Roumlssler SwedenDiego Saez-Chillon Gomez SpainTakao Sakaguchi USAJuan Joseacute Sanz-Cillero SpainEdward Sarkisyan-Grinbaum USASally Seidel USAGeorge Siopsis USALuca Stanco ItalyJouni Suhonen FinlandMariam Toacutertola SpainSmarajit Triambak South AfricaJose M Udiacuteas SpainElias C Vagenas KuwaitSunny Vagnozzi SwedenYau W Wah USA

Contents

Black Holes Insights and EnigmasIzzet Sakalli Eduardo Guendelman Douglas Singleton and S Habib MazharimousaviEditorial (2 pages) Article ID 5874973 Volume 2018 (2018)

P-v Criticality of a Specific Black Hole in f(R)Gravity Coupled with Yang-Mills FieldAli OumlvguumlnResearch Article (7 pages) Article ID 8153721 Volume 2018 (2018)

Field Equations for Lovelock Gravity An Alternative RouteSumanta ChakrabortyResearch Article (6 pages) Article ID 6509045 Volume 2018 (2018)

Discussion of a Possible Corrected Black Hole EntropyMiao He Ziliang Wang Chao Fang Daoquan Sun and Jianbo DengResearch Article (7 pages) Article ID 2315084 Volume 2018 (2018)

TheQuantum Effect on Friedmann Equation in FRWUniverseWei Zhang and Xiao-Mei KuangResearch Article (5 pages) Article ID 6758078 Volume 2018 (2018)

Hawking Radiation-Quasinormal Modes Correspondence for Large AdS Black HolesDao-Quan Sun Zi-Liang Wang Miao He Xian-Ru Hu and Jian-Bo DengResearch Article (6 pages) Article ID 4817948 Volume 2017 (2018)

EditorialBlack Holes Insights and Enigmas

Izzet Sakalli 1 Eduardo Guendelman2 Douglas Singleton 3

and S Habib Mazharimousavi 1

1Physics Department Arts and Sciences Faculty Eastern Mediterranean University Famagusta Northern Cyprus Mersin 10 Turkey2Physics Department Ben-Gurion University of the Negev 84105 Beer-Sheva Israel3Physics Department California State University Fresno Fresno CA 93740 USA

Correspondence should be addressed to Izzet Sakalli izzetsakalliemuedutr

Received 28 February 2018 Accepted 4 March 2018 Published 26 June 2018

Copyright copy 2018 Izzet Sakalli et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

The study of black holes has been a central feature ofgeneral relativity since its birth in 1915 Shortly after Einsteinwrote down the field equations of general relativity KarlSchwarzschild discovered the first and simplest solutionswhich represented the gravitational field of a massive bodywith spherical symmetry but without any other featureslike angular momentum or charge It was noticed thatthis Schwarzschild solution had a place where the metricapparently became singular the Schwarzschild radius whichfor a spherical body of mass 119872 occurred at the distance2Gmc2 It was initially hoped that ldquorealisticrdquo objects couldnever be compressed to the extent that the material ofthe body would be less than this Schwarzschild radius Inthe late 1930s starting with the work of Oppenheimer andSnyder it became clear that it was possible for a materialobject to be compressed past its Schwarzschild radius andbecome a black hole But what of the apparent singularityat the Schwarzschild radius In the early 1960s Kruskaland Szekeres were able to find a coordinate system whichshowed that the Schwarzschild singularity at 119903 = 2Gmc2was an artifact of the coordinates that had been used bySchwarzschild and others The only real singularity in theSchwarzschild solution was the one at the origin namely119903 = 0 Thus the study of black holes has been a prompt forpushing forward the understanding of the physical meaningof general relativity as well as better understanding what thecomplex math was saying physically

In the mid-1970s research led by Bekenstein Hawk-ing and others showed that there was a deep connection

between black hole physics and quantum mechanics andthermodynamics Bekenstein was the first to argue that ablack hole should have an entropy associated with it andthat this entropy should be proportional to the surfacearea of the event horizon as defined by the Schwarzschildradius Following on this work Hawking showed by applyingquantum field theory in the background of a black hole thatblack holes emitted thermal radiation and had a temperaturenow known as the Hawking temperature These seminalworks have led to researchers viewing black holes as atheoretical laboratory to giving hints as to the proper pathtoward a formulation of the as yet undiscovered theory ofquantum gravity This has led to the concepts like blackhole information paradox the holographic principle and thefirewall puzzle and a host of other interesting conjectures andpuzzles

Finally with the advent of more powerful telescopes andobserving equipment both on the ground and in spaceresearchers have begun to gather observational evidence forthe existence of black holes including finding that in theheart of most galaxies there are enormous black holes ofmillion plus solar masses In our galaxy this ldquocenter of thegalaxyrdquo black hole is called Sagittarius AlowastMore recently withthe detection of gravitational waves by the LIGO scientificcollaboration we have very strong evidence of binary blackhole coalescence which resulted in the awarding of the 2017Physics Nobel Prize

This special issue is devoted to works which carry onthe tradition of studying various aspects of black holes to

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 5874973 2 pageshttpsdoiorg10115520185874973

2 Advances in High Energy Physics

understand the workings of gravity as well as other branchesof physics

In the paper by D-Q Sun et al entitled ldquoHawkingRadiation-Quasinormal Modes Correspondence for LargeAdS Black Holesrdquo the authors investigate the not exactlythermal nature of the Hawking radiation emitted by a blackhole They find a connection between Hawking radiationand the quasi-normal modes of black holes They apply theiranalysis to Schwarzschild Kerr and nonextremal Reissner-Nordstrom black holesThey pay particular attention to theseblack holes in anti-de Sitter spacetime

In the paper ldquoP-V Criticality of a Specific Black Hole in119891(119877)Gravity Coupled with Yang-Mills Fieldrdquo by A Ovgun astudy is made of specific charge anti-de Sitter black holes in119891(119877) gravity with a Yang-Mills field Using thermodynamicsanalogies it is found that this complex gravitational systembehaves like the thermodynamic system of a van der Waalsgas at critical points of the system It is also found that thephase transition between small and large specific charge AdSblack holes is a first-order phase transition

In the paper by S Chakraborty entitled ldquoField Equationsfor Lovelock Gravity An Alternative Routerdquo the authorstudies an alternative derivation of the gravitational fieldequations for Lovelock gravity starting from Newtonrsquos lawwhich is closer in spirit to the thermodynamic descrip-tion of gravity Projecting the Riemann curvature tensorappropriately and taking a cue from Poissonrsquos equationthe generalized Einsteinrsquos equations in the Lanczos-Lovelocktheories of gravity are derived

In the paper byW Zhang and X-M Kuang entitled ldquoTheQuantum Effect on Friedmann Equation in FRW Universerdquothe quantummechanically modified Friedmann equation forthe Friedmann-Robertson-Walker universe is derived Theauthors also analyze themodified Friedmann equations usingthe conjecture by Padmanabhan of a modified entropy-arearelation

In the paper by M He et al entitled ldquoDiscussion of aPossible Corrected Black Hole Entropyrdquo by using T Pad-manabhanrsquos local formalism which avoids addressing theasymptotically flat structure of spacetimes the authors derivethe analogue of the first law of thermodynamics for the AdSblack holes in Eddington-inspired Born-Infeld (EiBI) gravitywith and without matter fields The same formalism has alsobeen used to express Einsteinrsquos field equations in the formof the first law of thermodynamics for a static sphericallysymmetric spacetime near any horizon of a definite radiusIn accordance with their results the authors conclude thatsince Einstein gravity and EiBI gravity give the same resultsfrom thermodynamicsrsquo point of view both of these theoriesof gravity could be equivalent on the event horizon

The present volume collects together works which useblack holes as a theoretical laboratory for understandinghow gravity works and how gravity might fit in with theother theories of modern physics and in particular quantummechanics

Izzet SakalliEduardo Guendelman

Douglas SingletonS Habib Mazharimousavi

Research Article119875-V Criticality of a Specific Black Hole in 119891(119877) Gravity Coupledwith Yang-Mills Field

Ali Oumlvguumln 123

1 Instituto de Fısica Pontificia Universidad Catolica de Valparaıso Casilla Postal 4950 Valparaıso Chile2Physics Department Arts and Sciences Faculty Eastern Mediterranean University Famagusta Northern Cyprus Mersin 10 Turkey3TH Division Physics Department CERN 1211 Geneva 23 Switzerland

Correspondence should be addressed to Ali Ovgun aliovgunpucvcl

Received 17 November 2017 Revised 18 January 2018 Accepted 30 January 2018 Published 25 June 2018

Academic Editor Douglas Singleton

Copyright copy 2018 Ali Ovgun This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We study the 119875-V criticality of a specific charged AdS type black hole (SBH) in 119891(119877) gravity coupled with Yang-Mills fieldIn the extended phase space we treat the cosmological constant as a thermodynamic pressure After we study the variousthermodynamical quantities we show that the thermodynamic properties of the SBH behave as a Van der Waals liquid-gas systemat the critical points and there is a first-order phase transition between small-large SBH

1 Introduction

Important contribution on black holesrsquo thermodynamics inanti-de-Sitter (AdS) spacetime is made by Hawking and Page[1] where a first-order phase transition is discovered betweenthe Schwarzschild-anti-de-Sitter (SAdS) black holes that isknown as the Hawking-Page transition Then Chamblin etal and Cvetic et al show that the first-order phase transitionamong Reissner Nordstrom (RN) AdS black holes and thesimilarities between charged AdS black holes and liquid-gassystems in grand canonical ensemble [2ndash4] Moreover in theseminal papers of Kubiznak and Mann [5] the cosmologicalconstant Λ is used as dynamical pressure [6]

119875 = minus Λ8120587 = 3

81205871198972 (1)

for the RN-AdS black holes in the extended phase spaceinstead of treating the Λ as a fixed parameter (in standardthermodynamic) and its conjugate variable has dimension ofvolume

119881 = (120597119872120597119875 )119878119876

(2)

Calculating the critical components and finding the phasetransition of the RN-AdS black holes it is shown that RN-AdS

black holes behave similar to the Van der Waals fluid in theextended phase space where a first-order smalllarge blackholersquos phase transition occurs at a critical temperature below[7ndash11] The Van der Waals equation

(119875 + 119886V2

) (V minus 119887) = 119896119879 (3)

where its pressure is 119875 its temperature is 119879 its specificvolume is V = 119881119873 the Boltzmann constant is 119896 andthe positive constants are 119886 and 119887 takes into account theattractive and repulsive forces between molecules and givesan improved model for ideal gas behaviour to describe thebasic properties of the liquid-gas phase transition with theratio of 119875119888119881119888119879119888 = 38 at critical points [12ndash17] Afterwardsapplications of the thermodynamical lawrsquos to the black holersquosphysics have gain attention Different researches are done byusing the variation of the first law of thermodynamics of blackholes and also application of the 119875-V criticality on black holes[18ndash88] Furthermore AdS-CFT correspondence is the otherreason for studying the AdS black hole

In this paper we use a black holersquos solution in the Yang-Mills field which is the one of themost interesting nonabeliangauge theories By using the string theory models they findthe Yang-Mills fields equations in low energy limit and then



Yasskin found the first black hole solution in the theory ofYang-Mills coupled to Einstein theory [89]

Our main aim is to check 119875-119881 criticality of a specificcharged AdS type black hole (SBH) in 119891(119877) gravity coupledwith Yang-Mills field (YMF) [90] by comparing its resultwith the Van der Waals system The Yang-Mills field isacted inside the nuclei with short range and 119891(119877) gravitywhich is an extension of Einsteinrsquos General Relativity with thearbitrary function of Ricci scalar 119891(119877) [91ndash94] It would beof interest to study the 119875-V criticality of SBH in 119891(119877) gravitycoupled with YMF in the extended phase space treating thecosmological constant as a thermodynamic pressure In thispaper we first study the thermodynamics in the extendedphase space and then we obtain its critical exponents to showthe existence of the Van der Waals like small-large black holephase transitions

The paper is organized as follows in Section 2 we willbriefly review the SBH in 119891(119877) gravity coupled with YMF InSection 3 119875-V criticality of the SBH in 119891(119877) gravity coupledwith YMF will be studied in the extended phase space bycalculating its critical exponents In Section 4 we concludewith final remarks

2 SBH in 119891(119877) Gravity Coupled with YMF

In this section we briefly present a solution of SBH in 119891(119877)gravity coupled with YMF with a cosmological constant in119889-dimensions [90]Then we discuss its temperature entropyand other thermodynamic quantities The action of the 119891(119877)gravity minimally coupled with YMF (119888 = 119866 = 1) is [90]

119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)

where 119891(119877) is a function of the Ricci scalar 119877 and L(119865)stands for the Lagrangian of the nonlinear YMF with 119865 =(14)tr(119865(119886)120583] 119865(119886)120583]) where the 2-form components of the YMFareF(119886) = (12)119865(119886)120583] 119889119909120583and119889119909] Here is the internal index (119886) forthe degrees of freedomof the nonabelian YMF It is noted thatthis nonlinear YMF can reduce to linear YM field (L(119865) =minus(14120587)119865119904) for 119904 = 1 and 119891119877 = 119889119891(119877)119889119877 = 120578119903 which 120578 isan integration constant Solving Einstein field equations forthe 119891(119877) gravity coupled with YMF gives to the sphericallysymmetric black hole metrics (see equation (36) in [90])

1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)

+ 1199032 (11988912057921 + Σ119889minus2119894=2 Π119894minus1119895=1sin21205791198951198891205792119894 )(5)

with 0 le 120579119889minus2 le 2120587 0 le 120579119894 le 120587 1 le 119894 le 119889 minus 3 in which themetric function 119891(119903) is

119891 (119903)= 119889 minus 3

119889 minus 2 minus Λ1199032 minus 119898119903119889minus2

minus (119889 minus 1) (119889 minus 2)(119889minus1)2 (119889 minus 3)(119889minus1)42(119889minus5)2120578119889

119876(119889minus1)2 ln 119903119903119889minus2

(6)

Note that Λ = minus11198972 119872 is the mass of the black holeand 120578 is a constant Furthermore for the limit of 11987672 rarr0 it becomes well-known solutions in 119891(119877) gravity TheBekenstein-Hawking temperature [95ndash98] of the black holeis calculated by 119879 = (14120587)(120597119891(119903)119889119903)|119903=119903ℎ

119879 = (minus1 + (119889 minus 2) ln (119903)) 1198761198892minus12 (119889 minus 1) (119889 minus 2)1198892minus12 (119889 minus 3)1198894minus14 + 11988921198892minus52 (minus21199032Λ119903119889minus2 + 119898 (119889 minus 2)) 1205784120587119903119889minus212057811988921198892minus52119903 (7)

where 119903ℎ is the horizon of the black hole and solving theequation 119891(119903ℎ) = 0 the total mass of the black hole isobtained as

119898

= minus41198761198892minus12 (1198892 minus 1)1198892 ln (119903)radic2 (119889 minus 1) (119889 minus 2) (119889 minus 3)1198894minus14 + (((119891 minus 1) 119889 minus 2119891 + 3) 119903119889minus2 + 119903119889Λ (119889 minus 2)) 119889radic119889 minus 2120578120578119889 (119889 minus 2)32 (8)

The entropy of the black hole can be derived as

119878 = 119860ℎ4 120578119903ℎ (9)

where119860ℎ = ((119889minus1)Γ((119889+1)2))120587(119889minus1)2119903119889minus2ℎ is the area of theblack holersquos event horizonThen in the extended phase spacewe calculate the pressure in terms of cosmological constant

119875 = minus Λ8120587 (10)

and its thermodynamic volume is

119881 = Ω119889minus2119903119889minus1ℎ 120578119899 minus 1 (11)

Advances in High Energy Physics 3

where Ω119889minus2 is the volume of the unit sphere Now the masscan be also written in terms of 119875 as follows

119898

= minus41198761198892minus12 (1198892 minus 1)1198892 ln (119903)radic2 (119889 minus 1) (119889 minus 2) (119889 minus 3)1198894minus14 + (((119891 minus 1) 119889 minus 2119891 + 3) 119903119889minus2 minus 8119903119889119875120587 (119889 minus 2)) 119889radic119889 minus 2120578(119889 minus 2)32 120578119889 (12)

The first law of the black hole thermodynamics in theextended phase space is

119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)

where the thermodynamic variables can be obtained as 119879 =(120597119898120597119878)119876119875 Φ = (120597119898120597119876)119878119875 and 119881 = (120597119898120597119875)119878119876

Thenwewrite the generalized Smarr relation for the blackhole which can be derived also using the dimensional scalingas

119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)

We introduce the cosmological constant as thermody-namic pressure in the extended phase space in (10) and it isseen that the first law of the black holersquos thermodynamics andthe Smarr relations is matched well

3 119875-V CriticalityIn this section we investigate the critical behaviour of theSBH in the extended phase space The critical point can bedefined as

120597119875120597V = 1205972119875

120597V2 = 0 (15)

Now we consider the case of four dimensions (119889 = 4) wherethe metric function becomes

119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)

and corresponding mass of the black hole is calculated as

119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)

The temperature of the four dimensional SBH is

119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)

Thenwewrite the temperature in terms of119875 (119875 = minusΛ8120587)as follows

119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)

Afterwards one can easily obtain the pressure 119875 in terms ofthe temperature 119879

119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)

To consider the 119875-V criticality using the extended phasespace we write the black hole radius in terms of the specificvolume V as 119903ℎ = (119889 minus 2)V4 Using the condition of (15)we derive the critical Bekenstein-Hawking temperature 119879119888critical pressure 119875119888 and critical specific volume V119888 as follows

119879119888 = minus24 11987632V3120578120587 + 1

V120587

V119888 = 24radic2radic11987632radic120578

119875119888 = 120578115211987632120587

(21)

One can also find this relation which is same with a Vander Waals fluid

120588119888 = 119875119888V119888119879119888 = 3

8 (22)

It is noted that Figures 1 and 2 show that 119875-119903 diagram isthe same with the diagram of the Van der Waals liquid-gassystem

Let us now analyze the Gibbs free energy of the systemWe first use the mass as enthalpy instead of internal energyand the Gibbs free energy in the extended phase space forthe SBH in 119891(119877) gravity coupled with Yang-Mills field iscalculated as

119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)

We plot the change of the free energy119866with119879 in Figure 3There is a small-large black hole phase transition as seen inFigure 3

4 Conclusion

In this paper we first treat the cosmological constant Λas a thermodynamical pressure 119875 and the thermodynamics


T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r

Figure 1 119875-119903 diagram of a SBH in a 119891(119877) gravity coupled with YMFfor 119876 = 01 and 120578 = 1

005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc

Figure 2119875-119879 diagramof a SBH in a119891(119877) gravity coupledwith YMFfor 119876 = 01 and 120578 = 1

and 119875-V criticality of the SBH in 119891(119877) gravity coupled withYMF is studied in the extended phase space It is shownthat there is a phase transition between small-large blackholes Furthermore after we obtain the critical exponentsthe critical behaviour of SBH in 119891(119877) gravity coupled withYMF in the extended space behaves also similarly as Van derWaals liquid-gas systems with the ratio of 119875119888V119888119879119888 = 38at critical points Hence it would be of great importance toobtain the 119875-119881 criticality of SBH in 119891(119877) gravity coupledwith YMF Hence the critical ratio 119875119888V119888119879119888 = 38 is universaland independent from the modified gravitiesThe YMF has aparameter of 120578 but has no effect on the universal ratio of 38

It is also interesting to study the holographic duality ofSBH in 119891(119877) gravity coupled with YMF It is noted thatwithout thermal fluctuations black hole is holographic dualwith Van der Waals fluid given by (119875 + 1198861198812)(119881 minus 119887) = 119879where 119896 is the Boltzmann constant [84 85] 119887 gt 0 is nonzero

P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc

Figure 3119866-119879 diagramof a SBH in a119891(119877) gravity coupledwith YMFfor different values of 119875 (119875 lt 119875119888 119875 = 119875119888 and 119875 gt 119875119888) with 119876 = 01and 120578 = 1

constant which is the size of the molecules of fluid and theconstant 119886 gt 0 is a value of the interaction measurementbetween molecules We leave this problem for the futureprojects

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Chilean FONDECYT Grantno 3170035 (Ali Ovgun) Ali Ovgun is grateful to theCERN Theory (CERN-TH) Division Waterloo UniversityDepartment of Physics and Astronomy and also PerimeterInstitute forTheoretical Physics for hosting him as a researchvisitor where part of this work was done It is a pleasure tothank Professor Robert B Mann for valuable discussions

References

[1] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 198283

[2] A Chamblin R Emparan C V Johnson and R C MyersldquoCharged AdS black holes and catastrophic holographyrdquo Physi-cal Review D Particles Fields Gravitation and Cosmology vol60 no 6 1999

[3] A Chamblin R Emparan C V Johnson and R C MyersldquoHolography thermodynamics and fluctuations of charged


AdS black holesrdquo Physical Review D Particles Fields Gravita-tion and Cosmology vol 60 no 10 Article ID 104026 1999

[4] M Cvetic G W Gibbons D Kubiznak and C N Pope ldquoBlackhole enthalpy and an entropy inequality for the thermodynamicvolumerdquo Physical Review D vol 84 2011

[5] D Kubiznak and R B Mann ldquo119875 minus 119881 criticality of charged AdSblack holesrdquo Journal of High Energy Physics vol 1207 article 332012

[6] B PDolan ldquoThe cosmological constant and black-hole thermo-dynamic potentialsrdquoClassical andQuantumGravity vol 28 no12 Article ID 125020 2011

[7] S Gunasekaran D Kubiznak and R BMann ldquoExtended phasespace thermodynamics for charged and rotating black holesand Born-Infeld vacuum polarizationrdquo Journal of High EnergyPhysics vol 2012 no 11 article 110 2012

[8] D Kubiznak and R B Mann ldquoBlack hole chemistryrdquo Can JPhys vol 93 no 9 p 999 2015

[9] D Kubiznak R B Mann and M Teo ldquoBlack hole chemistrythermodynamics with lambdardquoClassical andQuantumGravityvol 34 no 6 2017

[10] R A Hennigar R B Mann and E Tjoa ldquoSuperfluid blackholesrdquo Physical Review Letters vol 118 no 2 Article ID 0213012017

[11] A M Frassino R B Mann and F Simovic httpsarxivorgabs161103525

[12] R A Hennigar E Tjoa and R B Mann ldquoThermodynamics ofhairy black holes in Lovelock gravityrdquo Journal of High EnergyPhysics vol 2017 no 2 article no 70 2017

[13] S H Hendi R B Mann S Panahiyan and B Eslam PanahldquoVan der Waals like behavior of topological AdS black holes inmassive gravityrdquo Physical ReviewD Particles Fields Gravitationand Cosmology vol 95 no 2 Article ID 021501 2017

[14] D Hansen D Kubiznak and R B Mann ldquoUniversality of P-V Criticality in Horizon Thermodynamicsrdquo Journal of HighEnergy Physics vol 1701 no 47 2017

[15] R B Mann The Chemistry of Black Holes vol 170 SpringerProcPhys 2016

[16] R A Hennigar W G Brenna and R B Mann ldquoP-v criticalityin quasitopological gravityrdquo Journal of High Energy Physics vol2015 no 7 article 77 pp 1ndash31 2015

[17] T Delsate and R Mann ldquoVan Der Waals black holes in ddimensionsrdquo Journal of High Energy Physics vol 2015 no 2article no 70 2015

[18] S Chen X Liu C Liu and J Jing ldquoP-V criticality of AdSblack hole in f (R) gravityrdquo Chinese Physics Letters vol 30 no6 Article ID 060401 2013

[19] R-G Cai L-M Cao L Li and R-Q Yang ldquoP-V criticality inthe extended phase space of Gauss-Bonnet black holes in AdSspacerdquo Journal of High Energy Physics vol 2013 no 9 article 52013

[20] A Belhaj M Chabab H E Moumni and M B Sedra ldquoOnthermodynamics of AdS black holes in arbitrary dimensionsrdquoChinese Physics Letters vol 29 no 10 Article ID 100401 2012

[21] K Bhattacharya B R Majhi and S Samanta ldquovan der Waalscriticality in AdS black holes A phenomenological studyrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 96 no 8 2017

[22] K Jafarzade and J Sadeghi ldquoThe thermodynamic efficiencyin static and dynamic black holesrdquo International Journal ofTheoretical Physics vol 56 no 11 pp 3387ndash3399 2017

[23] J Sadeghi and A S Kubeka ldquoP-V criticality of modified BTZblack holerdquo International Journal of Theoretical Physics vol 55no 5 pp 2455ndash2459 2016

[24] J Liang ldquoThe P-v Criticality of a Noncommutative Geometry-Inspired Schwarzschild-AdS Black Holerdquo Chinese Physics Let-ters vol 34 no 8 Article ID 080402 2017

[25] P Pradhan ldquoEnthalpy geometric volume and logarithmiccorrection to entropy for van der Waals black holerdquo EPL (Euro-physics Letters) vol 116 no 1 Article ID 10001 2016

[26] X-X Zeng and L-F Li ldquoHolographic phase transition probedby nonlocal observablesrdquo Advances in High Energy Physics vol2016 Article ID 6153435 2016

[27] J-X Mo G-Q Li and X-B Xu ldquoCombined effects off(R) gravity and conformally invariant Maxwell field on theextended phase space thermodynamics of higher-dimensionalblack holesrdquo The European Physical Journal C vol 76 no 10article no 545 2016

[28] B RMajhi and S Samanta ldquoP-V criticality of AdS black holes ina general frameworkrdquoPhysics Letters B Particle Physics NuclearPhysics and Cosmology vol 773 pp 203ndash207 2017

[29] Z-Y Fan ldquoCritical phenomena of regular black holes in anti-deSitter space-timerdquoThe European Physical Journal C vol 77 no4 article no 266 2017

[30] Y G Miao and Y M Wu ldquoThermodynamics of the Schwarz-schild-AdS black hole with a minimal lengthrdquoAdvances in HighEnergy Physics vol 2017 14 pages 2017

[31] S He L F Li and X X Zeng ldquoHolographic van der waals-likephase transition in the GaussndashBonnet gravityrdquo Nuclear PhysicsB vol 915 pp 243ndash261 2017

[32] A Mandal S Samanta and B R Majhi ldquoPhase transition andcritical phenomena of black holes A general approachrdquoPhysicalReview D Particles Fields Gravitation and Cosmology vol 94no 6 Article ID 064069 2016

[33] D Chen G qingyu and J Tao ldquoThe modified first laws ofthermodynamics of anti-de Sitter and de Sitter spacendashtimesrdquoNuclear Physics B vol 918 pp 115ndash128 2017

[34] A Dehyadegari A Sheykhi and A Montakhab ldquoCriticalbehavior and microscopic structure of charged AdS black holesvia an alternative phase spacerdquo Physics Letters B vol 768 pp235ndash240 2017

[35] S H Hendi S Panahiyan B Eslam Panah M Faizal and MMomennia ldquoCritical behavior of charged black holes in Gauss-Bonnet gravityrsquos rainbowrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 2 Article ID 0240282016

[36] Y-B Ma R Zhao and S Cao ldquoQndashΦ criticality in the extendedphase space of (n+ 1) -dimensional RN-AdS black holesrdquo TheEuropean Physical Journal C vol 76 no 12 article no 669 2016

[37] H Liu and X-H Meng ldquoP-V criticality in the extended phase-space of charged accelerating AdS black holesrdquoModern PhysicsLetters A vol 31 no 37 Article ID 1650199 2016

[38] M B J Poshteh and N Riazi ldquoPhase transition and ther-modynamic stability in extended phase space and chargedHoravandashLifshitz black holesrdquo General Relativity and Gravita-tion vol 49 no 5 article no 64 2017

[39] X Guo H Li L Zhang and R Zhao ldquoThe critical phenomenaof charged rotating de Sitter black holesrdquoClassical andQuantumGravity vol 33 no 13 Article ID 135004 2016

[40] H-F Li M-S Ma and Y-Q Ma ldquoThermodynamic propertiesof black holes in de Sitter spacerdquoModern Physics Letters A vol32 no 2 1750017 12 pages 2017


[41] M-S Ma and R-H Wang ldquoPeculiar P-V criticality of topolog-ical Horava-Lifshitz black holesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 96 no 2 Article ID024052 2017

[42] S Upadhyay B Pourhassan and H Farahani ldquoP-V criticalityof first-order entropy corrected AdS black holes in massivegravityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 95 no 10 Article ID 106014 2017

[43] S Fernando ldquoPhase transitions of black holes in massivegravityrdquo Modern Physics Letters A vol 31 no 16 Article ID1650096 2016

[44] S Fernando ldquoP-V criticality in AdS black holes of massivegravityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 94 no 12 Article ID 124049 2016

[45] J Sadeghi B Pourhassan and M Rostami ldquoP-V criticality oflogarithm-corrected dyonic charged AdS black holesrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 94no 6 Article ID 064006 2016

[46] Y-G Miao and Z-M Xu ldquoPhase transition and entropyinequality of noncommutative black holes in a new extendedphase spacerdquo Journal of Cosmology and Astroparticle Physicsvol 2017 no 3 article no 046 2017

[47] P Cheng S-W Wei and Y-X Liu ldquoCritical phenomena inthe extended phase space of Kerr-Newman-AdS black holesrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 94 no 2 Article ID 024025 2016

[48] X Guo H Li L Zhang and R Zhao ldquoThe phase transitionof higher dimensional charged black holesrdquo Advances in HighEnergy Physics vol 2016 Article ID 7831054 10 pages 2016

[49] J Liang C-B Sun and H-T Feng ldquoP-V criticality in theextended phase space of charged f(R) black holes in AdS space-timerdquo EPL (Europhysics Letters) vol 113 no 3 Article ID 300082016

[50] L C Zhang and R Zhao ldquoThe critical phenomena ofSchwarzschild-de Sitter black holerdquo EPL (Europhysics Letters)vol 113 no 1 Article ID 10008 2016

[51] J-X Mo G-Q Li and X-B Xu ldquoEffects of power-lawMaxwellfield on the critical phenomena of higher dimensional dilatonblack holesrdquo Physical ReviewD Particles Fields Gravitation andCosmology vol 93 no 8 Article ID 084041 2016

[52] MH Dehghani A Sheykhi and Z Dayyani ldquoCritical behaviorof Born-Infeld dilaton black holesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 93 no 2 Article ID024022 2016

[53] X X Zeng and L F Li ldquoVan der waals phase transition in theframework of holographyrdquo Physics Letters B vol 764 pp 100ndash108 2017

[54] RMaity P Roy and T Sarkar ldquoBlack hole phase transitions andthe chemical potentialrdquo Physics Letters B vol 765 pp 386ndash3942017

[55] S H Hendi R M Tad Z Armanfard and M S TalezadehldquoExtended phase space thermodynamics and PndashV critical-ity bransndashdickendashbornndashinfeld vs einsteinndashbornndashinfeld-dilatonblack holesrdquo The European Physical Journal C vol 76 no 5article 263 2016

[56] A Karch and B Robinson ldquoHolographic black hole chemistryrdquoJournal of High Energy Physics vol 2015 no 12 article no 73pp 1ndash15 2015

[57] S-WWei P Cheng and Y-X Liu ldquoAnalytical and exact criticalphenomena of d -dimensional singly spinning Kerr-AdS blackholesrdquo Physical Review D Particles Fields Gravitation andCosmology vol 93 no 8 Article ID 084015 2016

[58] J Xu L-M Cao and Y-P Hu ldquoP-V criticality in the extendedphase space of black holes in massive gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 91 no 12Article ID 124033 2015

[59] J-X Mo and W-B Liu ldquohase transitions geometrothermody-namics and critical exponents of black holes with conformalanomalyrdquo Advances in High Energy Physics vol 2014 Article ID739454 10 pages 2014

[60] S H Hendi and Z Armanfard ldquoExtended phase space ther-modynamics and P - V criticality of charged black holes inBransndashDicke theoryrdquoGeneral Relativity and Gravitation vol 47no 10 article no 125 2015

[61] Z Sherkatghanad B Mirza Z Mirzaiyan and S A H Man-soori ldquoCritical behaviors and phase transitions of black holes inhigher order gravities and extended phase spacesrdquo InternationalJournal of Modern Physics D vol 26 no 3 Article ID 17500172017

[62] M Zhang Z-Y Yang D-C Zou W Xu and R-H Yue ldquoPndashVcriticality of AdS black hole in the EinsteinndashMaxwellndashpower-YangndashMills gravityrdquo General Relativity and Gravitation vol 47no 2 2015

[63] M H Dehghani S Kamrani and A Sheykhi ldquoP-V criticalityof charged dilatonic black holesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 90 no 10 Article ID104020 2014

[64] S H Hendi S Panahiyan and B Eslam Panah ldquoP - V criticalityand geometrical thermodynamics of black holes with Born-Infeld type nonlinear electrodynamicsrdquo International Journal ofModern Physics D vol 25 no 1 Article ID 1650010 2016

[65] D-C Zou S-J Zhang and BWang ldquoCritical behavior of Born-Infeld AdS black holes in the extended phase space thermo-dynamicsrdquo Physical Review D Particles Fields Gravitation andCosmology vol 89 Article ID 044002 2014

[66] D C Zou Y Liu and B Wang ldquoCritical behavior of chargedGauss-Bonnet-AdS black holes in the grand canonical ensem-blerdquo Physical Review D Particles Fields Gravitation and Cos-mology vol 90 no 4 Article ID 044063 2014

[67] M Zhang D-C Zou and R-H Yue ldquoReentrant phase transi-tions and triple points of topological AdS black holes in Born-Infeld-massive gravityrdquo Advances in High Energy Physics ArtID 3819246 11 pages 2017

[68] D Zou Y Liu and R Yue ldquoBehavior of quasinormalmodes andVan der Waals-like phase transition of charged AdS black holesin massive gravityrdquoThe European Physical Journal C vol 77 no6 2017

[69] D Zou R Yue and M Zhang ldquoReentrant phase transitions ofhigher-dimensional AdS black holes in dRGT massive gravityrdquoThe European Physical Journal C vol 77 no 4 2017

[70] C-Y Zhang S-J Zhang D-C Zou and B Wang ldquoChargedscalar gravitational collapse in de Sitter spacetimerdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 93no 6 Article ID 064036 2016

[71] D-C Zou Y Liu C-Y Zhang and BWang ldquoDynamical probeof thermodynamical properties in three-dimensional hairy AdSblack holesrdquo EPL (Europhysics Letters) vol 116 no 4 Article ID40005 2016

[72] D-C Zou Y Liu B Wang and W Xu ldquoThermodynamicsof rotating black holes with scalar hair in three dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 90 no 10 Article ID 104035 2014

[73] H-H Zhao L-C Zhang M-S Ma and R Zhao ldquoP-V critical-ity of higher dimensional charged topological dilaton de Sitter


black holesrdquo Physical ReviewD Particles Fields Gravitation andCosmology vol 90 no 6 Article ID 064018 2014

[74] G-Q Li ldquoEffects of dark energy on P-V criticality of chargedAdS black holesrdquo Physics Letters B vol 735 pp 256ndash260 2014

[75] J-X Mo and W-B Liu ldquoP-V criticality of topological blackholes in Lovelock-Born-Infeld gravityrdquo The European PhysicalJournal C vol 74 article 2836 2014

[76] R Zhao H H ZhaoM SMa and L C Zhang ldquoOn the criticalphenomena and thermodynamics of charged topological dila-ton AdS black holesrdquo The European Physical Journal C vol 73no 12 pp 1ndash10 2013

[77] X Kuang and O Miskovic ldquoThermal phase transitions ofdimensionally continued AdS black holesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 95 no 4 2017

[78] X-M Kuang and J-P Wu ldquoEffect of quintessence on holo-graphic fermionic spectrumrdquoThe European Physical Journal Cvol 77 no 10 article no 670 2017

[79] C-J Luo X-M Kuang and F-W Shu ldquoCharged Lifshitz blackhole and probed Lorentz-violation fermions from holographyrdquoPhysics Letters B vol 769 pp 7ndash13 2017

[80] L Aranguiz X-M Kuang and OMiskovic ldquoTopological blackholes in pure Gauss-Bonnet gravity and phase transitionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 93 no 6 Article ID 064039 2016

[81] D Kastor S Ray and J Traschen ldquoEnthalpy and the mechanicsof AdS black holesrdquo Classical and Quantum Gravity vol 26 no19 Article ID 195011 2009

[82] M Azreg-Aınou ldquoBlack hole thermodynamics no inconsis-tency via the inclusion of the missing P-V termsrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 91no 6 Article ID 064049 2015

[83] M Azreg-Aınou ldquoCharged de Sitter-like black holes quint-essence-dependent enthalpy and new extreme solutionsrdquo TheEuropean Physical Journal C vol 75 no 1 pp 1ndash13 2015

[84] E Caceres P H Nguyen and J F Pedraza ldquoHolographicentanglement entropy and the extended phase structure of STUblack holesrdquo Journal of High Energy Physics vol 2015 no 9article no 184 2015

[85] C V Johnson ldquoHolographic heat enginesrdquo Classical and Quan-tum Gravity vol 31 no 20 p 205002 2014

[86] M Momennia ldquoReentrant phase transition of Born-Infeld-dilaton black holesrdquo httpsarxivorgabs170909039

[87] S H Hendi and M Momennia ldquoAdS charged black holes inEinsteinndashYangndashMills gravityrsquos rainbowThermal stability and Pminus V criticalityrdquo Physics Letters B vol 777 pp 222ndash234 2018

[88] S H Hendi B Eslam Panah S Panahiyan and M MomennialdquoThree dimensional magnetic solutions in massive gravity with(non)linear fieldrdquo Physics Letters B Particle Physics NuclearPhysics and Cosmology vol 775 pp 251ndash261 2017

[89] P B Yasskin ldquoSolutions for gravity coupled to massless gaugefieldsrdquo Physical Review D Particles Fields Gravitation andCosmology vol 12 no 8 pp 2212ndash2217 1975

[90] S HMazharimousavi andMHalilsoy ldquo Black hole solutions inrdquo Physical Review D Particles Fields Gravitation and Cosmol-ogy vol 84 no 6 2011

[91] S Chakraborty and S SenGupta ldquoSpherically symmetric branespacetimewith bulk f (R) gravityrdquoTheEuropeanPhysical JournalC vol 75 no 1 article no 11 2015

[92] S Nojiri and S D Odintsov ldquoAnti-evaporation of Schwarz-schild-de Sitter black holes in F(R) gravityrdquo Classical andQuantum Gravity vol 30 no 12 Article ID 125003 2013

[93] S Nojiri and S D Odintsov ldquoInstabilities and anti-evaporationof ReissnerndashNordstrom black holes in modified F(R) gravityrdquoPhysics Letters B vol 735 pp 376ndash382 2014

[94] S Nojiri and S Odintsov ldquoRegular multihorizon black holesin modified gravity with nonlinear electrodynamicsrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 96no 10 2017

[95] I Sakalli A Ovgun and K Jusufi ldquoGUP assisted Hawkingradiation of rotating acoustic black holesrdquo Astrophysics andSpace Science vol 361 no 10 article no 330 2016

[96] I Sakalli and A Ovgun ldquoBlack hole radiation of massive spin-2 particles in (3+1) dimensionsrdquoThe European Physical JournalPlus vol 131 no 6 article no 184 2016

[97] I Sakalli and A Ovgun ldquoQuantum tunneling of massive spin-1 particles from non-stationary metricsrdquo General Relativity andGravitation vol 48 no 1 article no 1 pp 1ndash10 2016

[98] I Sakalli and A Ovgun ldquoTunnelling of vector particles fromLorentzian wormholes in 3+1 dimensionsrdquoThe European Phys-ical Journal Plus vol 130 no 6 article 110 2015

Research ArticleField Equations for Lovelock Gravity An Alternative Route

Sumanta Chakraborty

Department of Theoretical Physics Indian Association for the Cultivation of Science Kolkata 700032 India

Correspondence should be addressed to Sumanta Chakraborty sumantacphysicsgmailcom

Received 20 December 2017 Revised 9 February 2018 Accepted 20 February 2018 Published 19 April 2018

Academic Editor Izzet Sakalli

Copyright copy 2018 Sumanta Chakraborty 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

We present an alternative derivation of the gravitational field equations for Lovelock gravity starting from Newtonrsquos law whichis closer in spirit to the thermodynamic description of gravity As a warm up exercise we have explicitly demonstrated thatprojecting the Riemann curvature tensor appropriately and taking a cue from Poissonrsquos equation Einsteinrsquos equations immediatelyfollowThe above derivation naturally generalizes to Lovelock gravity theories where an appropriate curvature tensor satisfying thesymmetries as well as the Bianchi derivative properties of the Riemann tensor has to be used Interestingly in the above derivationthe thermodynamic route to gravitational field equations suited for null hypersurfaces emerges quiet naturally

1 Introduction

Equivalence principle acts as the guiding lighthouse tounderstand how matter fields behave in a curved spacetimebackground Unfortunately there exists no such principlewhich helps to answer the opposite namely howmatter fieldscurve the spacetime [1ndash7] Lack of such principle resultedin the vast landscape of various alternative gravity theoriesamong which general relativity remains the most useful oneDespite this large variety of alternative gravitational theoriesone can do better by imposing some physical requirementson the systems of interest In particular the restriction to theclass of theories having utmost second-order derivatives ofthe metric is a judicious one since this helps to overcome thewell known Ostrogradski instability by evading the existenceof any ghost modes in the theory [8ndash10] Surprisingly enoughthe above criteria turn out to be very interesting ones as theysingle out a very specific class of unique gravitational theoriesknown in the literature as the Lanczos-Lovelock models ofgravity [11ndash14] In particular all these models satisfy thecriterion that Bianchi derivative of the field tensor identicallyvanishes [15 16] One can add to this list a large numberof interesting properties a few among them are as follows(a) the action functional for Lanczos-Lovelock gravity can bebroken into a bulk part and a surface contribution which isclosely related to the Wald entropy associated with the black

hole horizon [17] (b) the field equations in Lanczos-Lovelockgravity can be expressed as a thermodynamic identity onan arbitrary null surface [18ndash26] (c) the difference betweena suitably defined surface and bulk degrees of freedom inthe context of Lanczos-Lovelock gravity can be interpretedas the evolution of the spacetime [27] (d) the surface andbulk terms in the Lanczos-Lovelock gravity are related by aholographic relation between them [28 29]

In the standard textbook treatment one first introducesthe gravitational Lagrangian119871 which isradicminus119892119877 for general rel-ativity while the Lagrangian is a polynomial in the Riemanntensor for general Lanczos-Lovelock gravity Variation of theaction functional due to arbitrary variation of the metricwith appropriate boundary conditions [30 31] leads to thecorresponding field equations for gravity namely the rule bywhich matter tells the spacetime how to curve (see also [32])However as emphasized earlier there is no physical principlewhich guides us to the precise mathematical structure of thegravitational action what ismore the field equations obtainedequate matter which is intrinsically quantum with spacetimegeometry describing gravity classically Despite this concep-tual discomfort there exist two additional representations ofthe gravitational dynamics which are intrinsically observerdependent The first one uses timelike observers with four-velocity 119906119886 and equates two quantities where this observer



measures in the geometric as well as in the matter sectorleading to a scalar equation

119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)

Here 120581 stands for the gravitational constant in four dimen-sions and enforcing this equation for all observers with four-velocity 119906119894 leading to Einsteinrsquos equations 119866119886119887 = 120581119879119886119887 Theother approach uses null vectors ℓ119886 (ie ℓ2 = 0) and leads tothe following scalar equation

119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)

In this situation as well validity of the above expressionfor all null vectors ℓ119886 along with two times contractedBianchi identity and covariant conservation of matter energymomentum tensor amounts to furnishing the ten compo-nents of Einsteinrsquos equations Following [33] in this articlewe would like to derive (1) and (2) from a geometrical pointof view for Lanczos-Lovelock theories of gravity We wouldalso argue why the route taken in this work is a natural onecompared to the standard derivation of Einsteinrsquos equationsfrom the gravitational action We have organized the paperas follows In Section 2 we briefly review the derivation ofEinsteinrsquos equations which will be generalized in Section 3 toLanczos-Lovelock theories of gravity Finally we provide theconcluding remarks before presenting a derivation regardingthe curvature tensor associated with the Lanczos-Lovelocktheory of gravity in Appendix

2 Einsteinrsquos Equations followingNewtonrsquos Path

To set the stage for Lanczos-Lovelock theories of gravitywe would like to briefly review the corresponding situationin general relativity which will mainly follow from [33]In Newtonian theory one describes gravity using the grav-itational potential 120601(x 119905) Given the matter density 120588(x 119905)the dynamics of the gravitational field is being determinedthrough Poissonrsquos equation nabla2120601 = (1205812)120588 When one invokesprinciple of equivalence the potential 120601 is readily identifiedwith components of themetric in particular 11989200 = minus(1+2120601)One then interprets Poissonrsquos equation in metric language asminusnabla211989200 = 12058111987900 where matter energy density is interpreted asthe time-time component of the divergence free symmetricmatter energy momentum tensor 119879119886119887 In order to have arelativistic generalization one naively makes the followingreplacement nabla2 rarr ◻ This immediately suggests looking forsecond derivatives of themetric which is a second rank tensoras well as divergence free paving the way to the Einsteintensor On the other hand for the right hand side the naturalchoice being the matter energy momentum tensor 119879119886119887 oneends up equating Einsteinrsquos tensor with the matter energymomentum tensor resulting in Einsteinrsquos equations (for amore detailed discussion see [33])

However we could have just followed Newtonrsquos path andlook for relativistic generalization of Poissonrsquos equation itselfpossibly leading to a scalar equation describing gravity ingeneral relativity As a first step one must realize that the

right hand side of Poissonrsquos equation is intrinsically observerdependent as it is the energy density of some matter fieldmeasured by an observer with some four-velocity 119906119894 Giventhe symmetric and conserved matter energy momentumtensor 119879119886119887 the matter energy density 120588 as measured byan observer with four-velocity 119906119886 is 119879119886119887119906119886119906119887 Hence thesame observer dependence must continue to exist in generalrelativity and the right hand side of the desired generalrelativistic equation for gravity should be 119879119886119887119906119886119906119887 [33]

To derive the left hand side that is analogue of nabla2120601we note that this requires deriving a scalar object whichinvolves two spatial derivatives of the metric (since this iswhat is responsible for the nabla2120601 term in nonrelativistic limit)and necessarily depends on the four-velocity 119906119894 The onlytensor depending on two derivatives of the metric that canbe constructed corresponds to the curvature tensor 119877119886119887119888119889 Inorder to obtain double spatial derivatives acting on themetricone has to project all the components of the curvature tensoron the plane orthogonal to 119906119886 using ℎ119886119887 = 120575119886119887 + 119906119886119906119887 andhence construct a scalar thereof To keep generality insteadof looking for a timelike vector 119906119894 we will concentrate ona particular vector ℓ119886 with norm ℓ119886ℓ119886 equiv ℓ2 Then one canimmediately introduce the following tensor

P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)

for which the final property ensures that it is a projectorThen one can define a projected Riemann curvature withrespect to the vector ℓ119886 by projecting all the indices of theRiemann curvature tensor leading to (by the very definitionthe Riemann tensor 119877119886119887119888119889 has a generic form 120597119887120597119888119892119886119889 in localinertial frame thus the above projection ensures that 119892119886119887 hasonly double spatial derivatives when ℓ119886 is a timelike vectorall time derivatives appearing in R119886119887119888119889 are single in nature andhence vanish in the local inertial frame all together this is theprime motivation of introduction of this projected Riemanntensor the same can be ascertained from the Gauss-Codazziequation as well see [1 5])

R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)

The scalar constructed out of this projected Riemann tensorbecomes

R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)

Note that we have thrown away another term involving119877119886119887119888119889ℓ119886ℓ119887ℓ119888ℓ119889 since due to antisymmetry properties of theRiemann tensor the above quantity identically vanishesFurthermore the above scalar by construction has only twoderivatives of the metric and is completely spatial in theinstantaneous rest frame of the timelike observer Hence R

is the relativistic generalization of nabla2120601 and thus must beequal to the corresponding matter energy density which in


this case corresponds to minus(2120581ℓ2)119879119886119887ℓ119886ℓ119887 where 119879119886119887 is thematter energymomentum tensorThus finally one obtains thefollowing equation

119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)

Surprisingly through the above exercise we have achievedtwo results in one go Firstly for ℓ2 = minus1 that is forunit normalized timelike vectors we get back the equation119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 Then requiring these equations to holdfor all timelike observers will lead to119866119886119887 = 120581119879119886119887 At this stageit will be worthwhile to mention that a similar approach asthe above one was taken in [34] to derive Einsteinrsquos equationsHowever the key difference between the approach in [34] andin this work is the use of the projection tensorP119886119887 judiciouslyFor our approach the use of projection tensor is of utmostimportance in contrast to [34]

Finally the limit ℓ2 rarr 0 would lead to the null versionof the above equation which reads 119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887In this case besides demanding the validity of the aboveequation for all null vectors it is important to use contractedBianchi identity as well as covariant conservation of energymomentum tensor leading to 119866119886119887 + Λ119892119886119887 = 120581119879119886119887 Notethat in the second situation a cosmological constant hasautomatically come into existence

3 Newton Leads the Way to Lovelock

In view of the above result one immediately asks for thecorresponding situation in Lovelock gravity can field equa-tions of Lovelock gravity be derived following the aboveprocedure This is what we will explore in this section andshall show that one can indeed derive the field equations forLovelock gravity following an equivalent procedure Beforejumping into the details let us mention some structuralaspects of Lovelock gravity Briefly speaking Lovelock gravitycorresponds to a class of gravitational Lagrangians whichare polynomial in the Riemann curvature tensor yieldingfield equations which are of second order in the metric Theanalytical form of such a polynomial (also called a pureLovelock term) of order 119898 involves 119898 Riemann curvaturetensors contracted appropriately such that

119871(119898) = 12119898 12057511988611198871 sdotsdotsdot11988611989811988711989811988811198891 sdotsdotsdot119888119898119889119898

1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)

The above relation defines the tensor 119875119886119887119888119889 (119898) associated withthe 119898th order Lanczos-Lovelock gravity having all the sym-metries of the Riemann tensor with the following algebraicstructure

119875119886119887119888119889 (119898) = 1198982119898 12057511988611988711988621198872 sdotsdotsdot11988611989811988711989811988811988911988821198892 sdotsdotsdot119888119898119889119898

1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)

The tensor 119875119886119887119888119889 (119898) satisfies an additional criterion nabla119886119875119886119887119888119889 (119898) =0 which ensures that the field equations derived from thisLagrangian are of second order In what follows we willexclusively concentrate on the 119898th order Lanczos-Lovelock

gravity and hence shall remove the symbol ldquo(119898)rdquo from thesuperscripts of various geometrical expressions

One can now start from Poissonrsquos equation and fix theright hand side to be minus119879119886119887ℓ119886ℓ119887ℓ2 which is the matter energydensity associated with the vector field ℓ119886 In order to get theleft hand side we must construct an appropriate curvaturetensor suited for the Lovelock gravity and project it usingthe projector P119886119887 introduced in (3) before constructing ascalar out of it There are two possible choices for such acurvature tensor among which we will discuss the simplerone in the following while the complicated one is deferredto Appendix 4 The curvature tensor described here was firstintroduced in [15 16] and is defined as follows

R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)

minus (119898 minus 1)(119889 minus 1) (119889 minus 2) (120575119886119888120575119887119889 minus 120575119886119889120575119887119888) 119871

(9)

where 119871 is the 119898th order Lovelock polynomial and 119875119886119887119888119889 =120597119871120597119877119888119889119886119887 Since the 119898th order Lanczos-Lovelock Lagrangiandepends on 119898 powers of Riemann the above definition for119875119886119887119888119889 ensures that it depends on (119898 minus 1) powers of Riemanntensor and hence exactly coincides with (8) Furthermorethe tensor 119875119886119887119888119889 can be easily generalized to the full Lovelockpolynomial by just adding over different 119898 values but wewill concentrate on a single term in the full Lanczos-LovelockLagrangian Also note that the above defined ldquoLovelockrdquoRiemann tensor has all the symmetries of the original ldquoEin-steinrdquo Riemann tensor 119877119886119887119888119889 Further it satisfies the contractedBianchi identity which will be sufficient for our purposeGiven the above 119898th order ldquoLovelockrdquo Riemann one canproject it in the plane orthogonal to ℓ119886 and obtain thefollowing scalar

R = R119886119887119888119889P119888119886P119889119887 (10)

Explicit evaluation of the above scalar can be performedkeeping in mind that the tensor R119886119887119888119889 is antisymmetric underexchange of the indices (119886 119887) and (119888 119889) respectively leadingto

R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)

(119889 minus 1) (119889 minus 2) (119889120575119886119888 minus 120575119886119888 ) 119871

sdot ℓ119886ℓ119888 = 119898119871 minus 119889 (119898 minus 1)119889 minus 2 119871 minus 2

ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2

sdot 119898 minus 1119889 minus 2 119871 = minus 2

ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)

Thus equating (11) to the matter energy density throughminus(2120581ℓ2)119879119886119887ℓ119886ℓ119887 we finally obtain

119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)


For unit normalized timelike vectors ℓ119886 = 119906119886 and ℓ2 = minus1leading to 119864119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 where 119864119886119887 equiv 119875119886119901119902119903119877119887119901119902119903 minus(12)119871119892119886119887 is the analogue of Einstein tensor in Lovelockgravity On the other hand for the null vectors we arrive atR119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 where the tensor R119886119887 equiv 119875119886119901119902119903119877119887119901119902119903is the analogue of Ricci tensor in Lovelock gravity Henceeven in the case of Lovelock gravity if one assumes that119864119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 holds for all timelike observers thefield equations for Lovelock gravity 119864119886119887 = 120581119879119886119887 followWhile in the null case besides demanding the validity of119864119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 for all null vectors one has to use theBianchi identity associated with Lovelock theories as well ascovariant conservation of matter energy momentum tensorto arrive at the Lovelock field equations 119864119886119887 + Λ119892119886119887 = 120581119879119886119887which inherit the cosmological constant as well Note that theabove result has been derived in the context of 119898th orderLanczos-Lovelock gravity which can be generalized to thegeneral Lanczos-Lovelock Lagrangian in a straightforwardmanner This results in sum119898 119864(119898)119886119887 + sum119898 Λ(119898)119892119886119887 = 120581119879119886119887Interestingly the cosmological constant besides being gen-erated as an integration constant of the field equations alsogets contribution from the 119898 = 0 term in the Lanczos-Lovelock gravity Therefore one may choose this term in theLagrangian appropriately to arrive at the present small valueof the cosmological constant Therefore we conclude thatthe derivation of field equations for gravity can always beachieved starting from Poissonrsquos equation and subsequentlyprojecting a suitable curvature tensor whether it is Einsteingravity or Lovelock

4 Concluding Remarks

By equivalence principle gravity manifests itself by curv-ing the spacetime which the material particles follow Inparticular one can invoke special relativity in locally freelyfalling frame and hence write down the laws of motion incurvilinear coordinates thus describing motion in curvedspacetimeThe notion of locally freely falling observer bringsin intrinsic observer dependence in the theory and introducesobservers for whom a local spacetime region is causallyinaccessible known as a local Rindler observer Remarkablythe local vacuum state of a test quantum field (as fit forlocal inertial observers) will appear as thermal to the localRindler observer [35 36] If any matter field (characterizedby matter energy momentum tensor 119879119886119887 ) crosses the localRindler horizon it will appear to be thermalized by theRindler observer (since the matter will take infinite time toreach the horizon) and the corresponding heat density isbeing given by119879119887119886ℓ119886ℓ119887 (for a perfect fluid the above quantity isgiven by 120588 + 119901 which by Gibbs-Duhem relation is the matterheat density) where ℓ119886 is the null normal to the horizon Notethat the above heat density for matter is invariant under thetransformation 119879119886119887 rarr 119879119886119887 + (constant)120575119886119887

At this stage one can ask a natural question ldquowhat aboutheat density of gravityrdquo Surprisingly one can answer thesame in the above setting Considering a general null surfaceit turns out that one can interpret 119877119886119887ℓ119887ℓ119886 as the heat densityof the spacetime This originates from the fact that one can

have a one to one correspondence between 119877119886119887ℓ119886ℓ119887 and theviscous dissipation term 2120578120590119886119887120590119886119887+1205771205792 where 120590119886119887 is the shearof the null congruence ℓ119886 and 120579 is its expansion with 120578 and120577 being shear and bulk viscous coefficients Thus the term119877119886119887ℓ119886ℓ119887 is related to heating of the spacetime [37] Giventhis thermodynamic backdrop it is clear that the Einsteinrsquosequations when written as (2) not only yield the geometricalinput of the gravitational theory but are also physically well-motivated since the equality of (2) can be thought of as anequilibrium situation where the heat produced by gravity isbeing compensated by that of matter

Thus the equation 2119866119886119887ℓ119886ℓ119887 = 119879119886119887ℓ119886ℓ119887 arises morenaturally from the relativistic generalization of Newtonrsquos lawand the usefulness of writing Einsteinrsquos equations in thismanner stems from the fact that one might interpret bothsides of these equations independently and the equationsthemselves follow due to a balancing act performed byspacetime itself [33 38 39] We would like to reiteratethat the field equations derived in this context are purelygeometrical and follow Newtonrsquos path One first realizesthat energy density associated with any material body isintrinsically observer dependent and surprisingly one canconstruct a tensor (again dependent on observer) whichcontains spatial derivatives of the metric alone Keeping thisas a curved spacetime generalization of Newtonrsquos law oneuniquely arrives at Einsteinrsquos equations when one makes useof all observers (or all the null surfaces) This shows that themost natural generalization of Newtonrsquos law to curved space-time is 119877119886119887ℓ119886ℓ119887 = 8120587119879119886119887ℓ119886ℓ119887 (of course leading to Einsteinrsquosequations but at a secondary level) bolstering the claim thatgravity is intrinsically a thermodynamic phenomenon

Appendix

An Alternative Riemann Tensor forLovelock Gravity

In Lovelock gravity it is possible to define two tensors havingthe symmetry properties of Riemann and satisfying Bianchiidentity The first one corresponds to defining a (2119898 times 2119898)tensor for 119898th order Lovelock polynomial by multiplying 119898such curvature tensors and then an alternating tensor of rank(2119898 times 2119898) as [40]

R11988711198872 sdotsdotsdot119887211989811988611198862 sdotsdotsdot1198862119898

= 12057511988711198872 sdotsdotsdot119887211989811988811198882 sdotsdotsdot119888211989812057511988911198892 sdotsdotsdot119889211989811988611198862 sdotsdotsdot1198862119898

1198771198881119888211988911198892

sdot sdot sdot 1198771198882119898minus111988821198981198892119898minus11198892119898

(A1)

Note that the above tensor is completely antisymmetric inboth upper and lower indices

Let us now project all the indices on a lower dimensionalspacelike hypersurface using the projection tensor P119886119887 = 120575119886119887 minus(1ℓ2)ℓ119886ℓ119887 such that one obtains another (2119898 times 2119898) tensorbut whose inner product with the normal ℓ119886 identicallyvanishes Hence one arrives at

119898R11990111199012sdotsdotsdot119901211989811990211199022sdotsdotsdot1199022119898= P11990111198871

sdot sdot sdotP11990121198981198872119898

P11988611199021

sdot sdot sdotP11988621198981199022119898R11988711198872 sdotsdotsdot119887211989811988611198862 sdotsdotsdot1198862119898 (A2)


One can immediately construct a scalar out of the projected(2119898 times 2119898) tensor leading to

119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)

There would be two terms contributing to the above expres-sion one when all the projectors are related by Kroneckerdeltas and when one of the projector P119886119887 is replaced by ℓ119886ℓ119887while all the others are replaced by Kronecker deltas In anyother case for example if two projectors are replaced bythe normal then it would identically vanish thanks to thecompletely antisymmetric nature of R11988711198872 sdotsdotsdot119887211989811988611198862 sdotsdotsdot1198862119898

Thus finallywe obtain

119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898


minus 2119898ℓ2 ℓ1198861ℓ119887112057511988621198872 sdot sdot sdot 12057511988621198981198872119898R11988711198872 sdotsdotsdot119887211989811988611198862 sdotsdotsdot1198862119898


1198771198881119888211988911198892


minus 2119898ℓ2 ℓ1198861ℓ119887112057511988711198872 sdotsdotsdot1198872119898c11198882 sdotsdotsdot1198882119898 12057511988911198892 sdotsdotsdot119889211989811988611198872 sdotsdotsdot1198872119898

1198771198881119888211988911198892


(A4)

One can now use the following identities

12057511988711198872 sdotsdotsdot119887211989811988911198892sdotsdotsdot1198892119898

12057511988811198882 sdotsdotsdot119888211989811988711198872 sdotsdotsdot1198872119898

= 1198982119898 12057511988811198882 sdotsdotsdot119888211989811988911198892 sdotsdotsdot1198892119898

(A5)

as well as

12057511988711198872 sdotsdotsdot119887211989811988811198882 sdotsdotsdot119888211989812057511988911198892 sdotsdotsdot119889211989811988611198872 sdotsdotsdot1198872119898

1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)

such that one arrives at

R = 12119898 12057511988911198892 sdotsdotsdot119889211989811988811198882 sdotsdotsdot1198882119898

1198771198881119888211988911198892


minus 2119898ℓ2

12119898 ℓ1198861ℓ119887112057511988711198881 12057511988911198892 sdotsdotsdot119889211989811988611198882 sdotsdotsdot1198882119898

1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)

whereR119886119887 stands for the analogue of Ricci tensor associatedwith the Lovelock gravitational action Hence this particularLovelock Riemann tensor reproduces the Lovelock fieldequations for gravity as well if the procedure outlined aboveis being followed The above exercise explicitly demonstratesthe robustness of the idea presented here


The author states that there are no conflicts of interest

Acknowledgments

Sumanta Chakraborty gratefully acknowledges the help ofT Padmanabhan for suggesting this project and also for

helpful discussions throughout it He also thanks NareshDadhich Kinjalk Lochan andKrishnamohanParattu for var-ious fruitful discussions A part of this work was completedwhile the author was visiting Albert Einstein Institute inGolm Germany and the author gratefully acknowledges thewarm hospitality there Research of Sumanta Chakrabortyis supported by the SERB-NPDF Grant (PDF2016001589)from DST Government of India

References

[1] T PadmanabhanGravitation Foundations and Frontiers Cam-bridge University Press Cambridge UK 2010

[2] L D Landau and E M Lifshitz The Classical Theory of Fieldsvol 2 of Course of Theoretical Physics Series Butterworth-Heinemann 4th edition 1980

[3] A S Eddington The Mathematicl Theory of Relativity Cam-bridge University Press Cambridge UK 2nd edition 1960

[4] W K Misner S Thorne and J A Wheeler Gravitation W HFreeman and Company 3rd edition 1973

[5] E Poisson A Relativist Toolkit Cambridge University PressCambridge UK 1st edition 2004

[6] R M Wald General Relativity University of Chicago Press 1stedition 1984

[7] S W Hawking and G F R Ellis The Large Scale Structure ofSpace-Time Cambridge Monographs onMathematical PhysicsCambridge University Press London UK 1973

[8] T-j Chen M Fasiello E A Lim and A J Tolley ldquoHigherderivative theories with constraints exorcising Ostrogradskirsquosghostrdquo Journal of Cosmology and Astroparticle Physics vol 2013no 2 2013

[9] H Motohashi K Noui T Suyama M Yamaguchi and DLanglois ldquoHealthy degenerate theories with higher derivativesrdquoJournal of Cosmology and Astroparticle Physics vol 2016 2016

[10] R Klein and D Roest ldquoExorcising the Ostrogradsky ghost incoupled systemsrdquo Journal of High Energy Physics no 7 2016

[11] C Lanczos ldquoA remarkable property of the Riemann-Christoffeltensor in four dimensionsrdquo Annals of Mathematics vol 39 no4 pp 842ndash850 1938

[12] C Lanczos ldquoElectricity as a natural property of Riemanniangeometryrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 39 no 4 pp 716ndash736 1932

[13] D Lovelock ldquoThe Einstein tensor and its generalizationsrdquoJournal of Mathematical Physics vol 12 pp 498ndash501 1971

[14] T Padmanabhan and D Kothawala ldquoLanczos-Lovelockmodelsof gravityrdquo Physics Reports vol 531 no 3 pp 115ndash171 2013

[15] N Dadhich ldquoCharacterization of the Lovelock gravity byBianchi derivativerdquo PramanamdashJournal of Physics vol 74 no 6pp 875ndash882 2010

[16] X O Camanho and N Dadhich ldquoOn Lovelock analogs of theRiemann tensorrdquo The European Physical Journal C vol 76 no3 2016

[17] D Kothawala T Padmanabhan and S Sarkar ldquoIs gravitationalentropy quantizedrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 78 no 10 Article ID 104018 5 pages2008

[18] A Paranjape S Sarkar and T Padmanabhan ldquoThermodynamicroute to field equations in Lanczos-LOVelock gravityrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 74no 10 Article ID 104015 2006


[19] M Akbar and R G Cai ldquoFriedmann equations of FRWuniverse in scalarndashtensor gravity f(R) gravity and first law ofthermodynamicsrdquo Physics Letters B vol 635 no 1 pp 7ndash102006

[20] RGCai and S PKim ldquoFirst lawof thermodynamics andFried-mann equations of Friedmann-Robertson-Walker universerdquoJournal of High Energy Physics vol 2005 article 050 2005

[21] X H Ge ldquoFirst law of thermodynamics and Friedmann-likeequations in braneworld cosmologyrdquo Physics Letters B vol 651no 1 pp 49ndash53 2007

[22] Y Gong and A Wang ldquoFriedmann Equations and Thermody-namics of Apparent Horizonsrdquo Physical Review Letters vol 99no 21 2007

[23] S Wu B Wang and G Yang ldquoThermodynamics on the appar-ent horizon in generalized gravity theoriesrdquo Nuclear Physics Bvol 799 no 3 pp 330ndash344 2008

[24] S Chakraborty ldquoLanczos-Lovelock gravity from a thermody-namic perspectiverdquo Journal of High Energy Physics vol 29 2015

[25] D Hansen D Kubiznak and R Mann ldquoHorizon Thermody-namics from Einsteins Equation of Staterdquo Physics Letters B vol771 pp 277mdash280 2017

[26] R Dey S Liberati and A Mohd ldquoHigher derivative gravityField equation as the equation of staterdquo Physical Review DParticles Fields Gravitation and Cosmology vol 94 no 4 2016

[27] T Padmanabhan ldquoEquipartition of energy in the horizondegrees of freedom and the emergence of gravityrdquo ModernPhysics Letters A vol 25 no 14 pp 1129ndash1136 2010

[28] G A Mukhopadhyay and T Padmanabhan ldquoHolography ofgravitational action functionalsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 74 no 12 Article ID124023 15 pages 2006

[29] D Dieks J van Dongen and S de Haro ldquoEmergence in holo-graphic scenarios for gravityrdquo Studies in History and Philosophyof Science Part B Studies in History and Philosophy of ModernPhysics vol 52 pp 203ndash216 2015

[30] K Parattu S Chakraborty B R Majhi and T PadmanabhanldquoA boundary term for the gravitational action with null bound-ariesrdquo General Relativity and Gravitation vol 48 no 7 Art 9428 pages 2016

[31] S Chakraborty ldquoBoundary Terms of the EinsteinndashHilbertActionrdquo in Gravity and the Quantum vol 187 of FundamentalTheories of Physics pp 43ndash59 Springer International Publish-ing 2017

[32] S Deser ldquoGravity from self-interaction reduxrdquo General Relativ-ity and Gravitation vol 42 no 3 pp 641ndash646 2010

[33] T Padmanabhan ldquoThe atoms of space gravity and the cos-mological constantrdquo International Journal of Modern Physics DGravitation Astrophysics Cosmology vol 25 no 7 Article ID1630020 33 pages 2016

[34] N StraumannGeneral relativity Springer Zurich Switzerland2013

[35] P C Davies ldquoScalar production in Schwarzschild and Rindlermetricsrdquo Journal of Physics A Mathematical and General vol 8no 4 pp 609ndash616 1975

[36] W G Unruh ldquoNotes on black-hole evaporationrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 14no 4 pp 870ndash892 1976

[37] S Chakraborty andT Padmanabhan ldquoThermodynamical inter-pretation of the geometrical variables associated with nullsurfacesrdquo Physical Review D Particles Fields Gravitation andCosmology vol 92 no 10 Article ID 104011 30 pages 2015

[38] Z Wang W Ai H Chen and J Deng ldquoCosmological modelfrom emergence of spacerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 92 no 2 2015

[39] C E Navia ldquoOn the Davies-Unruh effect in a wide range oftemperaturesrdquo httpsarxivorgabs170103442

[40] D Kastor ldquoThe Riemann-Lovelock curvature tensorrdquo Classicaland Quantum Gravity vol 29 no 15 155007 9 pages 2012

Research ArticleDiscussion of a Possible Corrected Black Hole Entropy

Miao He ZiliangWang Chao Fang Daoquan Sun and Jianbo Deng

Institute of Theoretical Physics Lanzhou University Lanzhou 730000 China

Correspondence should be addressed to Jianbo Deng dengjblzueducn

Received 21 November 2017 Revised 31 January 2018 Accepted 12 February 2018 Published 11 March 2018

Academic Editor Habib Mazharimousavi

Copyright copy 2018 Miao He et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

Einsteinrsquos equation could be interpreted as the first law of thermodynamics near the spherically symmetric horizon Throughrecalling the Einstein gravitywith amore general static spherical symmetricmetric we find that the entropywould have a correctionin Einstein gravity By using this method we investigate the Eddington-inspired Born-Infeld (EiBI) gravity Without matter fieldwe can also derive the first law in EiBI gravity With an electromagnetic field as the field equations have a more general sphericallysymmetric solution in EiBI gravity we find that correction of the entropy could be generalized to EiBI gravity Furthermore wepoint out that the Einstein gravity and EiBI gravity might be equivalent on the event horizon At last under EiBI gravity with theelectromagnetic field a specific corrected entropy of black hole is given

1 Introduction

Black hole thermodynamics has been proposed for manyyears since the entropy and temperature were found byBekenstein andHawking [1 2] even getting many interestingresults like the four laws of black hole thermodynamics Itestablished the connection between the gravity and thermo-dynamics

The entropy is assumed to be proportional to its horizonarea [1] and it is well-known that the so-called area formulaof black hole entropy holds only in Einstein gravity Howeverwhen some higher order curvature terms appear in somegravity theory the area formula has to be modified [3]A logarithmic term often occurs in the correction likethe black hole entropy in loop quantum gravity (quantumgeometry) [4ndash6] and thermal equilibrium fluctuation [7 8]The correction of entropy has been studied in Gauss-Bonnetgravity [9] Lovelock gravity [10] and 119891(119877) gravity [11] Inthe apparent horizon of FRW universe the entropy also has acorrection [12]

Einsteinrsquos equation can be derived from the thermody-namics [13] on the other side the thermodynamic routeto the gravity field equation which could get the first lawof thermodynamics in Einstein gravity was proposed byPadmanabhan [14ndash16] It indicated a generic connection

between thermodynamics of horizons and gravity although itis not yet understood at a deeper level [17]This technique hasbeen used in Gauss-Bonnet gravity and Lanczos-Lovelockgravity [15] the corrected entropy is the same in [18 19]respectively

The EiBI gravity was inspired by Banados and Ferreira[20] It is completely equivalent to the Einstein gravity invacuum but in the presence of matter it would show manyinteresting results like an alternative theory of Big Bang sin-gularity in early universe [21] and the mass inflation in EiBIblack holes [22 23] However there are few investigations forthe thermodynamic properties of black hole in EiBI gravityas it has a complicated spherically symmetric solution whenthe electromagnetic field is considered [24]

In this paper inspired by Padmanabhan [14ndash16] wederive the first law of black hole thermodynamics withthe commonly accepted thermodynamics quantities fromEinsteinrsquos equation We also use this technique in EiBIgravity and get the known first law the results show amore general formula of entropy which also holds for AdSSchwarzschild black hole and AdS R-N black hole Motivedby this supposing amore general static spherically symmetricmetric we get the same result in Einstein gravity

This paper is organized as follows In Section 2 there isa derivation of the thermodynamic identity from Einstein



gravity we also get a more general formula of entropy InSection 3 we investigate the EiBI gravity by using the ther-modynamic route to the field equation and get the formula ofentropy in EiBI gravity Conclusions and discussion are givenin Section 4

2 Black Hole Thermodynamic Identity fromEinsteinrsquos Equation

Einsteinrsquos equation can be derived from the thermodynamics[13] On the other side it is possible to interpret Einsteinrsquosequation near the spherical symmetric event horizon as thefirst law of thermodynamics which was proposed by Pad-manabhan [14ndash16] However the thermodynamic quantitiesmight not be consistent with the normal ones especially thepressure and internal energy Fortunately through restudyingthe field equation we can also derive the first law of ther-modynamics with the commonly accepted thermodynamicquantities [25 26]

Considering a static spherically symmetric space-time

1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)

and the event horizon 119903 = 119903119867 satisfying 119891(119903119867) = 0 then onecan get its thermodynamic quantities

119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)

If we consider AdS space-time with a negative cosmologicalconstant Λ there would be a pressure 119875 = minusΛ8120587 [25] Themass of black hole was treated as enthalpy [26] and the firstlaw of black hole thermodynamics is

119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)

Through the Legendre transformation one can get the inter-nal energy

119889119880 = 119879119889119878 minus 119875119889119881 (4)

Particularly the internal energies for AdS Schwarzschildblack hole and AdS R-N black hole are

119880Schwarzschild = 1199031198672 119880R-N = 1199031198672 + 11987622119903119867

(5)

Einsteinrsquos equation with a cosmological constant is

119866120583] + Λ119892120583] = 8120587119879120583] (6)

If the metric has the form of (1) in the case of 119879120583] = 0 onecan obtain the 120579120579 component of the equation

minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)

Setting 119903 = 119903119867 and then multiplying the above equation by119889119903119867 we can rewrite (7) as

1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)

Noticing (2) the above equation can be regarded as the firstlaw of black hole thermodynamics since 119880 = 1199031198672 for theSchwarzschild solution

For a charged AdS black hole the metric also takes theformof (1)The energy-momentum tensor of electromagneticfield is

119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)

Its nonzero components are

119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)

where 1198640 = 1198761199032 and 119876 represents the charge of black holeAccording to Einsteinrsquos equations one can also get

minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 minus 119903211986420 (11)

By the same technique and treating 119876 as a constant we get

1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)

which can be also treated as the first law with the thermody-namic quantities in (2) and one can verify that 119880 = 1199031198672 +11987622119903119867 for the AdS R-N black hole

Here we keep 119876 as a constant which means a chargelessparticle falls into the AdS R-N black hole (12) is consistentwith the first lawWhile a charged particle falls into theAdSR-N black hole the event horizon 119903119867would arise due to changesof 119889119872 and 119889119876 then (12) could be rewritten as [15]

1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)

Then it would adopt to the first law with the formulation

119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)

We should point out that 119879120583] contributes to the internalenergy 119880


Thus Einsteinrsquos equation can be interpreted as the firstlaw of thermodynamic near the event horizonThe techniquewas first proposed by Padmanabhan and some relevantcomments about the meaning of thermodynamic quantitiesfor this result were given [14ndash17] Since we used the differentthermodynamics quantities in our derivation compared withPadmanabhanrsquos work there are some comments we wouldlike to add Firstly this method can be applied to singlehorizon and multiple horizons space-time but this is just alocal description of horizon thermodynamics which meansthe temperature entropy and energy are local quantities justfor one horizon In addition this method still has someproblem to solve when it was applied to cosmic horizon or deSitter horizon because the definition of the temperatures forcosmic horizon is quite different Secondly it seems that thereis a manifest arbitrariness or freedom in the derivations Onecould multiply the entire equation by an arbitrary functionthen the expressions for entropy internal energy and volumewould be another one In fact if we consider the initial valuethat is 119878 = 1198604 119881 = 412058711990331198673 for Schwarzschild space-timethis problemwould disappear Finally the perturbation of thestatic space-time can be interpreted in terms of the physicalprocess of black hole evaporation or hawking radiation Onecan also interpret the relations by say dropping test particlesinto the black hole

Note that the structure of the equation itself allows usto read off the expression for entropy This technique hasbeen used for Gauss-Bonnet gravity and Lovelock gravity[15] in which their entropy formulas are the same as [18 19]respectively We would like to consider a more general staticspherical symmetric metric

1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)

One can calculate its nonzero components of Ricci tensor

119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)

And Einsteinrsquos equation with a cosmological constant can bewritten as

119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)

For the Schwarzschild vacuum 119879120583] = 0 one can get120595 = 119862 a constant We can always have 120595 = 1 by choosingthe coordinate time 119889 = 119862119889119905 without changing the killingvector then the metric would return back to the sphericallysymmetric metric (1) For the charged black hole we have gotthe Reissner-Nordstrom metric which also implies 120595 = 1thus the first law could be obtained The reason of this resultmight be that the matter field leads to 120595 = 1

To get a general case we just consider the 120579120579 componentof (17) which can be expressed as

1 minus 119903119891 (119903)2 (1198911015840 (119903)119891 (119903) + (1205952119891 (119903))10158401205952119891 (119903) ) minus 119891 (119903)= Λ1199032 + 8120587(119879120579120579 minus 121198791199032)

(18)

We assume the metric satisfies 119891(119903119867) = 0 and 120595(119903119867) = 0on the event horizon By setting 119903 = 119903119867 and considering thematter field contributes to the internal energy 119880 and thenmultiplying 119889119903119867 one can get

119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)

One can verify that for the AdS Schwarzschild black hole itreduces to 119880 = 1199031198672 and for the AdS R-N black hole it gives119880 = 1199031198672 + 11987622119903119867 So the internal energy expression is justa generalization

The Hawking temperature on the event horizon becomes

119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)

One would find the entropy has to satisfy

119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)

Thus we generalize the corrected entropy formula to Einsteingravity for static spherically symmetric metric equation (15)Once 120595 = 1 it is obvious that 119878 = 1205871199032119867 = 1198604 which is thewell-known Bekenstein-Hawking black hole entropy For theSchwarzschild black hole and R-N black hole which all have120595 = 1 119878 = 1198604


In fact Matt Visser has proposed that the entropy ofldquodirtyrdquo black holes might not equal quarter of area of eventhorizon [27 28] Generically a ldquodirtyrdquo black hole is a blackhole with various classical matter fields higher curvatureterms in the gravity Lagrangian or some other versionsof quantum hair For the Einstein gravity a more generalmetric (15) could be caused by some special matter fields likeelectromagnetismwithDilaton fields [29] So the entropy alsoshould be corrected in Einstein gravity In the next section wewould like to consider the EiBI gravity [20] its Lagrangianincludes the higher curvature terms in (25) which could beinterpreted as the self-gravity we will show our discussionof thismethod in the Eddington-inspired Born-Infeld gravityand get its thermodynamic quantities

3 The Entropy in Eddington-InspiredBorn-Infeld Gravity

The Eddington-inspired Born-Infeld theory of gravity isbased on the Palatini formulation which treats the metricand connection as independent fields [20] Its action can bewritten as

119878 = 18120587120581 int1198894119909 [radic10038161003816100381610038161003816119892120583] + 120581119877120583] (Γ)10038161003816100381610038161003816 minus 120582radic119892]+ 119878119872 (119892 Γ Ψ)

(25)

where 119892120583] is the metric of space-time and its determinantis 119892 119877120583] is the symmetric Ricci tensor related to Γ thedimensionless parameter 120582 = 1 + 120581Λ and the parameter 120581has the inverse dimension of cosmological constant Λ

By varying the action with respect to 119892120583] and Γ oneobtains the equation of motion

119902120583] = 119892120583] + 120581119877120583] (26)

radic10038161003816100381610038161199021003816100381610038161003816119902120583] = 120582radic10038161003816100381610038161198921003816100381610038161003816119892120583] minus 8120587120581radic10038161003816100381610038161198921003816100381610038161003816119879120583] (27)

where 119902120583] is the auxiliary metric compatible with the connec-tion to

Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)

By combining (26) and (27) and then expanding the fieldequations to 2nd order of 120581 [20]

119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)

where 119878120583] = 119879120572120583119879120572]minus(12)119879119879120583] one can find that the equationis the 1st-order correction to Einsteinrsquos equation On the otherhand EiBI gravity can be interpreted as a correction of thematter term compared with Einstein gravity Even the EiBIgravity is fully equivalent to the Einstein gravity in vacuum

Let us consider the thermodynamics from the fieldequation in this gravity model Generally a static sphericallysymmetric metric 119892120583] could be

1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)

and the auxiliary metric 119902120583] is assumed as [24]

1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)

The Ricci tensor was calculated as follows

119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840

119877119903119903 = minus211986710158401015840119867 minus 11986510158401198671015840119865119867 minus 32 11986610158401198651015840119866119865 minus 11986610158401015840119866 minus 119865101584010158402119865

119877120579120579 = 1 minus 11986711986710158401198651015840 minus 1198661015840119866 1198671198671015840119865 minus 11986710158402119865 minus 11986711986710158401015840119865119877120601120601 = sin2 120579119877120579120579

(32)

where1198831015840 = 120597119883120597119903 which we will use in the rest of this paperWithout the matter fields (27) reduces to

1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)

and thus one can obtain

119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)

Plugging these into (26) then the 120579120579 component is

1 minus 1199031198911015840 (119903) minus (1199031205951015840120595 + 1)119891 (119903) = 1120581 (120582 minus 1) 1199032 (35)

Near the event horizon 119903 = 119903119867 (119891(119903119867) = 0) we have119889(1199031198672 ) minus 1198911015840 (119903119867)4120587 119889 (1205871199032119867) = minus119875119889(412058711990331198673 ) (36)

As the EiBI gravity is fully equivalent to the Einstein gravity invacuum [20] the above equation could imply the first law In


fact the black hole solution to EiBI gravity with no source isthe same as Schwarzschild-de Sitter metric which illustrates120595 = 1 and 119891 = 1 minus 2119898119903 + Λ11990323 [20] The thermodynamicquantities in (36) also hold in EiBI gravity Thus the first lawcan also be got from the EiBI gravity Next wewould considerthe EiBI gravity with electromagnetic field

The energy-momentum tensor of electromagnetic fieldcould be

119879119905119905 = (1205952119891)minus1 119864208120587 119879119903119903 = minus119891119864208120587 119879120579120579 = 119903minus2119864208120587 119879120601120601 = 119903minus2sinminus2 120579119864208120587

(37)

where 1198640 = 1198761199032 and 119876 represents the charge of black holeThen (27) becomes

1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get

119866 = (120582 minus 12058111986420) 120595119865 = (120582 minus 12058111986420)minus1 119891

1198672 = (120582 + 12058111986420) 1199032(39)

The 120579120579 component of (26) is written as

1 minus 120582 + 12058111986420 + 120581119903119864011986410158400120582 minus 12058111986420 sdot 1199031198911015840 minus 119884120582 minus 12058111986420119891= 1120581 (120582 minus 1) 1199032 + 119864201199032

(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)

If we assume that the event horizon satisfies 119891(119903119867) = 0 and120595(119903119867) = 0 then set 119903 = 119903119867 in (40) and multiply it by 119889119903119867noting 11986410158400 = minus21198640119903 it gives119889(1199031198672 + 11987622119903119867) minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(4120587119903

31198673 ) (42)

This equation should be the first law since it can go back to(36) when 119876 = 0 And one can also confirm this by noticing

that 119889119880 = 119889(1199031198672+11987622119903119867) and 119889119881 = 119889(412058711990331198673) Moreoverit gives the same result in Einstein gravity So we shouldidentify that

119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)

However as the metric takes the form of (30) the surfacegravity could be [29]

120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)

One can obtain the temperature on the event horizon

119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)

Then (43) would imply that

119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)

Obviously when 120595 = 1 one can get 119878 = 1205871199032119867Thus we get the entropy formula in EiBI gravity Surpris-

ingly we find it also holds in Einstein gravity once metrictakes the form of (30) Therefore we can also get the first lawfor amore general static spherically symmetricmetric in EiBIgravity Moreover one can find that (12) is the same as (42)It implies that the Einstein gravity and EiBI gravity might beequivalent on the event horizon from the view of black holethermodynamics

In fact the black hole solution with electromagnetic fieldin EiBI gravity has been found while 119891(119903119867) = 0 and 120595(119903119867) =0 [20 24] It is given as follows

120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)

With this result we can get the corrected entropy

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 = 120587int 11199032119867radic1199034119867 + 12058112058211987621198891199032119867= 120587radic1199034119867 + 1205811205821198762 minus 120587radic 120581120582 |119876|sdot ln(radic120581120582 |119876|1199032119867 + radic1 + 120581120582 119876

2

1199034119867) (49)

A logarithmic term occurs in this formula as a correctedentropy When 119876 = 0 one gets 119878 = 1205871199032119867 which is consistentwith the vacuum case and so it is the same as Einstein gravityWhen 120581 rarr 0 EiBI gravity would reduce to the Einsteingravity and entropy becomes the Bekenstein-Hawking one


4 Conclusion and Discussion

In this paper we restudied Padmanabhanrsquos work that it ispossible towrite Einsteinrsquos equation for spherically symmetricspace-time in the form of the first law of thermodynamics[14ndash17] but the thermodynamic quantities might not beconsistent with the normal ones especially the pressure andinternal energy By using this technique we reproducedthe first law with the commonly accepted thermodynamicquantities in the AdS space-time and this technique providesan effective approach to read off the thermodynamic quanti-ties Next we investigated a more general static sphericallysymmetric metric taking form (15) in Einstein gravity It issurprising to find that the entropy might have a correction inEinstein gravity

Since it provided a convenient approach to study theblack hole thermodynamics just from the field equation weinvestigated the black hole thermodynamic in EiBI gravityWe found that there is nothing different fromEinstein gravityin vacuum but entropy could be different from the R-N blackhole when the electromagnetic field was considered Thecorrected entropy fromEiBI gravity should be (47) which canreduce to the Bekenstein-Hawking entropy when 120595 = 1 andit is also the same as Einstein gravity when 119876 = 0 withoutthe matter field [20] Thus the corrected entropy in Einsteingravity could be generalized to EiBI gravity

Moreover as the Einstein gravity and EiBI gravity holdthe same result we remarked that these two theories ofgravity could be equivalent on the event horizon from theview of thermodynamics

At last as an example a specific corrected entropy of thecharged black hole in EiBI gravity was given The entropyform would lead to something different for the black holethermodynamics in EiBI gravity like the phase transitionand critical phenomenonThese would be left for our furtherresearch


The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the National NaturalScience Foundation of China (Grant no 11571342) for sup-porting them in this work

References

[1] J D Bekenstein ldquoBlack holes and entropyrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 7 no 8 pp2333ndash2346 1973

[2] S W Hawking ldquoParticle creation by black holesrdquo Communica-tions in Mathematical Physics vol 43 no 3 pp 199ndash220 1975

[3] RMWald ldquoBlack hole entropy is the Noether chargerdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 48no 8 pp R3427ndashR3431 1993

[4] C Rovelli ldquoBlack hole entropy from loop quantum gravityrdquoPhysical Review Letters vol 77 no 16 pp 3288ndash3291 1996

[5] A Ashtekar J Baez A Corichi and K Krasnov ldquoQuantumgeometry and black hole entropyrdquo Physical Review Letters vol80 no 5 pp 904ndash907 1998

[6] R K Kaul and P Majumdar ldquoLogarithmic correction to theBekenstein-Hawking entropyrdquo Physical Review Letters vol 84no 23 pp 5255ndash5257 2000

[7] S Das P Majumdar and R K Bhaduri ldquoGeneral logarithmiccorrections to black-hole entropyrdquo Classical and QuantumGravity vol 19 no 9 pp 2355ndash2367 2002

[8] A Chatterjee and P Majumdar ldquoUniversal canonical black holeentropyrdquo Physical Review Letters vol 92 no 14 article 1413012004

[9] M Cvetic S Nojiri and S D Odintsov ldquoBlack hole thermo-dynamics and negative entropy in de Sitter and anti-de SitterEinstein-Gauss-Bonnet gravityrdquo Nuclear Physics B vol 628 no1-2 pp 295ndash330 2002

[10] R Banerjee and S K Modak ldquoQuantum tunneling blackbodyspectrum and non-logarithmic entropy correction for Lovelockblack holesrdquo Journal of High Energy Physics vol 2009 no 11article 073 2009

[11] M Akbar and R-G Cai ldquoThermodynamic behavior of fieldequations for gravityrdquo Physics Letters B vol 648 no 2-3 pp243ndash248 2007

[12] R-G Cai L-M Cao and Y-P Hu ldquoCorrected entropy-arearelation and modified Friedmann equationsrdquo Journal of HighEnergy Physics vol 2008 no 8 article no 090 2008

[13] T Jacobson ldquoThermodynamics of spacetime the Einsteinequation of staterdquo Physical Review Letters vol 75 no 7 pp1260ndash1263 1995

[14] D Kothawala S Sarkar and T Padmanabhan ldquoEinsteinrsquosequations as a thermodynamic identity The cases of stationaryaxisymmetric horizons and evolving spherically symmetrichorizonsrdquo Physics Letters B vol 652 no 5-6 pp 338ndash342 2007

[15] A Paranjape S Sarkar and T Padmanabhan ldquoThermodynamicroute to field equations in Lanczos-Lovelock gravityrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 74no 10 Article ID 104015 2006

[16] T Padmanabhan ldquoClassical and quantum thermodynamics ofhorizons in spherically symmetric spacetimesrdquo Classical andQuantum Gravity vol 19 no 21 pp 5387ndash5408 2002

[17] T Padmanabhan ldquoGravity and the thermodynamics of hori-zonsrdquo Physics Reports vol 406 no 2 pp 49ndash125 2005

[18] R C Myers and J Z Simon ldquoBlack-hole thermodynamics inLovelock gravityrdquo Physical Review D Particles Fields Gravita-tion and Cosmology vol 38 no 8 pp 2434ndash2444 1988

[19] R-G Cai ldquoGauss-Bonnet black holes in AdS spacesrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 65no 8 Article ID 084014 2002

[20] M Banados and P G Ferreira ldquoEddingtonrsquos theory of gravityand its progenyrdquo Physical Review Letters vol 105 no 1 article011101 2010

[21] P P Avelino and R Z Ferreira ldquoBouncing Eddington-inspiredBorn-Infeld cosmologies An alternative to inflationrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 86no 4 Article ID 041501 2012

[22] P P Avelino ldquoInner structure of black holes in Eddington-inspired Born-Infeld gravity The role of mass inflationrdquo Phys-ical Review D Particles Fields Gravitation and Cosmology vol93 no 4 Article ID 044067 2016


[23] P P Avelino ldquoMass inflation in Eddington-inspired Born-Infeldblack holes Analytical scaling solutionsrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 93 no 10Article ID 104054 2016

[24] S-W Wei K Yang and Y-X Liu ldquoBlack hole solution andstrong gravitational lensing in Eddington-inspired BornndashInfeldgravityrdquo The European Physical Journal C vol 75 no 6 articleno 253 2015



[27] M Visser ldquoDirty black holes Entropy as a surface termrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 48 no 12 pp 5697ndash5705 1993

[28] M Visser ldquoDirty black holes entropy versus areardquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 48no 2 pp 583ndash591 1993

[29] M Visser ldquoDirty black holes thermodynamics and horizonstructurerdquo Physical Review D Particles Fields Gravitation andCosmology vol 46 no 6 pp 2445ndash2451 1992

Research ArticleThe Quantum Effect on Friedmann Equation in FRW Universe

Wei Zhang1 and Xiao-Mei Kuang 2

1Department of General Studies Nanchang Institute of Science amp Technology Nanchang 330108 China2Center for Gravitation and Cosmology College of Physical Science and Technology Yangzhou University Yangzhou 225009 China

Correspondence should be addressed to Xiao-Mei Kuang xmeikuanggmailcom

Received 2 December 2017 Revised 4 February 2018 Accepted 5 February 2018 Published 4 March 2018


Copyright copy 2018 Wei Zhang and Xiao-Mei Kuang This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3

We study the modified Friedmann equation in the Friedmann-Robertson-Walker universe with quantum effect Our modifiedresults mainly stem from the new entropy-area relation and the novel idea of Padmanabhan who considers the cosmic space tobe emerging as the cosmic time progresses so that the expansion rate of the universe is determined by the difference of degreesof freedom between the holographic surface and the bulk inside We also discuss the possibility of having bounce cosmologicalsolution from the modified Friedmann equation in spatially flat geometry

1 Introduction

In the 1970s the thermodynamic property of black holeshas been proposed [1ndash3] and it reveals that the gravitationaldynamics is entwined with thermodynamics Inspired byBekensteinrsquos entropy-area theorem [1] Bardeen et al putforward the four thermodynamical laws of black hole systems[2] In 1995 Jacobson considered Einsteinrsquos field equation asan equation of state Afterwards he reproduced Einsteinrsquosfield equation by demanding that the fundamental relation120575119876 = 119879119889119878 holds for all local Rindler causal horizons througheach space-time point and 120575119876 and 119879 are treated as theenergy flux and Unruh temperature respectively felt by anaccelerated observer inside the horizon [4] In 2010 Verlindedefined gravity as an entropic force due to the changes ofthe information related to the positions of the materials andthe space is emergent based on the holographic principle inhis discussions [5] Moreover Verlindersquos proposal has beenapplied to reproduce the Friedmann equation into branecosmology [6] and Friedmann-Robertson-Walker (FRW)universe [7] respectively

On the other hand it was addressed in [8ndash12] that theFriedmann equation can be modified by a bounce solutionof the universe as

1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)

in loop quantum cosmology (LQC) Then the authors of [13]attempted to derive the Friedmann equation by borrowingthe Clausius relation that is 120575119876 = 119879119889119878 and an entropy-area relation with quantum correction however they failedto reproduce the same modified Friedmann equation as thatin LQC with bounce solution Luckily the difficulty wasovercome by the authors of [14] where they proposed amodified dispersion relation at quantum phenomenologicallevel and then obtained the modified Friedmann equationfor a bounce solution to the flat FRW universe in LQC Itwas found that the role of their modified dispersion relationis to explicitly modify the Clausius relation This proposalhas also been extended into the spatially curved cases andthe correspondingmodified entropy-area relations have beenderived [15]

However the above studies are only involved in thegravity as an emergent phenomenon rather than the space-time itself as an emergent structure This situation has beenimproved by Padmanabhan In detail he proposed in [16]that the accelerated expansion of the universe is related tothe difference between the surface degrees of freedom (119873sur)and the bulk degrees of freedom (119873bulk) in a region of spacethat is Δ119881 = Δ119905(119873sur minus 119873bulk) where 119881 is the Hubblevolume and 119905 is the cosmic time in Planck units Moreoverthe standard evolution of the universe was also reproduceddirectly from the proposed relation This proposal inspired



plenty of related studies and remarkable progress [17ndash33]However in the framework of Padmanabhanrsquos conjecture thestudy of quantum effect is missing So it is interesting tointroduce the quantum effect to Padmanabhanrsquos conjectureand study the related cosmology

Thus in this paper we will introduce the modifieddispersion relation in the framework of Padmanabhanrsquos con-jecture and derive the modified Friedmann equation Thenwewill analyzewhether the quantumeffect in Padmanabhanrsquosconjecture will bring in modified Friedmann equation (1)with bounce solution It is notable that starting from theClausius relation to the apparent horizon along with themodified dispersion relation one can easily get the modifiedFriedmann equation with bounce solution to the FRW uni-verse [14 15] but the answer is not direct in Padmanabhanrsquosconjecture in the emergent universe Our study will give aninsight into the answer

Our paper is organized as follows In next section webriefly review Padmanabhanrsquos idea that the cosmic space isemergent as cosmic time progresses and give the standardFriedmann equation governing the dynamical evolution ofthe FRW universe Then in Section 3 we will analyze themodified Friedmann equation from Padmanabhanrsquos conjec-ture based on modified entropy-area relation Finally we willgive our summary and discussions in Section 4 In this paperwe use the natural units with 119888 = ℏ = 119896119861 = 12 The Emergence of Cosmic Space ofthe FRW Universe

In this section we will give a brief review on the processof obtaining standard Friedmann equation in the emergentuniverse which was addressed by Padmanabhan in [16]The main idea is that the expansion of the universe (orthe emergence of space) tends to fulfill the holographicequipartition condition which stated that the number ofdegrees of freedom (119873bulk) inside the Hubble volume is equalto the number of degrees of freedom (119873sur) on the sphericalsurface of Hubble radius that is 119873bulk = 119873sur So in ourasymptotic de Sitter universe the natural law governing theemergence of space in an infinitesimal interval 119889119905 is

119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)

where 119881 = 412058731198673 is the Hubble volume and 119905 is the cosmictime

For a spatially flat FRW universe with Hubble constant119867and apparent horizon 119903119860 = 1119867 we have

119873sur = 4119878 = 41205871198661198672 (3)

where 119878 = 1198604119866 = 1205871198661198672 is the entropy of the apparenthorizon and

119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)

where in the second equality we recalled the horizon temper-ature 119879 = 1198672120587 and Komar energy |119864| = minus(120588 + 3119901)119881 for

accelerating part with dark energy having 120588 + 3119901 lt 0 (it isnotable that in [16] the author discussed the contributions ofthematter with |119864| = (120588+3119901)119881 in the bulk degrees of freedomand the derivation of Friedmann equation was unaffected)Subsequently one can reduce (2) into

119886119886 = minus41205871198663 (120588 + 3119901) (5)

which is the standard dynamical Friedmann equation of flatFRWuniverse in general relativity Furthermore recalling thecontinuity equation

120588 + 3119867 (120588 + 119901) = 0 (6)

and integrating (5) gives us the standard Friedmann equation

1198672 = 81205871198661205883 (7)

Note that in [13] the integration result is1198672 +1198961198862 = 81205871198661205883with general geometry where the authors interpreted theintegration constant 119896 as the spatial curvature of the FRWuniverse

3 The Quantum Effect onFriedmann Equation in Cosmic Spaceof the FRW Universe

In this section we will apply the proposal described inlast section to study the quantum effect on the Friedmannequation We only consider the quantum effect at the phe-nomenological level and borrow the modified dispersionrelation (MDR) [14]

sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)

Here 119901 and 119864 are the momentum and energy of a particlewith mass119898 respectively The Planck length is 119897119901 = radic8120587119866 =1119872119901 where 119872119901 is the Planck mass 120578 is a dimensionlessparameter and 120578 rarr 0 goes to the standard dispersion relation1198642 = 1199012 + 1198982

With the use of thermodynamical description on theapparent horizon the authors of [14] derived the modifiedFriedmann equation of a spatially flat universe from MDR(8) Later the extended study in general FRW universe with119896 = 0 plusmn1 was presented in [15]

Here we will derive the modified Friedmann equation byfollowing the steps of emergent cosmic space shown in lastsection According to the study in [15] MDR (8) modifiedthe entropy for the first energy branch as

119878119872 = 1198604119866radic1 minus 412058712057821198972119901119860+ 12058712057821198972119901119866 ln[radic 119860412058712057821198972119901 + radic 119860412058712057821198972119901 minus 1]

(9)


where 119860 = 41205871199032119860 = 4120587(1198672 + 1198961198862) is the area of theapparent horizon at the classical level To proceed we definean effective apparent horizon area with the quantum effect

119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860

+ 412058712057821198972119901 ln[radic 119860412058712057821198972119901 + radic 119860412058712057821198972119901 minus 1]

= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)

Note that when 120578 rarr 0119860 is equal to119860 and recovers the usualresult

Moreover the volume (119881) and the area (119860) of the apparenthorizon of an 119899-sphere with radius 119903119860 satisfy [34]

119889119881119889119860 = 119903119860119899 minus 1 (11)

Then one can think that the change of the effective volumemainly stems from the change of the effective area so that wehave the time evolution of the effective volume of the FRWuniverse [34]

119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)

from which we can obtain the effective volume

= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)

Also when 120578 rarr 0 = 412058711990331198603 is the usual Hubble volumeWe move on to calculate119873bulk in the bulk and119873sur in the

boundary Considering the Hawking temperature (similar to(12) we ignore the direct correction to the radius in theHawking temperature and the changes of numbers of degreesof freedom directly stem from the corrections of the area ofthe apparent horizon) 119879 = 12120587119903119860 and 119864 = minus(120588 + 3119901) withdark energy in the bulk we obtain

119873bulk = 2119864119879= minus161205872 (120588 + 3119901) 120578311989731199011199031198603 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (14)

The statistical physics has shown that 119873sur can be calculatedfrom the entropy [18]

119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)

Substituting (12) (14) and (15) into (2) we get

41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1= 41205871199032119860radic1 minus 120578211989721199011199032119860 + 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]+ 161205872119866 (120588 + 3119901) 120578311989731199011199031198603 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1

(16)

The expression above looks very complicated howeverwith 119896 = 0 we have 119903119860 = 1 minus ( 119886119886)1199032119860 so that (16) can bereduced into (with 119896 = plusmn1 we have 119903119860 = 1 minus ((119867 minus 119896 1198861198863 +radic(1198672 + 1198961198862)3)radic(1198672 + 1198961198862))1199032119860 which makes it difficult tosimplify (16) we hope to solve this problem in near future)

119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)

where in the third line we have approximately expanded theexpression to the order 120578211989721199011199032119860 because it is a small quantityForm (17) is the modified dynamical Friedmann equationfor the flat FRW universe which reduces to the standarddynamical Friedmann equation when 120578 rarr 0 Furthercombining continuity equation (6) with (17) we obtain theother modified Friedmann equation

11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)

Here we also set the integration constant to be vanishedAgain 120578 rarr 0 in (18) reproduces the standard result of flatFRW universe


In order to analyze whether (18) admits a bouncesolution we define

120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)

so that (18) can be rewritten as (1) for bounce solution Theunsolved integral in 120588119888 makes it difficult to give a reliableconclusion however we can at least give some discussionsFirst without the quantum correction that is 120578 rarr 0 120588119888goes to infinity so bounce case (1) recovers standard case(7) without bounce Then when 120588119888 in (19) is positive (17)and (18) fulfill the bouncing conditions that is 119886 gt 0119886 = 0 and 119886 gt 0 then (18) admits a bounce solutionFinally when 120588119888 is nonpositive we can not have any bouncesolution

Wenote that for the second energy branchwhose entropyis minus119878119872 [15] the procedures above are straightforward and themodified Friedmann equation is the same as (17) and (18)However for this energy branch the effective volume () andthe area (119860) of the apparent horizon are negative which arenot physical

4 Summary and Discussions

In this paper we studied the quantum effect on the Fried-mann equation for the flat FRW universe with the use ofPadmanabhanrsquos conjecture in the emergent universe Weobtained modified Friedmann equations (17) and (18) withthe quantum correction on dispersion relation (8) For theclosed (119896 = 1) and open (119896 = minus1) universes we found it isdifficult to simplify the dynamical equation in our processbut we still see the quantum effect on (16) which is supposedto be the modified Friedmann equation We also argued thecondition under which modified Friedmann equation (18)admits a bounce solution in the flat universe

It is worth pointing out that in our paper the modifiedFriedmann equation was only obtained in the flat FRWuniverse with 119896 = 0 which may imply that key equation(2) is not the basic equation of the emergent universe andit may have to be corrected at the quantum level This isan interesting point we will study in the near future Onthe other hand the experimental testing of quantum effectin bouncing cosmology is another interesting aspect Thereare many literatures discussed this topic for example [3536] comparing the quantum theory with experimental dataand [37 38] probing the quantum gravity with modifieddispersion relation by cold atoms


The authors declare that they have no conflicts of interest

Acknowledgments

The authors appreciate Yi Ling and Wen-Jian Pan for helpfuldiscussion This work is supported by the National Natural

Science Foundation of China under Grant no 11705161 andNatural Science Foundation of Jiangsu Province under Grantno BK20170481

References

[1] J D Bekenstein ldquoBlack holes and entropyrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 7 pp 2333ndash2346 1973

[2] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973



[5] E Verlinde ldquoOn the origin of gravity and the laws of NewtonrdquoJournal of High Energy Physics vol 29 2011

[6] Y Ling and J Wu ldquoA note on entropic force and branecosmologyrdquo Journal of Cosmology and Astroparticle Physics vol2010 no 08 article 017 2010

[7] R-G Cai L-M Cao and N Ohta ldquoFriedmann equations fromentropic forcerdquo Physical Review D Particles Fields Gravitationand Cosmology vol 81 Article ID 061501 2010

[8] M Bojowald ldquoLoop quantum cosmologyrdquo Living Reviews inRelativity vol 8 no 11 2005

[9] A Ashtekar T Pawlowski and P Singh ldquoQuantum nature ofthe big bangrdquo Physical Review Letters vol 96 no 14 Article ID141301 2006

[10] A Ashtekar T Pawlowski and P Singh ldquoQuantum nature ofthe big bang Improved dynamicsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 74 no 8 Article ID084003 2006

[11] A Ashtekar T Pawlowski P Singh and K Vandersloot ldquoLoopquantum cosmology of k=1 FRW modelsrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 75 no 2Article ID 024035 2007

[12] A Ashtekar and E Wilson-Ewing ldquoCovariant entropy boundand loop quantum cosmologyrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 78 no 6 Article ID064047 2008

[13] R-G Cai L-M Cao and Y-P Hu ldquoCorrected entropy-arearelation and modified Friedmann equationsrdquo Journal of HighEnergy Physics vol 2008 2008

[14] Y Ling W Li and J Wu ldquoBouncing universe from a modifieddispersion relationrdquo Journal of Cosmology and AstroparticlePhysics vol 2009 no 11 pp 016-016 2009

[15] W-J Pan and Y-C Huang ldquoBouncing universe with modifieddispersion relationrdquo General Relativity and Gravitation vol 48no 144 2016


[16] T Padmanabhan ldquoEmergence and Expansion of CosmicSpace as due to the Quest for Holographic Equipartitionrdquohttpsarxivorgabs12064916

[17] K Yang Y Liu and Y Wang ldquoEmergence of cosmic space andthe generalized holographic equipartitionrdquo Physical Review DParticles Fields Gravitation andCosmology vol 86 no 10 2012

[18] F-Q Tu andY-X Chen ldquoEmergence of spaces and the dynamicequations of FRW universes in the f(R) theory and deformedHorava-Lifshitz theoryrdquo Journal of Cosmology and AstroparticlePhysics vol 1305 no 024 2013

[19] Y Ling and W Pan ldquoNote on the emergence of cosmic spacein modified gravitiesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 88 no 4 2013

[20] M Eune and W Kim ldquoEmergent Friedmann equation fromthe evolution of cosmic space revisitedrdquo Physical Review DParticles Fields Gravitation andCosmology vol 88 no 6 article067303 2013

[21] F-Q Tu and Y-X Chen ldquoEmergence of space and cosmicevolution based on entropic forcerdquo General Relativity andGravitation vol 47 no 8 2015

[22] R-G Cai ldquoEmergence of space and spacetime dynamics ofFriedmann-Robertson-Walker universerdquo Journal ofHigh EnergyPhysics vol 88 no 8 article 084019 2012

[23] M Hashemi S Jalalzadeh and S Vasheghani Farahani ldquoHawk-ing temperature and the emergent cosmic spacerdquo GeneralRelativity and Gravitation vol 47 no 4 2015

[24] A Sheykhi M H Dehghani and S E Hosseini ldquoFriedmannequations in braneworld scenarios from emergence of cosmicspacerdquo Physics Letters B vol 726 no 1-3 pp 23ndash27 2013

[25] C-Y Ee and D Lee ldquoFriedmann equation and the emergenceof cosmic spacerdquo Journal of High Energy Physics vol 1404 no125 2014

[26] A Sheykhi M H Dehghani and S E Hosseini ldquoEmergenceof spacetime dynamics in entropy corrected and braneworldmodelsrdquo Journal of Cosmology and Astroparticle Physics vol2013 2013

[27] A Sepehri F Rahaman A Pradhan and I H Sardar ldquoEmer-gence and expansion of cosmic space in BIonic systemrdquo PhysicsLetters B vol 741 pp 92ndash96 2015

[28] Z-L Wang W-Y Ai H Chen and J-B Deng ldquoCosmologicalmodel from emergence of spacerdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 92 no 2 article 0240512015

[29] A Sepehri F Rahaman S Capozziello A F Ali and APradhan ldquoEmergence and oscillation of cosmic space by joiningM1-branesrdquo The European Physical Journal C vol 76 no 5article no 231 2016

[30] F-F Yuan and P Huang ldquoEmergent cosmic space in Rastalltheoryrdquo Classical and Quantum Gravity vol 34 no 7 article077001 2017

[31] N Komatsu ldquoCosmological model from the holographicequipartition law with a modified Renyi entropyrdquoThe EuropeanPhysical Journal C vol 77 no 4 2017

[32] P Krishna and T K Mathew ldquoHolographic equipartition andthe maximization of entropyrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 96 no 6 2017

[33] F Q Tu Y X Chen B Sun and Y C Yang ldquoAccelerated Expan-sion of the Universe based on Emergence of Space andThermo-dynamics of the Horizonrdquo httpsarxivorgabs170706461

[34] R-G Cai ldquoEmergence of space and spacetime dynamics ofFriedmann-Robertson-Walker universerdquo Journal ofHigh EnergyPhysics vol 1211 no 016 2012

[35] G Amelino-Camelia ldquoIntroduction to quantum-gravity phe-nomenologyrdquo Lecture Notes in Physics vol 669 no 59 2005

[36] G Amelino-Camelia C Lammerzahl A MacIas and HMuller ldquoThe search for Quantum gravity signalsrdquo in AIPConference Proceedings 758 2005

[37] G Amelino-Camelia C Lammerzahl F Mercati and G MTino ldquoConstraining the Energy-Momentum Dispersion Rela-tion with Planck-Scale Sensitivity Using Cold Atomsrdquo PhysicalReview Letters vol 103 article 171302 2009

[38] F Mercati D Mazon G Amelino-Camelia et al ldquoProbingthe quantum-gravity realm with slow atomsrdquo Classical andQuantum Gravity vol 27 article 215003 2010

Research ArticleHawking Radiation-Quasinormal Modes Correspondence forLarge AdS Black Holes

Dao-Quan Sun Zi-LiangWang Miao He Xian-Ru Hu and Jian-Bo Deng


Correspondence should be addressed to Jian-Bo Deng dengjblzueducn

Received 22 September 2017 Accepted 8 November 2017 Published 26 November 2017


Copyright copy 2017 Dao-Quan Sun et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

It is well-known that the nonstrictly thermal character of the Hawking radiation spectrum generates a natural correspondencebetween Hawking radiation and black hole quasinormal modes This main issue has been analyzed in the framework ofSchwarzschild black holes Kerr black holes and nonextremal Reissner-Nordstrom black holes In this paper by introducing theeffective temperature we reanalyze the nonstrictly thermal character of large AdS black holes The results show that the effectivemass corresponding to the effective temperature is approximatively the average one in any dimension And the other effectivequantities can also be obtained Based on the known forms of frequency in quasinormal modes we reanalyze the asymptoticfrequencies of the large AdS black hole in three and five dimensions Then we get the formulas of the Bekenstein-Hawking entropyand the horizonrsquos area quantization with functions of the quantum ldquoovertonerdquo number 119899

1 Introduction

Ever since Hawkingrsquos astounding discovery [1] that blackholes radiate thermally many studies have been carriedout [2ndash11] One of the works done by Parikh and Wilczekconsidering the conservation of energy shows that an isolatedradiation black hole cannot have an emission spectrum thatis precisely thermal [2 3] Nonstrictly thermal characterof Hawking radiation spectrum implies that the spectrumis not strictly continuous and thus generates a naturalcorrespondence between Hawking radiation and black holequasinormal modes [12ndash16] It is very important for the phys-ical interpretation of black holes quasinormal modes Hodrsquosintriguing idea [17 18] suggested that black hole quasinormalmodes carry important information about black holersquos areaquantization Afterwards Hodrsquos influential conjecture wasrefined and clarified by Maggiore [19] In addition it is alsobelieved that quasinormal modes would be connected withthe microstructure of space-time [20] In [12ndash16] it wasshown that quasinormal modes can be naturally interpretedin terms of quantum levels where the emission or absorptionof a particle is interpreted as a transition between two distinctlevels on the discrete energy spectrum These results are

important to realize the unitary quantum gravity theory asblack holes are considered as theoretical laboratories fortesting models of quantum gravity

Based on the nonstrictly thermal character the effectiveframework of black holes was introduced by [12ndash16] ForSchwarzschild black holes [12 14] the physical solutionsfor the absolute values of the frequencies can be used toanalyze important properties and quantities like the horizonarea quantization the area quanta number the Bekenstein-Hawking entropy and the number of microstates [21] that isquantities which are considered fundamental to realize theunderlying unitary quantum gravity theory

Research showed that asymptotical AdS solutions arestable [22] The quantum mechanical and thermodynamicproperties of black holes inAdS space-time have been consid-ered [22 23] Some researches also show that the quasinormalfrequencies of AdS black holes could be interpreted in termsof dual conformal field transformation [24ndash26] In this paperwe would like to focus on the quasinormal frequencies oflarge AdS black holes We apply the effective framework ofnonstrictly thermal character to large AdS black holes indifferent dimensions We introduce the effective temperature



and show that the effective mass is approximatively theaverage one in any dimension As examples we will givethree- and five-dimensional cases Based on the knownforms of frequency in quasinormal modes we reanalyzethe asymptotic frequencies of the large AdS black hole inthree and five dimensions and calculate the Bekenstein-Hawking entropy the horizon area quantization the areaquanta number and the number of microstates

The paper is organized as follows in Section 2 we willstudy the effective temperature for large AdS black holes andthe corresponding quasinormal modes Section 3 is reservedfor conclusions and discussions

2 Effective Application of QuasinormalModes to the Large AdS Black Hole

In this section we will apply the nonstrictly thermal blackhole effective framework of [12ndash16] to large AdS black holesWorking with 119866 = 119888 = 119896B = ℏ = 141205871205760 = 1 (Planck units) isadopted in the followingThe metric of a 119889-dimensional AdSblack hole can be written as

1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903) + 1199032119889Ω2119889minus2119891 (119903) = 11990321198772 + 1 minus 120603119889minus1119872119903119889minus3

(1)

where119877 is the AdS radius and119872 is themass of the black holeFor a large black hole the metric function 119891(119903) is simplifiedto

119891 (119903) = 11990321198772 minus 120603119889minus1119872119903119889minus3 (2)

where120603119889minus1 = 16120587(119889minus2)119860119889minus2 119860119889minus2 = 2120587(119889minus1)2Γ((119889minus1)2)is the volume of a unit (119889 minus 2)-sphere

In general the rate of particle emission from the horizonis as follows [2]

Γ sim 119890minusΔ119878BH (3)

There are two different tunneling methods about calculatingthe Hawking temperature Both formulas come from asemiclassical approximation with a scalar field on a curvedbackground to calculate the tunneling amplitude but theydiffer by a factor of 2 in the resulting temperature The onemethodwhich is called canonically invariant tunneling uses aparticular ansatz for the action and then solves theHamilton-Jacobi equations to find the imaginary part for details see[5ndash11] In this paper we will focus on the method proposedby Parikh and Wilczek [2] to calculate the relevant physicalquantities And the form of the probability of the emissionfor the particle in a 119889-dimensional AdS black hole is given bythe following [27]

Im 119878 = 18119860119889minus2 [119903119889minus2ℎ (119872) minus 119903119889minus2ℎ (119872 minus 120596)] (4)

For a large AdS black hole the radius of the horizon is 119903ℎ =[1198772120603119889minus1119872]1(119889minus1) Then the probability will be

Γ = exp [minus 4120587119889 minus 2 (120603119889minus1)minus1(119889minus1)sdot 119877(2119889minus4)(119889minus1) (119872(119889minus2)(119889minus1) minus (119872 minus 120596)(119889minus2)(119889minus1))] (5)

By making the second-order Taylor expansion of Δ119878BHabout 120596

Γ = exp[minus4120587 (120603119889minus1)minus1(119889minus1)sdot 119877(2119889minus4)(119889minus1) ( 1119889 minus 1 1205961198721(119889minus1)+ 12 (119889 minus 1)2 1205962119872119889(119889minus1))]

(6)

For a large AdS black hole the Hawking temperature is 119879H =((119889 minus 1)4120587)(119903ℎ1198772) From (6) we transform Γ to getΓ = exp [minus 120596119879H (1 + 12 (119889 minus 1) 120596119872)] (7)

and the leading term gives the thermal Boltzmann factorsfor the emanating radiation The second term representscorrections from the response of the background geometryto the emission of a quantum

Then we introduce the effective temperature

119879E (120596) = 2 (119889 minus 1)1198722 (119889 minus 1)119872 + 120596119879H = 2 (119889 minus 1) 120574119872119889(119889minus1)2 (119889 minus 1)119872 + 120596 (8)

where 120574 = ((119889 minus 1)4120587)((1198772120603119889minus1)1(119889minus1)1198772) Then (7) can berewritten in Boltzmann-like form

Γ sim exp [minus120573E (120596) 120596] = exp(minus 120596119879E (120596)) (9)

where 120573E(120596) = 1119879E(120596) and exp[minus120573E(120596)120596] is the effectiveBoltzmann factor From theHawking temperature119879H = ((119889minus1)4120587)(119903ℎ1198772) and (8) we can introduce the effective mass119872E = [2(119889minus1)]119889minus1119872119889[2(119889minus1)119872+120596]119889minus1 and thenweTaylorexpand119872E in 120596119872 Thus for 119889-dimensional large AdS blackholes

119872E = 119872 minus 1205962 (10)

As one can see the effective mass is the average quantitybetween the states before and after the emission Here itshould be remarked that the original idea is that during theparticle emission the effective mass is the average quantitywhich has only been proved in Schwarzschild black holes andcan be generalized to cases of the large AdS black hole in anydimension Thus in any dimension we may also introducethe other effective quantities such as the large AdS blackhole the effective horizon 119903E effective horizon area 119860E the


Bekenstein-Hawking large AdS black hole effective entropy119878E and other quantitiesSince the asymptotic forms of quasinormal modes of the

3-dimensional and 5-dimensional large AdS black holes weregiven by [28] we will reanalyze the asymptotic frequencies ofthe large AdS black hole in the three and five dimensions byintroducing effective temperature

For the 5-dimensional large AdS black hole 119903ℎ =(11987721206034119872)14 119879H = (1120587)(119903ℎ1198772) = (11987721206034119872)141205871198772 From(8) we can easily get

119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)

where 1205745 = (11987721206034)141205871198772 The asymptotic form of quasinor-mal modes of the 5-dimensional large AdS black hole is givenas follows [28]

120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)

To consider the nonstrictly thermal spectrum of thelarge AdS black hole we will correct the asymptotic formof quasinormal modes of the 5-dimensional large AdS blackhole Namely we substitute the Hawking temperature 119879H in(12) by the effective temperature 119879E Now (12) yields

120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)

Now |120596119899| is the damped harmonic oscillatorrsquos proper fre-quency that is defined as follows [12 14 19]

10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)

By using the new expression (13) for the frequencies ofquasinormal modes we get

10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get

10038161003816100381610038161205961198991003816100381610038161003816 = minus4119872 plusmn 4119872radic1 + radic21205745120587119872minus34119899 (17)

Clearly only |120596119899| gt 0 has physical meaning So we can get

10038161003816100381610038161205961198991003816100381610038161003816 = 4119872radic1 + radic21205745120587119872minus34119899 minus 4119872 (18)

Since the emitted energy is much less than the black holemass that is |120596119899| lt 119872 thus we have

4119872radic1 + radic21205745120587119872minus34119899 minus 4119872 lt 119872 (19)

Then we give a maximum value for the quantum ldquoovertonerdquonumber 119899

119899 lt 119899max = 9119877321198723416radic2120603144 (20)

The above approach has had some important conse-quences on the quantum physics of black hole Then ourfollowing discussion will start with the quantization of theblack hole horizon area For the 5-dimensional large AdSblack hole the horizon area 119860 is related to the mass throughthe relation 119860 = 21205872(11987721206034119872)34 and a variation Δ119872 in themass generates a variation

Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)

In any case assuming a transition 119899 rarr 119899minus1 (13) gives anemitted energy

Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)

where we have defined

1198915 (119872 119899) = 4119872radic1 + radic21205745120587119872minus34119899minus 4119872radic1 + radic21205745120587119872minus34 (119899 minus 1) (23)

Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)

due to the black hole horizon area associated withBekenstein-Hawking entropy 119878 = 1198604 thus

Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)

As can be seen (24) and (25) give the corrected formula ofthe horizonrsquos area quantization related to the quantum ldquoover-tonerdquo number 119899 by introducing the effective temperature Inthe approximation of Taylor expansion

1198915 (119872 119899) asymp 2radic2120603144 1198721411987732 Δ119860 asymp 3radic212058721206034Δ119878 asymp 3radic24 12058721206034

(26)

We can see that under the condition of approximation thearea spectrum is equidistant Our results of (26) are the sameas (29) and (31) of [29] which calculated the area and entropy


spectra for 5-dimensional large AdS black holes by the Bohr-Sommerfeld quantization condition to the adiabatic invariantquantity [29 30]

Now assuming that the horizon area is quantized withquantum Δ119860 = 119886 where 119886 = (3120587211987732120603344 211987214)1198915(119872 119899)the total horizon area is 119860 = 119873Δ119860 = 119873119886 where 119873 is thenumber of quanta of area We give

119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)

The above analysis will have important consequences onentropy and microstates It is usually believed that anycandidate for quantum gravity must explain the microscopicorigin of the Bekenstein-Hawking entropy

From (27) we can give

119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)

and the formula of the famous Bekenstein-Hawking [1 3132] entropy becomes function of the quantum ldquoovertonerdquonumber 119899

According to [33] the number of microstates for thisradiation is simply

Ωradiation (120596) = 1Γ (120596) (29)

where Γ(120596) is the radiation probability for the an emission 120596Thus the number of microstates for a large AdS black holewith119872 is found to be

Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)

where Γ5(120596119894) = exp[minus(41205873)11987732(1206034)minus14(11987234minus(119872minus120596119894)34)]From (27) and (30) we can get

Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)

We can see that the number of microstates for the large AdSblack hole becomes a function of the quantum ldquoovertonerdquonumber 119899

According to discussion of the 5-dimensional AdS largeblack hole we will do the same approach to treat the 3-dimensional large AdS black hole

For the 3-dimensional AdS large black hole 119903ℎ =119877(1206032119872)12 119879H = 119903ℎ21205871198772 From (8) we can easily get

119879E = 41205743119872324119872 + 120596 (32)

where 1205743 = radic12059622120587119877 The asymptotic form of quasinormalmodes of the 3-dimensional large AdS black hole is given asfollows [28]

120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)

where 1199012 = 1199014120587119877119879H To consider the nonstrictly thermalspectrum of the large AdS black hole we will correct theasymptotic form of quasinormal modes of the 3-dimensionallarge AdS black hole Namely we substitute the Hawkingtemperature 119879H in (33) by the effective temperature 119879E Now(33) yields

120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where

119898119899 = plusmnradic4120587119901119877 2120574123 11987234radic4119872 + 10038161003816100381610038161205961198991003816100381610038161003816

119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)

By using the new expression (34) for the frequencies ofquasinormal modes then we get

10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)

where the solution of (36) in terms of |120596119899| will be the answerof |120596119899| Therefore given a quantum transition between 119899 and119899 minus 1 we define

Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)

For the 3-dimensional large AdS black hole the horizonarea 119860 = 2120587119877radic1206032119872 and a variation Δ119872 in the massgenerates a variation

Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)

From (40) we can give Bekenstein-Hawking entropy

119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)

which becomes the function of the quantum ldquoovertonerdquonumber 119899

Same as the technique with the 5-dimensional large AdSblack hole we can get the number of microstates

Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)

where Γ3(120596119894) = exp[minus4120587119877(1206032)minus12(11987212 minus (119872 minus 120596119894)12)]


3 Conclusion

In this paper by introducing the effective temperature theanalysis on the nonstrictly thermal character of the large AdSblack hole is presented Our results show that the effectivemass corresponding to the effective temperature is approx-imatively the average one in any dimension And the othereffective quantities can also be obtained Based on the knownforms of frequency in quasinormal modes we reanalyze theasymptotic frequencies of the large AdS black hole in threeand five dimensions Then the formulas of the Bekenstein-Hawking entropy and the horizonrsquos area quantization withfunctions of the quantum ldquoovertonerdquo number 119899 are gotMoreover the results we give in the five dimensions have agood consistency with the original one in the approximationof Taylor expansionWe can also see that under the conditionof approximation the area spectrum is equidistant



Acknowledgments

The authors would like to acknowledge the National NaturalScience Foundation of China (Grant no 11571342) for sup-porting them during this work

References


[2] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[3] M K Parikh ldquoA secret tunnel through the horizonrdquo GeneralRelativity and Gravitation vol 36 no 11 pp 2419ndash2422 2004

[4] M K Parikh ldquoEnergy conservation and Hawking radiationrdquoin On Recent Developments in Theoretical and ExperimentalGeneral Relativity Gravitation and Relativistic Eld TheoriesProceedings 10th Marcel Grossmann Meeting MG10 Rio deJaneiro Brazil July 20ndash26 2003 Pt A-C pp 1585ndash1590 2004

[5] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[6] E T Akhmedov V Akhmedova D Singleton and T PillingldquoThermal radiation of various gravitational backgroundsrdquo Inter-national Journal of Modern Physics A vol 22 no 8-9 pp 1705ndash1715 2007

[7] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo PramanamdashJournal of Physics vol 70 no 1 pp 3ndash26 2008

[8] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classicl calculation of hawking radationrdquo InternationalJournal of Modern Physics D vol 17 no 13-14 pp 2453ndash24582008

[9] V Akhmedova T Pilling A de Gill and D Singleton ldquoTempo-ral contribution to gravitationalWKB-like calculationsrdquoPhysicsLetters B vol 666 no 3 pp 269ndash271 2008

[10] V Akhmedova T Pilling A de Gill and D SingletonldquoComments on anomaly versus WKBtunneling methods forcalculating Unruh radiationrdquo Physics Letters B vol 673 no 3pp 227ndash231 2009

[11] T Pilling ldquoBlack hole thermodynamics and the factor of 2problemrdquo Physics Letters B vol 660 no 4 pp 402ndash406 2008

[12] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 2011 no 8 article no 101 2011

[13] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[14] C Corda ldquoEffective temperature hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 2012

[15] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics vol 2013 no 6article no 008 2013

[16] C Corda ldquoNon-strictly black body spectrum from the tun-nelling mechanismrdquo Annals of Physics vol 337 pp 49ndash54 2013

[17] S Hod ldquoGravitation the quantum and Bohrrsquos correspondenceprinciplerdquo General Relativity and Gravitation vol 31 no 11 pp1639ndash1644 1999

[18] SHod ldquoBohrrsquos correspondence principle and the area spectrumof quantum black holesrdquo Physical Review Letters vol 81 no 20pp 4293ndash4296 1998

[19] M Maggiore ldquoPhysical interpretation of the spectrum of blackhole quasinormal modesrdquo Physical Review Letters vol 100 no14 Article ID 141301 2008

[20] M Casals and A Ottewill ldquoBranch cut and quasinormal modesat large imaginary frequency in Schwarzschild space-timerdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 86 no 2 Article ID 024021 2012

[21] J Zhang ldquoBlack hole quantum tunnelling and black holeentropy correctionrdquo Physics Letters B vol 668 no 5 pp 353ndash356 2008


[23] G T Horowitz and V E Hubeny ldquoQuasinormal modes of AdSblack holes and the approach to thermal equilibriumrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 62no 2 Article ID 024027 11 pages 2000

[24] J Maldacena ldquoThe large-N limit of superconformal fieldtheories and supergravityrdquo International Journal of TheoreticalPhysics vol 38 no 4 pp 1113ndash1133 1999

[25] E Witten ldquoAnti de Sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[26] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[27] S Hemming and E Keski-Vakkuri ldquoHawking radiation fromAdS black holesrdquo Physical Review D Particles Fields Gravita-tion and Cosmology vol 64 no 4 Article ID 044006 2001

[28] S Musiri and G Siopsis ldquoAsymptotic form of quasi-normalmodes of large AdS black holesrdquo Physics Letters B vol 576 no3-4 pp 309ndash313 2003


[29] S-W Wei and Y-X Liu Area spectrum of the large adsblack hole from quasinormal modes 2009 httpsarxivorgabs09060908

[30] G Kunstatter ldquod-dimensional black hole entropy spectrumfrom quasinormal modesrdquo Physical Review Letters vol 90 no16 161301 4 pages 2003

[31] J D Bekenstein ldquoBlack holes and the second lawrdquo Lettere AlNuovo Cimento Series 2 vol 4 no 15 pp 737ndash740 1972


[33] Q-Y Cai C-P Sun and L You ldquoInformation-carrying Hawk-ing radiation and the number of microstate for a black holerdquoNuclear Physics B vol 905 pp 327ndash336 2016







Editorial Board




Contents































Ali Oumlvguumln 123







1 Introduction


119875 = minus Λ8120587 = 3

81205871198972 (1)


119881 = (120597119872120597119875 )119878119876

(2)



(119875 + 119886V2

) (V minus 119887) = 119896119879 (3)










119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)


1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)



119891 (119903)= 119889 minus 3



119876(119889minus1)2 ln 119903119903119889minus2

(6)




119898



119878 = 119860ℎ4 120578119903ℎ (9)


119875 = minus Λ8120587 (10)





119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References








































Editorial Board




Contents































Ali Oumlvguumln 123







1 Introduction


119875 = minus Λ8120587 = 3

81205871198972 (1)


119881 = (120597119872120597119875 )119878119876

(2)



(119875 + 119886V2

) (V minus 119887) = 119896119879 (3)










119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)


1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)



119891 (119903)= 119889 minus 3



119876(119889minus1)2 ln 119903119903119889minus2

(6)




119898



119878 = 119860ℎ4 120578119903ℎ (9)


119875 = minus Λ8120587 (10)





119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References





































Editorial Board




Contents































Ali Oumlvguumln 123







1 Introduction


119875 = minus Λ8120587 = 3

81205871198972 (1)


119881 = (120597119872120597119875 )119878119876

(2)



(119875 + 119886V2

) (V minus 119887) = 119896119879 (3)










119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)


1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)



119891 (119903)= 119889 minus 3



119876(119889minus1)2 ln 119903119903119889minus2

(6)




119898



119878 = 119860ℎ4 120578119903ℎ (9)


119875 = minus Λ8120587 (10)





119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References



































Editorial Board




Contents































Ali Oumlvguumln 123







1 Introduction


119875 = minus Λ8120587 = 3

81205871198972 (1)


119881 = (120597119872120597119875 )119878119876

(2)



(119875 + 119886V2

) (V minus 119887) = 119896119879 (3)










119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)


1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)



119891 (119903)= 119889 minus 3



119876(119889minus1)2 ln 119903119903119889minus2

(6)




119898



119878 = 119860ℎ4 120578119903ℎ (9)


119875 = minus Λ8120587 (10)





119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References



































Contents































Ali Oumlvguumln 123







1 Introduction


119875 = minus Λ8120587 = 3

81205871198972 (1)


119881 = (120597119872120597119875 )119878119876

(2)



(119875 + 119886V2

) (V minus 119887) = 119896119879 (3)










119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)


1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)



119891 (119903)= 119889 minus 3



119876(119889minus1)2 ln 119903119903119889minus2

(6)




119898



119878 = 119860ℎ4 120578119903ℎ (9)


119875 = minus Λ8120587 (10)





119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References



























































Ali Oumlvguumln 123







1 Introduction


119875 = minus Λ8120587 = 3

81205871198972 (1)


119881 = (120597119872120597119875 )119878119876

(2)



(119875 + 119886V2

) (V minus 119887) = 119896119879 (3)










119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)


1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)



119891 (119903)= 119889 minus 3



119876(119889minus1)2 ln 119903119903119889minus2

(6)




119898



119878 = 119860ℎ4 120578119903ℎ (9)


119875 = minus Λ8120587 (10)





119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References














































Ali Oumlvguumln 123







1 Introduction


119875 = minus Λ8120587 = 3

81205871198972 (1)


119881 = (120597119872120597119875 )119878119876

(2)



(119875 + 119886V2

) (V minus 119887) = 119896119879 (3)










119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)


1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)



119891 (119903)= 119889 minus 3



119876(119889minus1)2 ln 119903119903119889minus2

(6)




119898



119878 = 119860ℎ4 120578119903ℎ (9)


119875 = minus Λ8120587 (10)





119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References




































Ali Oumlvguumln 123







1 Introduction


119875 = minus Λ8120587 = 3

81205871198972 (1)


119881 = (120597119872120597119875 )119878119876

(2)



(119875 + 119886V2

) (V minus 119887) = 119896119879 (3)










119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)


1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)



119891 (119903)= 119889 minus 3



119876(119889minus1)2 ln 119903119903119889minus2

(6)




119898



119878 = 119860ℎ4 120578119903ℎ (9)


119875 = minus Λ8120587 (10)





119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References









































119878 = int119889119889119909radicminus119892[119891 (119877)16120587 + L (119865)] (4)


1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903)



119891 (119903)= 119889 minus 3



119876(119889minus1)2 ln 119903119903119889minus2

(6)




119898



119878 = 119860ℎ4 120578119903ℎ (9)


119875 = minus Λ8120587 (10)





119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References





































119898



119889119898 = 119879119889119878 + Φ119889119876 + 119881119889119875 (13)



119898 = 2119879119878 + Φ119876 minus 2119881119875 (14)



120597119875120597V = 1205972119875

120597V2 = 0 (15)


119891 = 12 minus 119898

1199032 minus 1199032Λ minus 311987632 ln (119903)1199032120578 (16)


119898 = 11990322 minus 1199032Λ minus 311987632 ln (119903)

120578 (17)


119879 = minus311987632 minus 1199032120578 + 41199034120578Λ41205871199033120578 (18)


119879 = 8119903119875 minus 31198763241199033120578120587 + 1

4119903120587 (19)


119875 = 1198798119903 + 311987632

321199034120578120587 minus 1321199032120587 (20)


119879119888 = minus24 11987632V3120578120587 + 1

V120587


119875119888 = 120578115211987632120587

(21)


120588119888 = 119875119888V119888119879119888 = 3

8 (22)



119866 = 119898 minus 119879119878

= 611987632 + 1199032120578 minus 161198751205871199034120578 minus 1811987632 ln (119903)6120578 (23)


4 Conclusion



T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References




































T = 13Tc

T = Tc

T = 08Tc

0000

0005

0010

0015

0020

0025

0030

P

1 2 3 4 5 60r


005 010 015 020 025 030000T

001

002

003

004

P

r = 13rc

r = 08rc

r = rc




P = Pc

010 015 020005T

00

01

02

03

04

05

G

P = 02Pc

P = 2Pc





Acknowledgments


References









































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References








































































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References

































































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References































































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References




































Sumanta Chakraborty







1 Introduction







119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References





































119866119886119887119906119886119906119887 = 120581119879119886119887119906119886119906119887 (1)


119877119886119887ℓ119886ℓ119887 = 120581119879119886119887ℓ119886ℓ119887 (2)







P119886119887 = 120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887

P119886119887P119887119888 = (120575119886119887 minus 1

ℓ2 ℓ119886ℓ119887)(120575119887119888 minus 1

ℓ2 ℓ119887ℓ119888) = 120575119886119888 minus 1

ℓ2 ℓ119886ℓ119888

= P119886119888

(3)


R119898119899119903119904 = P119886119898P119887119899P119888119903P119889119904119877119886119887119888119889 (4)


R = P119898119903P119899119904119877119898119899119903119904 = 119877 minus 2ℓ2119877119886119887ℓ

119886ℓ119887 (5)





119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References





































119877119886119887ℓ119886ℓ119887 minus 12119877ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (6)






1198771198881119889111988611198871

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

equiv 1119898119875119886119887119888119889 (119898)119877119888119889119886119887 (7)



1198771198882119889211988621198872

sdot sdot sdot 119877119888119898119889119898119886119898119887119898

(8)




R119886119887119888119889 = 1

2 (119875119886119887119898119899119877119898119899119888119889 + 119875119898119899119888119889 119877119886119887119898119899)


(9)


R = R119886119887119888119889P119888119886P119889119887 (10)


R = R119886119887119888119889120575119888119886120575119889119887 minus 2

ℓ2 R119886119887119888119889ℓ119888ℓ119886120575119889119887 = 119875119886119887119888119889119877119888119889119886119887


minus 2ℓ2 119875

119886119887119898119899119877119898119899119888119887 minus (119898 minus 1)



ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 2


ℓ2119875119886119887119898119899119877119898119899119888119887 ℓ119886ℓ119888 + 119871

(11)


119875119886119887119898119899119877119887119898119899119888 ℓ119886ℓ119888 minus 12119871ℓ2 = 120581119879119886119887ℓ119886ℓ119887 (12)








Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References










































Appendix





1198771198881119888211988911198892


(A1)




sdot sdot sdotP11990121198981198872119898

P11988611199021




119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References





































119898R = P11988611198871

sdot sdot sdotP11988621198981198872119898


(A3)



119898R= 12057511988611198871

sdot sdot sdot 12057511988621198981198872119898




1198771198881119888211988911198892



1198771198881119888211988911198892


(A4)





(A5)

as well as


1198771198881119888211988911198892



1198771198881119888211988911198892


(A6)



1198771198881119888211988911198892


minus 2119898ℓ2


1198771198881119888211988911198892


= 119871 minus 2ℓ2 ℓ119888ℓ119889119877119888119901119902119903119875119902119903119889119901 = 119871 minus 2

ℓ2R119886119887ℓ119886ℓ119887

(A7)




Acknowledgments



References


















































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References


































































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References











































1 Introduction














1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References








































1198891199042 = minus119891 (119903) 1198891199052 + 1119891 (119903)1198891199032 + 1199032 (1198891205792 + sin2 1205791198891206012) (1)


119879 = 1205812120587 = 1198911015840 (119903119867)4120587 119878 = 1205871199032119867119881 = 41205873 1199033119867

(2)


119889119867 = 119889119872 = 119879119889119878 + 119881119889119875 (3)


119889119880 = 119879119889119878 minus 119875119889119881 (4)



(5)


119866120583] + Λ119892120583] = 8120587119879120583] (6)


minus1 + 119891 (119903) + 1199031198911015840 (119903) = minusΛ1199032 (7)


1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889 (1199031198672 ) = minus Λ8120587119889(412058711990331198673 ) (8)



119879120583] = 14120587 (119865120583120590119865120590] minus 14119892120583]119865120590120588119865120590120588) (9)


119879119905119905 = 119891119864208120587 119879119903119903 = minus119891minus1119864208120587 119879120579120579 = 1199032119864208120587 119879120601120601 = 1199032sin2 120579119864208120587

(10)




1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) = 119875119889(412058711990331198673 ) (12)



1198911015840 (119903119867)4120587 119889 (1205871199032119867) minus 119889(1199031198672 + 11987622119903119867) + 119876119903119867119889119876= 119875119889(412058711990331198673 )

(13)


119889119880 = 119879119889119878 minus 119875119889119881 + Φ119889119876 (14)





1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References






































1198891199042 = minus120595 (119903)2 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (15)


119877119905119905 = (1205952119891)10158401015840 1198912 minus (1205952119891)1015840 1198914 (minus1198911015840119891 + (1205952119891)10158401205952119891 )

+ (1205952119891)1015840 119891119903 119877119903119903 = minus(1205952119891)

10158401015840

21205952119891 + (1205952119891)101584041205952119891 (minus1198911015840119891 + (1205952119891)10158401205952119891 ) minus 1198911015840119903119891

119877120579120579 = 1 minus 1199031198912 (1198911015840119891 + (1205952119891)10158401205952119891 ) minus 119891119877120601120601 = sin2 120579119877120579120579

(16)


119877120583] minus Λ119892120583] = 8120587(119879120583] minus 12119879119892120583]) (17)




(18)


119889119880 minus 1199031198672 1198911015840 (119903119867) 119889119903119867 = minus119875119889(412058711990331198673 ) (19)

where

119889119880 = [12 + 4120587 (119879120579120579 minus 121198791199032119867)] 119889119903119867 (20)



119879 = 1205812120587 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (21)

Now rewrite (19) as

119889119880 minus 119879( 2120587119903119867120595 (119903119867)119889119903119867) = minus119875119889119881 (22)


119889119878 = 2120587119903119867120595 (119903119867)119889119903119867 (23)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (24)







(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References








































(25)



119902120583] = 119892120583] + 120581119877120583] (26)



Γ120582120583] = 12119902120582120590 (119902120583120590] + 119902]120590120583 minus 119902120583]120590) (28)


119877120583] ≃ Λ119892120583] + 8120587(119879120583] minus 12119879119892120583])+ 8120587120581 [119878120583] minus 14119878119892120583]]

(29)



1198891199042119892 = minus1205952 (119903) 119891 (119903) 1198891199052 + 1119891 (119903)1198891199032+ 1199032 (1198891205792 + sin2 1205791198891206012) (30)


1198891199042119902 = minus1198662 (119903) 119865 (119903) 1198891199052 + 1119865 (119903)1198891199032+ 1198672 (119903) (1198891205792 + sin2 1205791198891206012) (31)


119877119905119905 = 2119866119866101584011986710158401198652119867 + 119866211986511986510158401198671015840119867 + 3211986611986610158401198651198651015840 + 119866119866101584010158401198652+ 12119866211986511986510158401015840



(32)


1198672119866119865 = 1205821199032120595119891 1198661198672119865 = 1205821199032120595119891

119866 = 120582120595(33)


119866 = 120582120595119865 = 120582minus1119891

1198672 = 1205821199032(34)









(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References







































(37)


1198672119866119865 = (120582 + 12058111986420) 11990321205951198911198661198672119865 = (120582 + 12058111986420) 1199032120595119891

119866 = (120582 minus 12058111986420) 120595(38)

and one can get


1198672 = (120582 + 12058111986420) 1199032(39)



(40)

where

119884 = 2120581119903119864011986410158400 + 1198661015840119866 1198671198671015840 minus 11986710158402 (41)


31198673 ) (42)



119879119889119878 = 1199031198672 1198911015840 (119903119867) 119889119903119867 (43)


120581 = lim119903rarr119903119867

12 120597119903119892119905119905radic119892119905119905119892119903119903 = 120595 (119903119867) 1198911015840 (119903119867)2 (44)


119879 = 120595 (119903119867) 1198911015840 (119903119867)4120587 (45)


119889119878 = ( 2120587119903119867120595 (119903119867)) 119889119903119867 (46)

or

119878 = int 2120587119903119867120595 (119903119867)119889119903119867 (47)




120595 (119903) = radic1205821199032radic1205821199034 + 1205811198762 (48)



2

1199034119867) (49)










Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References











































Acknowledgments


References







































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References



















































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References











































1 Introduction



1198672 = 81205871198663 120588(1 minus 120588120588119888) (1)









119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References








































119889119881119889119905 = 119866 (119873sur minus 119873bulk) (2)



119873sur = 4119878 = 41205871198661198672 (3)


119873bulk = 2 |119864|119879 = minus2 (120588 + 3119901)119881119879 (4)



119886119886 = minus41205871198663 (120588 + 3119901) (5)


120588 + 3119867 (120588 + 119901) = 0 (6)


1198672 = 81205871198661205883 (7)




sin (120578119897119901119864)120578119897119901 = radic1199012 + 1198982 (8)





(9)



119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References





































119860 = 4119866119878119872= 119860radic1 minus 412058712057821198972119901119860


= 41205871199032119860radic1 minus 120578211989721199011199032119860+ 412058712057821198972119901 ln[[

119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(10)



119889119881119889119860 = 119903119860119899 minus 1 (11)


119889119889119905 = 1199031198602 119889119860119889119905 = 41205871199033119860 119903119860120578119897119901radic119903211986012057821198972119901 minus 1 (12)


= 4120587120578311989731199013 (2 + 119903211986012057821198972119901)radic 119903211986012057821198972119901 minus 1 (13)





119873sur = 4119878119872= 41205871199032119860119866 radic1 minus 120578211989721199011199032119860

+ 412058712057821198972119901119866 ln[[119903119860120578119897119901 + radic 119903211986012057821198972119901 minus 1]]

(15)




(16)


119886119886 = minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 minus 2120578411989741199011199034119860 )+ 12057821198972119901 [2 + ln (1205782119897211990141199032119860)]21199034119860

≃ minus4120587119866 (120588 + 3119901)3 (1 + 120578211989721199011199032119860 ) + 120578211989721199011199034119860 (17)


11988621198862 = 81205871198661205883 + 81205871198661205782119897211990131198862 int( 1198862 120588 + 2 1198863120588119886 )119889119905+ 2120578211989721199011198862 int 11988651198863 119889119905

(18)




120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References





































120588119888 = minus 1205882(120578211989721199011198862) int ( 1198862 120588 + 2 1198863120588119886) 119889119905 + (31205782119897211990141205871198661205881198862) int ( 11988651198863) 119889119905 (19)








Acknowledgments



References
















































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References



































































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References











































1 Introduction












(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References









































(1)












(6)








119872E = 119872 minus 1205962 (10)






119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References







































119879E (120596) = 81198728119872 + 120596119879H = 81205745119872548119872 + 120596 (11)


120596119899 = 2120587119879H119899 (plusmn1 minus 119894) (12)


120596119899 = plusmn2120587119879E119899 minus 1198942120587119879E119899 = 119898119899 + 119894119901119899 (13)

where

119898119899 = plusmn161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899119901119899 = minus161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899

(14)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992 (15)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic(161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)

2 + (161205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899)2

= 16radic21205871205745119872548119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(16)

Then one can get







119899 lt 119899max = 9119877321198723416radic2120603144 (20)


Δ119860 = 3120587211987732120603344211987214 Δ119872 (21)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198915 (119872 119899) (22)



Therefore

Δ119860 = 3120587211987732120603344211987214 1198915 (119872 119899) (24)


Δ119878 = 3120587211987732120603344811987214 1198915 (119872 119899) (25)



(26)





119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References






































119873 = 119860Δ119860 = 21205872 (11987721206034119872)34119886 = 411987231198915 (119872 119899) (27)



119878BH = 1198604 = 3119873120587211987732120603344811987214 1198915 (119872 119899) (28)



Ωradiation (120596) = 1Γ (120596) (29)


Ω5 = 119899prod119894=1

1Γ5 (120596119894) = 119890(12058722)(11987721206034119872)34 (30)


Ω5 = 119890(3119873120587211987732120603344 811987214)1198915(119872119899) (31)




119879E = 41205743119872324119872 + 120596 (32)


120596119899 = plusmn4120587119879H119901 minus 1198944120587119879H119899 (33)


120596119899 = plusmn4120587119879E119901E minus 1198944120587119879E119899 = 119898119899 + 119894119901119899 (34)

where


119901119899 = minus161205871205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 119899(35)


10038161003816100381610038161205961198991003816100381610038161003816 = radic1198981198992 + 1199011198992= radic 16120587119901119877 1205743119872324119872 + 10038161003816100381610038161205961198991003816100381610038161003816 +

2561205872120574231198723(4119872 + 10038161003816100381610038161205961198991003816100381610038161003816)2 1198992(36)


Δ119872 = 1003816100381610038161003816120596119899 minus 120596119899minus11003816100381610038161003816 = 1198913 (119872 119899) (37)


Δ119860 = 120587119877radic1206032119872Δ119872 = 120587119877radic12060321198721198913 (119872 119899) (38)

Thus

Δ119878 = 1205871198774 radic12060321198721198913 (119872 119899) (39)

119873 = 119860Δ119860 = 21198721198913 (119872 119899) (40)


119878BH = 1198604 = 1198734 120587119877radic12060321198721198913 (119872 119899) (41)



Ω3 = 119899prod119894=1

1Γ3 (120596119894) = 119890(12)120587119877radic1206032119872= 119890(1198734)120587119877radic12060321198721198913(119872119899)

(42)



3 Conclusion




Acknowledgments


References




































3 Conclusion




Acknowledgments


References









































Documents

Black Holes: Insights and Enigmasdownloads.hindawi.com/journals/specialissues/519381.pdf · Black Holes: Insights and Enigmas Lead Guest Editor: Izzet Sakalli Guest Editors: Eduardo