Research Article On Continuum Damage Modeling of Fiber

Research ArticleOn Continuum Damage Modeling of Fiber ReinforcedViscoelastic Composites with Microcracks in terms of Invariants

Melek Usal

Department of Manufacturing Engineering Suleyman Demirel University 32260 Isparta Turkey

Correspondence should be addressed to Melek Usal melekusalsduedutr

Received 30 December 2014 Accepted 23 February 2015

Academic Editor Carla Roque

Copyright copy 2015 Melek Usal 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

A continuum damage model is developed for the linear viscoelastic behavior of composites with microcracks consisting of anisotropic matrix reinforced by two arbitrarily independent and inextensible fiber families Despite the fact that the matrix materialis isotropic the model in consideration bears the characteristic of directed media included in the transverse isotropy symmetrygroup solely due to its fibers distributions and the existence of microcracks Using the basic laws of continuum damage mechanicsand equations belonging to kinematics and deformation geometries of fibers the constitutive functions have been obtained It hasbeen detected as a result of the thermodynamic constraints that the stress potential function is dependent on two symmetric tensorsand two vectors whereas the dissipative stress function is dependent on four symmetric tensors and two vectors To determinearguments of the constitutive functionals findings relating to the theory of invariants have been used as a method because of thefact that isotropy constraint is imposed on the material As a result the linear constitutive equations of elastic stress dissipativestress and strain energy density release rate have been written in terms of material coordinate description Using these expressionstotal stress has been found

1 Introduction

In various fields of industry fiber reinforced compositematerials are commonly used for load carrying componentsIn particular these materials have become indispensablein advanced aerospace and aviation structures [1] Fiberreinforced composites can display time dependent behavior(stress relaxation or creep) as a manifestation of one or bothconstituents deforming in a viscous manner [2] Therefore itis significant to develop a damagemodel that also includes therate dependent nature of the viscoelasticmaterial subjected toexternal loading Damage modeling of composites is not aneasy task given the evolution of various damage mechanisms(matrix cracking fiber breakage interfacial debondingtransverse ply cracking and ply delamination) in compositematerials [3] The effective stress theory assumes that thedamage is uniformly distributed where in fiber reinforcedcomposites the complex damage occurs in a distributedman-ner with the damage mechanisms occurring in experimentalwork [1] Some researches have conducted studies on the for-mulation of damage modeling on various composites [4ndash10]

Continuum damage mechanics (CDM) developed in thelast few decades provides a framework for incorporatingthe effects of damage induced stiffness softening anisotropyand so forth in constitutive equations [11] Phenomenologicalcontinuum damage mechanics (CDM) models have beenused extensively for modeling creep and fatigue damage byseveral authors as described in Lemaitre and Chaboche [12]Krajicinovic [13] and Voyiadjis and Kattan [14] Chabochehas proposed a thermodynamic framework for continuumdamage mechanics (CDM) for isotropic damage [15] Fur-thermore an updated continuum damage model was pro-posed in which the fracture energy density a function of thestress triaxiality temperature and strain rate in the transitionregion were taken as the critical damage factor [16] Li et alpresented a damage model and its numerical solution bymeans of fast Fourier transforms (FFT) [17] Much progresshas been [18] made in the development of a consistent con-tinuum framework for damage mechanics Darabi et al haveproposed in their paper to enhance the continuum damagemechanics theories in modeling the microdamage healing

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 624750 15 pageshttpdxdoiorg1011552015624750

2 Mathematical Problems in Engineering

120590y

120591

120590x

120591

A1

AK

Ak

A2

Alowast

2

Alowast

k

(a)

A Alowast

(b)

Figure 1 (a) A sketch of a representative volume element containing119873microcracks [21] (b) Open and closed surfaces of a microcrack

phenomenon in materials that tend to self-heal This frame-work extends the well-known Kachanovrsquos [19] effective con-figuration and the concept of the effective stress space to self-healing materials by introducing the healing natural con-figuration in order to incorporate the microdamage healingeffects [18]

Continuum damage mechanics (CDM) has been provento be a valuable tool for the determination of properties dete-rioration due to microcracks of composites In continuumdamage mechanics (CDM) models families of damage arerepresented as continuous internal state variables and con-cepts of irreversible thermodynamics are used in derivingconstitutive equations [20] A number of models of damagebehavior of fiber reinforced composites are based on themethods of continuum damage mechanics (CDM) For thecase of damage in viscoelastic materials two sets of internalvariables are needed Weitsman [21] presented a formulationto treat this case In this formulation damage in the formof microcracks is represented by two symmetric secondrank tensors related to the total area of active and passivemicrocracks in a representative volume element (RVE) of thematerial and viscoelasticity is introduced through 119899 scalarvalued internal state variables (ISVs) Since the scale ofmolecular rearrangement is much smaller than the scale ofmicrocracks the effect of damage on viscoelastic retardationtime is ignoredThe formulation uses Taylor series expansionof Gibbs free energy (Figure 1) [2]

Voyiadjis and Kattan [22] have presented a new formu-lation to link continuum damage mechanics with the con-cept of fabric tensors within the framework of classicalelasticity assuming small strains The new theory was calleddamagemechanicswith fabric tensors A fourth-rankdamagetensor was used and its exact relationship to the fabric ten-sors was illustrated A [23] model of damage mechanics fordirectional data was formulated using fabric tensors [22]Qiang et al have described accurately by a fourth-order fabric

tensor of the cracks within the framework of noninteractionapproximation the compliance increment due to the presenceof generally nonuniformly distributed cracks They haveexpressed that the compliance increment can be approxi-mated by the corresponding second-order fabric tensor [24]Spencer has formulated constitutive equations for flow offabric-reinforced composite materials which show viscousand viscoplastic response at the forming temperature [24 25]

The fabric tensor is a symmetric second rank tensorwhichcharacterizes the geometric arrangement of the porous mate-rial microstructure In developing this result it is assumedthat the matrix material of the porous elastic solid is isotropicand thus that the anisotropy of the porous elastic solid isdetermined by the fabric tensor [26] Fabric tensor has provedto be an effective tool statistically characterizing directionaldata in a smooth and frame-indifferent formDirectional dataarising from microscopic physics and mechanics can besummed up as tensor-valued orientation distribution func-tions [27] The subject of fabric tensor has been studied by anumber of different researchers [28ndash31]There are studies thatestablish a relationship between the fabric tensor with dam-age tensor However such studies are based on experimentaldata In these studies statistical distribution along a certainplane of the microcracks in the material has been consid-ered and an average density of microcracks has been usedOur study is not based on experimental data and is phe-nomenological

In this study in the scope of continuum damage me-chanics (CDM) considering representative volume ele-ment (RVE) the mechanical representation of damage wasexpressed with two interior conditional variables showingthe properties of a second degree symmetrical tensor It isassumed that an element from two different continuous fiberfamilies is placed on each point of the composite materialIn the scope of continuum mechanics balance equationshave been summarized and the combined form was stated

Mathematical Problems in Engineering 3

for the energy equation and entropy inequality Consideringnecessary constitutive axioms after determining the argu-ments affecting the stress potential it was further proceededto the formulation of the constitutive theory and a modelhas been formed for the damaged viscoelastic compositewith isotropic matrix material Considering fully orthogonaltransformations for the material coordinate system afterdetermining the common invariants affecting both the stresspotential and the dissipative stress constitutive equationshave been obtained related with the ldquoelastic stressrdquo theldquodissipative stressrdquo and the ldquostrain energy density release raterdquo

On the other hand inextensibility of the fibers andincompressibility of the composite are acceptable as broadrecognition in practice in terms of formulation Due to sometechnological requirements it is aspired that specific con-struction elements have rather elastic properties providedthat they have high durability in certain directions Fiber-reinforced composite materials are produced sticking fibersin a polymeric matrix which is elastic but with low strengthThese fibers are manufactured from high strength graphiteor bor They can be easily bent due to the very small size oftheir cross section Assume that inextensibility of the fibers isa reasonable approach since the rigidity of the fibers is veryhigh according to the rigidity of the matrix [32]

The constitutive model developed in this study is applica-ble for natural or biological fiber composites such as livingtissues or synthetic composites for implant applicationsbecause the incompressible (nearly incompressible) behaviorof composites is often used for natural or biological fibercomposites such as living tissues or synthetic compositesfor implant applications In our study [33] we are concernedwith developing the continuumdamagemechanicsmodel forelastic behavior of composites having microcracks consistingof an isotropic matrix reinforced by independent and inex-tensible two families of arbitrarily fibers

2 The Representation of Damage

In some researches in order to be able to define the damagevariable a representative volume element (RVE) has beenconsidered that has a 119896 number of microcracks While theopen or active part of any 119896th microcrack has been shown byA(119896) its closed or passive surface has been shown by Alowast(119896)Active or passive surfaces of a crack can switch positionsamong each other depending on stress temperature andhumidity percentage Despite thatWeitsman states that theseopen and closed surfaces can be selected as independentvariables characterizing the state of a material at a certaintime range [21 34]

Stress and strain at the macrolevel are average values overthe representative volume element (RVE) volume Infinites-imal deformations can also be considered among thesemacrovalues To fully consider the behaviors of representativevolume element (RVE) it is necessary to deal with a 119896numberof crack parameters representing A(119896) and Alowast(119896) (no sum on119896 119896 = 1 119873) surfaces Because the real shape of thesesurfaces is unknown on the mesoscale assuming them tobe equivalent plane surfaces Weitsman represented them by

vectors A(119896) = 119860(119896)n(119896) and Alowast(119896) = 119860

lowast(119896)n(119896) Here n(119896)stands for a unit normal vector of a microcrack surface [34]However since each microcrack possesses two equal andopposite surfaces the description should not discriminatebetween (+n119896) and (minusn119896) which is accomplished by choosingthe dyadic product as follows to represent the 119896th flaw Thischoice follows an analogous approach by Spencer [35]

H(119896) = A(119896) otimes A(119896) Hlowast(119896) = Alowast(119896) otimes Alowast(119896) or

119867(119896)

119894119895 = 119860(119896)

119894 119860(119896)

119895 119867lowast(119896)

119894119895 = 119860lowast(119896)

119894 119860lowast(119896)

119895

(1)

In view of the paucity of information regarding the posi-tions and sizes of the individual microflaws it is proposedto represent damage by the dyadic sums of all H(119896) (119896 =

1 2 119873) and Hlowast(119896) (119896 = 1 2 119873) contained in therepresentative volume element (RVE) namely by

H =

119873

sum

119896=1

H(119896) =119873

sum

119896=1

A(119896) otimes A(119896)

Hlowast =119873

sum

119896=1

Hlowast(119896) =119873

sum

119896=1

Alowast(119896) otimes Alowast(119896)

(2)

ObviouslyH andHlowast are symmetric second rank tensorsAs the constitutive variable in this study we are going todeal with only one damage tensor taking into considerationonly the effect of open microsurfaces In this study due tothe existence of fiber distributions and microvoids in thematerial it is assumed that the material has gained directedmedium characteristics that is an anisotropic structure hasappeared due to the damage and the fibers We assume thatinitially the material was isotropic and that the anisotropy isonly caused by the dispersion of microvoids and fibers For amedium like that the role of material description vectors willbe played by the vector A(X 119905) representing the mean valuesin the representative volume element (RVE) and the vectorA(X 119905) representing the change in time of the precedingvectorWe believe that by dividing these vectors by the area ofany characteristic surface pertaining to representative volumeelement (RVE) we render them dimensionless

On the other hand because thematerial will not be able todetect the positive and negative sides of microvoid surfaceswe had previously specified that the dependence on vectorsA(X 119905) and A(X 119905) can be expressed by a product of tensors

H equiv A otimes A 997904rArr H equiv A otimes A + A otimes A or

119867119870119871 equiv 119860119870119860119871 997904rArr 119870119871 equiv 119870119860119871 + 119860119870119871

(3)

3 Kinematics of Fibers Deformation andThermomechanic Balance Equations

Fiber families are represented by continuous vector fieldsB(X) and Z(X) before deformation and by continuous vector


fields b(x) and z(x) after deformation Relationships givenbelow are true for 119861 and 119885-fiber families [35 36]

119887119896 = 120582minus1

119887 119909119896119870119861119870 120582119887 equiv (119889119897

119889119871)

119887

1205822

119887 = 119862119870119871119861119870119861119871

119911119896 = 120582minus1

119911 119909119896119870119885119870 120582119911 equiv (119889119897

119889119871)

119911

1205822

119911 = 119862119870119871119885119870119885119871

(4)

where 119889119871 and 119889119897 are respectively arc length of fiber beforeand after deformation 119861119870 and 119885119870 are fiber vector compo-nents before deformation 119887119896 and 119911119896 are fiber vector com-ponents after deformation 119909119896119870 = 120597119909119896120597119883119870 is deformationgradient 120582119887 and 120582119911 are rates of extension of fiber familiesand 119862119870119871 = 119909119896119870119909119896119871 is Green deformation tensor

The local thermomechanical balance equations can besummarized as follows [32 37]

conservation of mass

120588 + 120588V119896119896 = 0 (5)

conservation of mass in material representation

120588 (x 119905) =1205880 (X)119869 (x 119905)

(6)

balance of linear momentum

120588V119901 = 120588119891119901 + 119905119903119901119903 (7)

balance of moment of momentum

120576119896119903119901119905119903119901 = 0 119905119903119901 = 119905119901119903 (8)

conservation of energy

120588 120576 = 119905119896119897119889119896119897 + 119902119896119896 + 120588ℎ (9)

second law of thermodynamics

120588120579 120578 minus nabla sdot q +1

120579q sdot nabla120579 minus 120588ℎ ge 0 (10)

Here k stands for the velocity field in a continuousmedium 1205880 for mass density before deformation 120588 for massdensity after deformation 119869 equiv det[119909119896119870] = 1205880120588(x 119905) forJacobian k for acceleration 119905119897119896 for stress tensor 119891119896 for themechanical volumetric force per unit of mass 120576 for internalenergy density per unit of mass 119902119896 for heat flux vector ℎ forheat source per unit of mass 120578 for entropy density per unit ofmass 120579(X 119905) for the absolute temperature of a material point119883 at a moment 119905 and 120576119894119895119896 for permutation tensor

4 Thermodynamic Constraints and ModelingConstitutive Equations

Local energy equation (9) is then suitably combined with theentropy inequality (10) and using a Legendre transformationsuch as 120595 equiv 120576 minus 120579120578 for free energy entropy inequality isobtained as follows in the material form

minus (Σ + 1205880120579120578) +

1

2119879119870119871119870119871 +

1

120579120579119870119876119870 ge 0 (11)

Terms relating to the new values have been provided asfollows

Σ equiv 1205880120595

119870119871 = 2119889119896119897119909119896119870119909119897119871 997904rArr 119889119896119897 =1

2119870119871119883119870119896119883119871119897

119879119870119871 equiv 119869119883119870119896119883119871119897119905119896119897 997904rArr 119905119896119897 = 119869minus1119909119896119870119909119897119871119879119870119871

119876119870 equiv 119869119883119870119896119902119896 997904rArr 119902119896 = 119869minus1119909119896119870119876119870

120579119870 = 119909119896119870120579119896 997904rArr 120579119896 = 119883119870119896120579119870

(12)

Here Σ stands for thermodynamic stress potential 120595for generalized free energy density 119889119896119897 for deformation(strain) rate tensor 119883119870119896 = 120597119883119870120597119909119896 for the deformationgradient of the reverse motion 119879119870119871 for the stress tensor onmaterial coordinates 119876119870 for the heat flux vector on materialcoordinates and 120579119870 for the temperature gradient onmaterialcoordinates

For the use of inequality (11) which is a general expressionof entropy generation we need to know on which inde-pendent variables the thermodynamic potential Σ dependsas well as the mechanism of such dependence Based onthe material selected the arguments of Σ and variables itdepends on have been found using constitutive axiomsUsing the results of the axioms of causality determinismobjectivity smooth neighborhood and admissibility axioms[32 37] the arguments for Σ allowing two fiber familiesand viscoelastic structures bearing cracks and where thesecracks are believed to be changing with time exposed tothermomechanical loading without heat transfer can besummarized as follows

Σ = Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579) (13)

120578 = minus1

1205880

120597Σ

120597120579 (14)

120576 =1

1205880

(Σ minus120597Σ

120597120579120579) (15)

119876119870 = 0 (16)

119864119879119870119871 equiv 2120597Σ

120597119862119870119871

(17)


119884119872119873 equiv minus120597Σ

120597119867119872119873

119884119870119871 equiv minus119884119870119871 119884119870119871 equiv120597Σ

120597119867119870119871

(18)

119863119879119872119873 = 119863119879119872119873 (119862119872119873 119872119873 119867119872119873 119872119873 119861119870 119885119870 120579)

(19)

1

2119863119879119872119873119872119873 minus 119884119872119873119872119873 ge 0 (20)

119863119879119870119871 (119862119872119873 0 119867119872119873 0 119861119870 119885119870 120579) = 0 (21)

119879119870119871 equiv 119864119879119870119871 + 119863119879119870119871 (22)

In expression (18) 119884119870119871 is called as the strain energydensity release The definition 119884119870119871 equiv minus119884119870119871 is used in orderto deal with a positive value In expression (22) dissipativestress tensor 119863119879119870119871 was defined with expression (19) 119864119879119870119871 isconsidered as an energetic stress All energetic stresses are notelastic stresses because a part of the energetic stress occurringin the material is spent for the formation of microcracksBut in this study 119864119879119870119871 is called the elastic stress and it wasdefined with expression (17)

It is expressly understood from (17) and (18) that theelastic stress and the strain energy density release rate arederived from the stress potentialΣ From (19) it is understoodthat the dissipative stress is obtained in form of a matrixwhose arguments are known In this case open forms of Σand 119863119879119870119871 which are known as constitutive functions withclear arguments should be found

However firstly we should consider the constraintsimposed by the material symmetry axiom onto the materialunder consideration Because the symmetry group of thematerial under consideration is the fully orthogonal groupproperty [119878119870119871]

minus1= [119878119870119871]

119879 det S = plusmn1 is true for the sym-

metry operation [119878119870119871] Therefore each material point con-versionmatches an orientation of the material medium Suchconversion should for every [119878119870119871] be in the following form

1198831015840

119870 = 119878119870119871119883119871 119883119871 = 119878119879

1198711198701198831015840

119870 [119878119870119871]minus1

= [119878119870119871]119879

(23)

and leave constitutive functionals form invariant Mathemat-ically this means the validity of the following conversions

Σ (119878119870119875119878119871119877119862119875119877 119878119870119875119878119871119877119867119875119877 119878119870119871119861119871 119878119870119871119885119871 120579)

= Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

(24)

119863119879119870119871 (119878119869119875119878119873119877119862119875119877 119878119869119875119878119873119877119875119877 119878119869119875119878119873119877119867119875119877

119878119869119875119878119873119877119875119877 119878119869119875119861119875 119878119869119875119885119875 120579)

= 119878119870119875119878119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(25)

On the other hand both incompressibility of themediumand inextensibility of the fiber are broadly recognized in

terms of formulation Assuming the medium to be incom-pressible and the fiber to be inextensible the followingrequirements should be satisfied [32]

119869 = 1 or detC = 119868119868119868 = 1

1205822

119887 = 119862119870119871119861119870119861119871 = 1 1205822

119911 = 119862119870119871119885119870119885119871 = 1

(26)

Thus the constitutive equation for the elastic stress isobtained as follows in material coordinates

119864119879119870119871 = minus119901119862minus1

119870119871 + Γ119887119861119870119861119871 + Γ119911119885119870119885119871 + 2120597Σ

120597119862119870119871

(27)

In this expression 119901 Γ119887 and Γ119911 are Lagrange coefficientsand are defined by field equations and boundary conditions119862minus1119870119871 equiv 119883119870119897119883119871119897 is the Piola deformation tensor

5 Determination of Elastic Stressand Strain Energy Density Release RateConstitutive Equations

Since the matrix has been assumed to be isotropic relation(24) is expressed as follows

Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

= Σ (119872119870119875119872119871119877119862119875119877119872119870119875119872119871119877119867119875119877119872119870119871119861119871119872119870119871119885119871 120579)

(28)

Here the orthogonal matrix indicating the symmetrygroup 119872119870119871 will be expressed for [119872119870119871] isin 119874(3) and prop-erty [119872119870119871]

minus1= [119872119870119871]

119879997904rArr detM = plusmn1 is true

On the other hand since Σ has been assumed to be theanalytical function of its arguments such arguments whichare expected to remain invariant under orthogonal transfor-mations belonging to the symmetry group should depend ona finite number of invariants

1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 = 119867119870119870

1198685 equiv 119861119870119861119870 1198686 equiv 119885119870119885119870

1198687 equiv 119861119870119885119870 1198688 equiv 119861119870119862119870119871119861119871 = 1205822

119887

1198689 equiv 119861119870119862119870119871119862119871119872119861119872 11986810 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986811 equiv 119885119870119862119870119871119862119871119872119885119872 11986812 equiv 119861119870119862119870119871119885119871

11986813 equiv 119861119870119862119870119871119862119871119872119885119872 11986814 equiv 119861119870119867119870119871119861119871

11986815 equiv 119861119870119862119870119871119867119871119872119861119872 11986816 equiv 119885119870119867119870119871119885119871

11986817 equiv 119885119870119862119870119871119867119871119872119885119872 11986818 equiv 119861119870119867119870119871119885119871

11986819 equiv 119861119870119862119870119871119867119871119872119885119872 11986820 equiv 119862119870119871119867119871119870

11986821 equiv 119862119870119871119862119871119872119867119872119870

(29)


Instead of the first three invariants of the Green deforma-tion tensor C we can use the principal invariants as follows

119868 = 1198681 119868119868 =1

2(1198682

1 minus 1198682)

119868119868119868 =1

6(1198683

1 minus 311986811198682 + 21198683) = detC(30)

Given the incompressibility of the composite inextensi-bility of the fiber families and the fact that B and Z are unitvectors the invariants 119868119868119868 1198688 11986810 1198685 and 1198686 in expressions (29)and (30) are equal to 1 thus eliminating the dependence ofΣ on these invariants As a result the invariants on which Σ

depends are expressed as follows

Σ = Σ (119868 119868119868 1198684 1198687 1198689 119868119898 120579) (119898 = 11 21) (31)

Taking the derivative of expression (31) according to 119862119875119877

and 119867119875119877 and substituting it into (27) and (18) the followingexpressions are obtained

119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868

120597119868

120597119862119875119877

+120597Σ

120597119868119868

120597119868119868

120597119862119875119877

+120597Σ

120597119868119894

120597119868119894

120597119862119875119877

)

(119894 = 9 11 12 13 15 17 19 20 21)

119884119875119877 =120597Σ

120597119868119898

120597119868119898

120597119867119875119877

(119898 = 4 14 15 21)

(32)

It is understood that as always repeated indices willundergo summation If derivatives of invariants appearing inthese equations according to 119862119875119877 and 119867119875119877 are taken fromexpressions (29) and (30) and substituted afterwards con-stitutive equation of the elastic stress in nonlinear form isobtained as follows

119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868+

120597Σ

120597119868119868119862119871119871)120575119875119877 minus

120597Σ

120597119868119868119862119875119877

+120597Σ

1205971198689

(119861119875119861119871119862119871119877 + 119862119875119871119861119871119861119877)

+120597Σ

12059711986811

(119885119875119885119871119862119871119877 + 119862119875119871119885119871119885119877)

+120597Σ

12059711986812

119861119875119885119877

+120597Σ

12059711986813

(119861119875119885119871119862119871119877 + 119862119875119871119861119871119885119877)

+120597Σ

12059711986815

(119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+120597Σ

12059711986817

(119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+120597Σ

12059711986819

(119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+120597Σ

12059711986820

119867119875119877 +120597Σ

12059711986821

(119862119875119871119867119871119877 + 119867119875119871119862119871119877)

(33)

And the strain energy density release rate in nonlinear formis obtained as follows

119884119875119877 =120597Σ

1205971198684

120575119875119877 +120597Σ

12059711986814

119861119875119861119877 +120597Σ

12059711986815

119862119875119870119861119870119861119877

+120597Σ

12059711986816

119885119875119885119877 +120597Σ

12059711986817

119862119875119871119885119871119885119877 +120597Σ

12059711986818

119861119875119885119877

+120597Σ

12059711986819

119862119875119871119861119871119885119877 +120597Σ

12059711986820

119862119875119877 +120597Σ

12059711986821

119862119875119871119862119871119877

(34)

More concrete form of the constitutive equations givenby (33) and (34) can be obtained provided that Lagrangecoefficients minus119901 Γ119887 and Γ119911 and the derivatives of Σ basedon its invariants are known It has been already stated thatminus119901 Γ119887 and Γ119911 can be obtained from field equations andboundary conditions To obtain the derivatives ofΣ accordingto its invariants how Σ depends on the invariants it is shownto depend on in expression (31) should be estimated Inthis study the matrix material has been considered as anisotropicmediumAccording to the fact thatΣ is an analyticalfunction of those invariants assuming that this function isanalytic the stress potential is expanded in the power seriesaround natural condition To obtain a quadratic theory theterms in this series expanding should be kept to secondorder therefore the stress potential can be represented by apolynomial [32 37] However the grade and number of termsof the polynomial representing Σ depend on the size of itsdeformation invariant and their shares of interaction in thecase [38ndash40]

In this study mechanical interactions and effect of dam-age have been assumed to be linear Furthermore consideringthat the material remains insensitive to directional changesalong fibers double components of fiber vectors have beenincluded in the operation Because mechanical interactionsand effect of damage are assumed to be linear the stressshould remain linear according to the deformation tensor andthe damage tensor Therefore function Σ could be repre-sented by a second degree polynomial according to theinvariants it depends on On the other hand because internalenergy is defined as a positive definite form for a polynomialto be positively defined and for the order of invariants notto affect Σ the polynomial must have symmetric coefficientsthat is it must be in a quadratic form Accordingly if polyno-mial approximation is selected the following expression can


be recorded for the stress potential Σ in terms of the existinginvariants

Σ = Σ119894119895119886119894119895119868119894119868119895 (119894 119895 = 1 2 4 7 9 11 21) 119886119894119895 = 119886119895119894

(35)

The derivatives of Σ based on its invariants in (33) and(34) are obtained from expression (35) as follows

120597Σ

120597119868= 2 (11988611119868 + 11988612119868119868 + 1198861119896119868119896)

120597Σ

120597119868119868= 2 (11988612119868 + 11988622119868119868 + 1198862119896119868119896)

120597Σ

120597119868119898

= 2 (1198861198981119868 + 1198861198982119868119868 + 119886119898119896119868119896)

(119898 = 4 9 11 21) (119896 = 4 7 9 11 21)

(36)

At this stage expressions (29) and (30) have shown onwhat the invariants in expression (36) depend Due to theexistence of the relationship 119862119870119871 = 120575119870119871 + 2119864119870119871 betweenthe Green deformation tensor and the strain tensor andassuming that mechanic interactions are linear (119864119870119871 cong 119864119870119871 =

(12)(119880119870119871+119880119871119870)) those invariants that depend on theGreendeformation tensor (119862119870119871) have been expressed in terms ofstrain tensor (119864119870119871) which is a more useful parameter

Terms after the third term on the right side of (33) and allterms of the right side of (34) have been calculated using thepartial derivatives given in expression (36) and invariants thatdepend on the strain tensor (119864119870119871) Due to the assumptionsmade in this study of the first grade components of the straintensor 119864119870119871 and the damage tensor 119867119870119871 and of the externalmultiplication components of vectors 119861119870 and 119885119870 thosewhose number is even have been taken into considerationThus in the beginning the elastic stress is expressed for thecondition without stress and without load (with the term1205721120575119875119877 assumed to be zero) by taking common coefficientsinto common parenthesis

119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877) + 12057222119867119875119877

(37)

The strain energy density release rate is expressed asfollows by taking common coefficients into parenthesis in thebeginning withoutmicrocracks (the term 1205741120575119875119877 is taken hereas zero) being obtained as follows

119884119875119877 = 1205742119864119870119870120575119875119877 + 1205743119867119870119870120575119875119877 + 1205744119861119870119867119870119873119861119873120575119875119877

+ 1205745119885119870119867119870119873119885119873120575119875119877 + 1205746119861119875119861119877 + 1205747119864119870119870119861119875119861119877

+ 1205748119867119870119870119861119875119861119877 + 1205749119885119870119867119870119873119885119873119861119875119861119877 + 12057410119864119875119873119861119873119861119877

+ 12057411119885119875119885119877 + 12057412119864119870119870119885119875119885119877 + 12057413119867119870119870119885119875119885119877

+ 12057414119861119870119867119870119873119861119873119885119875119885119877 + 12057415119864119875119873119885119873119885119877

+ 12057416119861119870119885119870119861119875119885119877 + 12057417119861119870119864119870119873119885119873119861119875119885119877

+ 12057418119861119870119867119870119873119885119873119861119875119885119877 + 12057419119861119870119885119870119864119875119873119861119873119885119877 + 12057420119864119875119877

(38)

Coefficients [120572119896 (119896 = 1 2 3 22) and 120574119896 (119896 = 1 2

3 20)] in (37) and (38) have been depending on themedium temperature 120579 and 119886119894119895

In a composite material that consists of an isotropicmatrix reinforced by two arbitrary independent and inexten-sible fiber families the medium is assumed to be incompress-ible and homogeneous has microcracks and shows linearviscoelastic behavior Equation (37) is the linear constitutiveequation of elastic stress First second and third terms of (37)are hydrostatic pressure and contributions of fiber tensionsto the elastic stress respectively fourth and eighth termscombined are the contribution of the elastic deformationfifth and twenty sixth terms combined are the contributionof the damage tensor sixth tenth and twenty-third termsare the stress arising of the interaction between the fiberdistribution B and the damage tensor seventh fifteenth andtwenty-fourth terms are the stress arising of the interactionbetween the fiber distributionZ and the damage tensor ninthand thirteenth terms are the stress arising of the interactionbetween the fiber distribution B and the elastic deformationeleventh sixteenth twenty-first and twenty-fifth terms arethe contribution produced by the triple interaction betweenthe fiber fields B and Z and the damage tensor twelfthterm is the contribution of fiber distribution B fourteenthand eighteenth terms are the stress arising of the interactionbetween the fiber distribution Z and the elastic deformationseventeenth term is the contribution of fiber distribution Znineteenth term is the stress produced by the interactionbetween the fiber field B and the fiber field Z and twentiethand twenty second terms are the contribution produced bythe triple interaction between the fiber fields B and Z and theelastic deformation field


Equation (38) is the linear constitutive equation of strainenergy density release rate First and nineteenth terms com-bined are the contribution of the elastic deformation secondterm is the contribution of the damage tensor third andseventh terms combined are the strain energy density releasearising of the interaction between the fiber distributionB andthe damage tensor fourth and twelfth terms are the strainenergy density release arising of the interaction betweenthe fiber distribution Z and the damage tensor fifth termis the contribution of fiber distribution B sixth and ninthterms are the strain energy density release arising of theinteraction between the fiber distribution B and the elasticdeformation eighth thirteenth and seventeenth terms arethe contribution produced by the triple interaction betweenthe fiber fields B and Z and the damage tensor tenth termis the contribution of fiber distribution Z eleventh andfourteenth terms are the strain energy density release arisingof the interaction between the fiber distribution Z and theelastic deformation fifteenth term is the strain energy densityrelease produced by the interaction between the fiber field Band the fiber field Z and sixteenth and eighteenth terms arethe contribution produced by the triple interaction betweenthe fiber fields B and Z and the elastic deformation field

6 Determination of Dissipative StressConstitutive Equation

It is assumed that the viscoelastic behavior of the mediumin consideration is in conformity with Kelvin-Voigt modelIt has been determined that the dissipative stress dependson deformation deformation rate damage damage rateand distributions of fibers yielding in expressions (19)-(21)Additional constraints imposed on the dissipative stress byconstitutive functions originate from the material symmetryof the medium The structure of the dissipative stress shouldbe in compliance with the following transformation for eachorthogonal matrix [119872119870119871] isin 119874(3) belonging to the symmetrygroup of the material

119863119879119870119871 (119872119869119875119872119873119877119862119875119877119872119869119875119872119873119877119875119877119872119869119875119872119873119877119867119875119877

119872119869119875119872119873119877119875119877119872119869119875119861119875119872119869119875119885119875 120579)

= 119872119870119875119872119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(39)

where the matrix is isotropic relation (39) is valid for eachorthogonal matrix of the fully orthogonal groupThe dissipa-tive stress is an isotropic function of the symmetric matrices119862119870119871 119870119871 119867119870119871 119870119871 and polar vectors 119861119870 and 119885119870 Forsimplicity of notation dependence of the dissipative stresstensor on 120579 has not been denoted To obtain the explicitexpression of the tensor component 119863119879119870119871 in terms of itsinvariant arguments the following way has been followedaccording to the theory of invariants [41] 119881119870 is an arbitraryvector 119863119879119870119871 and the vector 119881119870 are multiplied on the rightand on the left by scalar multiplication and the product isdefined by a scalar functionR

Consider

R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

equiv 119881119870119881119871 119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

(40)

Here the R scalar is an isotropic function of the sym-metric matrices 119862119870119871 119870119871 119867119870119871 119870119871 and absolute vectors119861119870 119885119870 and 119881119870 Taking the partial derivative of expression(40) according to 119881119870 and 119881119871 the following expression can berecorded [32]

119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

(41)

Because the left side of this expression is independentfrom the vector V V=0 should be true for inequality (41) Inthis situation isotropic tensor function 119863119879119870119871 is expressed asfollows

119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816119881119870=0

(42)

In this situation to find the tensor 119863119879119870119871 from relation-ship (42) structure of the scalarR should be determined thatdepends on the arguments 119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870

and second degree partial derivative of this function accord-ing to the vector V should be calculated at V = 0 Letus first remove the arbitrary vector V from the argumentsof the scalar function R and define a scalar function witharguments 119862119870119871 119870119871119867119870119871 119870119871 119861119870 119885119870

Ξ equiv Ξ (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870) (43)

ForΞ which is an isotropic function to keep the invariantunder orthogonal coordinate transformations its argumentsmust depend on a finite number of invariants Using themethods in the theory of invariants [41] 98 invariants of thefour symmetric tensors119862119870119871 119870119871119867119870119871 119870119871 and the twopolarvectors 119861119870 119885119870 which are independent of each other havebeen expressed in the following list

1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 equiv 119870119870

1198685 equiv 119870119871119871119870 1198686 equiv 119870119871119871119872119872119870

1198687 equiv 119867119870119870 1198688 equiv 119870119870


1198689 equiv 119862119870119871119871119870 11986810 equiv 119862119870119871119867119871119870

11986811 equiv 119862119870119871119871119870 11986812 equiv 119870119871119867119871119870

11986813 equiv 119870119871119871119870 11986814 equiv 119867119870119871119871119870

11986815 equiv 119862119870119871119862119871119872119872119870 11986816 equiv 119862119870119871119862119871119872119867119872119870

11986817 equiv 119862119870119871119862119871119872119872119870 11986818 equiv 119862119870119871119871119872119872119870

11986819 equiv 119870119871119871119872119867119872119870 11986820 equiv 119870119871119871119872119872119870

11986821 equiv 119862119870119871119871119872119867119872119870 11986822 equiv 119862119870119871119871119872119872119870

11986823 equiv 119862119870119871119867119871119872119872119870 11986824 equiv 119862119870119871119871119872119872119870

11986825 equiv 119862119870119871119862119871119872119872119873119867119873119870 11986826 equiv 119862119870119871119862119871119872119872119873119873119870

11986827 equiv 119862119870119871119862119871119872119867119872119873119873119870 11986828 equiv 119870119871119871119872119862119872119873119867119873119870

11986829 equiv 119870119871119871119872119862119872119873119873119870 11986830 equiv 119870119871119871119872119867119872119873119873119870

11986831 equiv 119862119870119871119862119871119872119872119873119873119870 11986832 equiv 119862119870119871119871119872119867119872119873119873119870

11986833 equiv 119861119870119861119870 11986834 equiv 119885119870119885119870

11986835 equiv 119861119870119885119870 11986836 equiv 119861119870119862119870119871119861119871 = 1205822

119887

11986837 equiv 119861119870119862119870119871119862119871119872119861119872 11986838 equiv 119861119870119870119871119861119871

11986839 equiv 119861119870119870119871119871119872119861119872 11986840 equiv 119861119870119867119870119871119861119871

11986841 equiv 119861119870119870119871119861119871 11986842 equiv 119861119870119862119870119871119871119872119861119872

11986843 equiv 119861119870119862119870119871119867119871119872119861119872 11986844 equiv 119861119870119862119870119871119871119872119861119872

11986845 equiv 119861119870119870119871119867119871119872119861119872 11986846 equiv 119861119870119870119871119871119872119861119872

11986847 equiv 119861119870119867119870119871119871119872119861119872

11986848 equiv 119861119870119862119870119871119871119872119867119872119873119861119873

11986849 equiv 119861119870119862119870119871119871119872119872119873119861119873

11986850 equiv 119861119870119862119870119871119867119871119872119872119873119861119873

11986851 equiv 119861119870119870119871119867119871119872119872119873119861119873

11986852 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11986853 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986854 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119861119878

11986855 equiv 119861119870119870119871119871119872119867119872119873119873119878119861119878

11986856 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119861119878

11986857 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986858 equiv 119885119870119862119870119871119862119871119872119885119872

11986859 equiv 119885119870119870119871119885119871 11986860 equiv 119885119870119870119871119871119872119885119872

11986861 equiv 119885119870119867119870119871119885119871 11986862 equiv 119885119870119870119871119885119871

11986863 equiv 119885119870119862119870119871119871119872119885119872 11986864 equiv 119885119870119862119870119871119867119871119872119885119872

11986865 equiv 119885119870119862119870119871119871119872119885119872 11986866 equiv 119885119870119870119871119867119871119872119885119872

11986867 equiv 119885119870119870119871119871119872119885119872 11986868 equiv 119885119870119867119870119871119871119872119885119872

11986869 equiv 119885119870119862119870119871119871119872119867119872119873119885119873

11986870 equiv 119885119870119862119870119871119871119872119872119873119885119873

11986871 equiv 119885119870119862119870119871119867119871119872119872119873119885119873

11986872 equiv 119885119870119870119871119867119871119872119872119873119885119873

11986873 equiv 119885119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986874 equiv 119885119870119862119870119871119862119871119872119872119873119873119878119885119878

11986875 equiv 119885119870119870119871119871119872119867119872119873119862119873119878119885119878

11986876 equiv 119885119870119870119871119871119872119867119872119873119873119878119885119878

11986877 equiv 119885119870119862119870119871119871119872119867119872119873119873119878119885119878

11986878 equiv 119861119870119862119870119871119885119871

11986879 equiv 119861119870119862119870119871119862119871119872119885119872 11986880 equiv 119861119870119870119871119885119871

11986881 equiv 119861119870119870119871119871119872119885119872 11986882 equiv 119861119870119867119870119871119885119871

11986883 equiv 119861119870119870119871119885119871 11986884 equiv 119861119870119862119870119871119871119872119885119872

11986885 equiv 119861119870119862119870119871119867119871119872119885119872 11986886 equiv 119861119870119862119870119871119871119872119885119872

11986887 equiv 119861119870119870119871119867119871119872119885119872 11986888 equiv 119861119870119870119871119871119872119885119872

11986889 equiv 119861119870119867119870119871119871119872119885119872

11986890 equiv 119861119870119862119870119871119871119872119867119872119873119885119873

11986891 equiv 119861119870119862119870119871119871119872119872119873119885119873

11986892 equiv 119861119870119862119870119871119867119871119872119872119873119885119873

11986893 equiv 119861119870119870119871119867119871119872119872119873119885119873

11986894 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986895 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986896 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119885119878

11986897 equiv 119861119870119870119871119871119872119867119872119873119873119878119885119878

11986898 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119885119878

(44)

However the main function that needs to be obtained isin the form of arguments of the scalar isotropic function Rthat is 119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870 Because the scalar


R is a bilinear function of the vector V it depends on thefollowing invariants in addition to those provided in (44)

1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)

In this situation the scalar functionR defined by expres-sion (40) must be bilinear in terms of invariants 119870119894 (119894 = 1 2

42) in expression (45) and linear in terms of invariants119870119894 (119894 = 43 44 64) In this case the following expressioncan be recorded for functionR

R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)

Coefficients 1205820 1205821 120582119898 119898 = 2 3 4 20 and120582120572120573 (120572 120573 = 1 2 3 42) in (46) are each a scalar function ofthe invariants provided in (44) Besides symmetry condition120582120572120573 = 120582120573120572 is true for the coefficients 120582120572120573 Using relationship(42) due to the assumptions made in this study concerninginteractions terms of tensors C C H and H have been onlyconsidered on the first grade of the external product of fiber


vectors B and Z only even number components have beenconsidered yielding the dissipative stress as follows

119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)

Because mechanical interactions and effect of damagehave been assumed to be linear coefficients in (47) are eacha scalar function of invariants that do not contain square orhigher grade terms of tensors C and C or terms in the forms(CC) (CH) (CH) (CH) (CH) and (HH) in (44) Besidestaking into account that values of the invariants 11986836 and 11986857 areequal to 1 due to the inextensibility of fiber families and valuesof the invariants 11986833 and 11986834 are equal to 1 because they are unitvectors pertaining to the distribution of fibers B and Z beforedeformation invariants onwhich the said coefficients dependhave been recorded as follows in terms of (119864119870119871) and (119870119871)

1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)

Coefficients in (47) can be recorded as a scalar functionof the invariants in (48) can be expressed as follows

120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)

Here the following definitions have been used(120572 = 0 rArr 120573 = 119886 120572 = 1 rArr 120573 = 119887 120572 = 2 rArr 120573 =

119888 120572 = 3 rArr 120573 = 119889 120572 = 4 rArr 120573 = 119891 120572 = 11 rArr 120573 = 119892120572 = 12 rArr 120573 = ℎ 120572 = 13 rArr 120573 = 119894 120572 = 14 rArr 120573 = 119896 120572 =

15 rArr 120573 = 119897 120572 = 16 rArr 120573 = 119898 120572 = 17 rArr 120573 = 119899 120572 = 18 rArr

120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =

26 rArr 120573 = 119906 120572 = 27 rArr 120573 = 119908 120572 = 28 rArr 120573 = V 120572 = 29 rArr

120573 = 119910 120572 = 2 10 rArr 120573 = 119911)Using expressions (49) in (47) expressing the tensors

119862119870119871 and 119870119871 in terms of the tensors 119864119870119871 and 119870119871 takingmechanical interactions effect of damage interactions andassumptions made concerning the fibers into account thefollowing has been expressed

119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)

Coefficients Γ119896 (119896 = 1 2 3 52) in (50) have beendepending on the medium temperature 120579 and 120582120572120573

Because the tensors 119862119870119871 and 119870119871 can be expressed interms of the tensors 119864119870119871 and 119870119871 expression (21) imposesa constraint on the coefficients in (50) Accordingly the fol-lowing expression can be recorded

0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)

Because according to expression (51) the above-men-tioned are arbitrary the necessary and sufficient condition forvalidity of this equation is Γ119910 = 0 coefficients being zero (119910 =

1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37

38 40 43 45 46 49 50) In this situation expression provid-ing the dissipative stress is obtained as follows

119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)

Equations of the elastic stress provided by expression (37)and of the dissipative stress provided by expression (52) are


substituted into (22) thus the total stress has been obtainedas follows

119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)

Kelvin-Voigt body composing a viscoelastic model isdefined as a continuum model where stress depends on therate of deformation together with the deformationThe stressconstitutive equation of the linear Kelvin-Voigt body hasbeen expressed as 119905119896119897 = Σ119896119897119898119899(120579 x)isin119898119899 + Γ119896119897119898119899(120579 x)isin119898119899Σ119896119897119898119899 coefficient bears the symmetry characteristics given asΣ119896119897119898119899 = Σ119897119896119898119899 = Σ119896119897119899119898 = Σ119898119899119896119897 whereas Γ119896119897119898119899 coefficientbears the symmetry characteristics given as Γ119896119897119898119899 = Γ119897119896119898119899 =

Γ119896119897119899119898 This symmetry characteristic decreases the number ofcomponents of material module Σ119896119897119898119899 to 21 and the numberof components of material module Γ119896119897119898119899 to 36 Besides as aresult of Onsager principle in the linear reversible thermody-namics it has been assumed that relaxation tensor Γ119896119897119898119899 hasa symmetry characteristic as Γ119896119897119898119899 = Γ119898119899119896119897 According to thissymmetry condition the number of components pertainingto material module Γ119896119897119898119899 is thus reduced to 21

7 Concluding Remarks

This paper presents a continuum damage model based onfundamental concepts of continuum mechanics for the lin-ear viscoelastic behavior of incompressible composites withmicrocracks that consist of an isotropic matrix reinforced byinextensible two families of fibers having an arbitrary dis-tribution ldquoDamagerdquo is expressed by two symmetric secondrank tensors which are related to the total areas of ldquoactiverdquoand ldquopassiverdquo microcracks within a representative volumeelement of the multifractured material The matrix materialhas been assumed to be an isotropicmedium however due tothe distributions of fibers and the existence of microcracks ithas gained the property of a directed object thus gaining theappearance of an anisotropic structure Inextensibility of thefibers is an assumptionwhich is widely used in practiceThusfiber families are assumed to be inextensible and compositemedium is assumed to be incompressible In this contextthe composite expresses itself behaviorally in terms of theelastic stress the dissipative stress and the strain energydensity release rate To obtain a more concrete expressionof nonlinear constitutive equations of the elastic stress andthe strain energy density release rate given by expressions(33) and (34) derivatives of Σ must be known according tothe arguments it depends on Thus stress potential Σ hasbeen represented by a second degree polynomial and itsderivatives according to its invariants have been calculated Inthis study mechanical interactions and effect of damage havebeen assumed to be linear Furthermore since the matrixmaterial has to remain insensitive to directional changesalong fibers even-numbered exterior products of vector fieldsrepresenting fiber distributions have been considered Thelinear constitutive equations of the elastic stress and the strainenergy density release rate have been given by expressions(37) and (38) From these equations it can be seen that thedeformation field the damage tensor distribution of fibers


and interactions of them all contribute to the creation ofelastic stress and strain energy density release rate

Equation (47) obtained by using the invariants of thearguments depends on the dissipative stress since the dis-sipative stress is a vector-valued isotropic function Coeffi-cients in this equation have been expressed in terms of theinvariants on which they depend and each term has beencalculatedWhen these calculations aremade themechanicalinteraction and the effect of damage have been assumed to belinear and even number vector components of fiber vectorshave been included in the operations since the compositeremains indifferent to changes of direction along the fibersThe linear constitutive equation of the dissipative stress hasbeen expressed by (52) As understood from this equationseparate and common interactions of the fiber distributionfields with the rate of deformation and the change in time ofthe damage tensor contribute to the formation of dissipativestress Substituting (37) and (52) into the expression (22) thetotal stress has been obtained in (53)

This paper is concerned with developing the continuumdamage mechanics model for viscoelastic behavior of com-posites having microcracks consisting of an isotropic matrixreinforced by independent and inextensible two families ofarbitrarily fibers The elastic stress and strain energy densityrelease rate are expressed in terms of the thermodynamicstress potential a function of the left Cauchy-Green tensorthe damage tensor and the fiber distributionsThe dissipativestress depends on these quantities and the rate of deformationand the change in time of the damage tensorThe elastic stressand strain energy density release rate and dissipative stress aretreated in separate sections The appropriate invariants usedas arguments for thermodynamic stress potential functionand dissipative stress function are introduced and the elasticstress strain energy density release rate and dissipative stressconstitutive equations are worked out This is followed byspecializations for incompressibility the special case whenthermodynamic stress potential is a quadratic polynomial inthe invariants and a linearization based on small strains

The resulting expressions show that the first is attributableto the ldquoinherentrdquo viscoelastic behavior of the undamagedmaterial while the second is due to the change in time of thedamage tensor As can be noted from constitutive equationsfor elastic stress strain energy density release rate and dis-sipative stress the coupling of damage and viscoelasticityintroduces very significant complexities in the compositematerialrsquos response and substantial work remains to be doneboth experimentally and analytically to attain a quantitativeand practical understanding of the phenomenon

In this paper as a result the constitutive equations relatingto the elastic stress strain energy density release rate anddissipative stress have been written in terms of materialcoordinate descriptions This paper develops a mathematicalmodel based on methods of continuum damage mechanicsAfter this paper practical problems will be solved by formingB(X) and Z(X) vector fields for various fiber distributionswhose parametric equations in the material medium are inthe form of X=X(119904) and necessary interpretations will bemade in a more concrete way Also in a future work we willstudy the development of numerical methods for this model

Conflict of Interests

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

References

[1] E-H Kim M-S Rim I Lee and T-K Hwang ldquoCompositedamage model based on continuum damage mechanics andlow velocity impact analysis of composite platesrdquo CompositeStructures vol 95 pp 123ndash134 2013

[2] R S Kumar and R Talreja ldquoA continuum damage model forlinear viscoelastic composite materialsrdquoMechanics of Materialsvol 35 no 3ndash6 pp 463ndash480 2003

[3] R Talreja Fatigue of Composite Materials Technomic Publish-ing Co Lancaster Pa USA 1987

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] G Z Voyiadjis and B Deliktas ldquoA coupled anisotropic damagemodel for the inelastic response of composite materialsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 183no 3-4 pp 159ndash199 2000

[6] G F Abdelal A Caceres and E J Barbero ldquoAmicro-mechanicsdamage approach for fatigue of compositematerialsrdquoCompositeStructures vol 56 no 4 pp 413ndash422 2002

[7] L Iannucci and J Ankersen ldquoAn energy based damage modelfor thin laminated compositesrdquoComposites Science andTechnol-ogy vol 66 no 7-8 pp 934ndash951 2006

[8] T E Tay G Liu A Yudhanto and V B C Tan ldquoA micro-macro approach to modeling progressive damage in compositestructuresrdquo International Journal of Damage Mechanics vol 17no 1 pp 5ndash28 2008

[9] P Maimı P P Camanho J A Mayugo and C G Davila ldquoAcontinuum damage model for composite laminates part Imdashconstitutive modelrdquo Mechanics of Materials vol 39 no 10 pp897ndash908 2007

[10] F Li and Z Li ldquoContinuum damagemechanics basedmodelingof fiber reinforced concrete in tensionrdquo International Journal ofSolids and Structures vol 38 no 5 pp 777ndash793 2001

[11] P Raghavan and S Ghosh ldquoA continuum damage mechanicsmodel for unidirectional composites undergoing interfacialdebondingrdquoMechanics of Materials vol 37 no 9 pp 955ndash9792005

[12] J Lemaitre and J L ChabocheMechanics of Solids CambridgeUniversity Press Cambridge UK 1990

[13] D Krajicinovic Damage Mechanics Elsevier Amsterdam TheNetherlands 1996

[14] G Z Voyiadjis and P I KattanAdvances in DamageMechanicsMetals and Metal Matrix Composites Elsevier New York NYUSA 1999

[15] J-L Chaboche ldquoContinuous damage mechanicsmdasha tool todescribe phenomena before crack initiationrdquo Nuclear Engineer-ing and Design vol 64 no 2 pp 233ndash247 1981

[16] Z X Wang J Lu H J Shi D F Li and X Ma ldquoAn updatedcontinuum damage model to investigate fracture process ofstructures in DBTT regionrdquo International Journal of Fracturevol 151 no 2 pp 199ndash215 2008

[17] J Li X-X Tian and R Abdelmoula ldquoA damagemodel for crackprediction in brittle and quasi-brittle materials solved by the


FFT methodrdquo International Journal of Fracture vol 173 no 2pp 135ndash146 2012

[18] M K Darabi R K Abu Al-Rub and D N Little ldquoA continuumdamage mechanics framework for modeling micro-damagehealingrdquo International Journal of Solids and Structures vol 49no 3-4 pp 492ndash513 2012

[19] L M Kachanov ldquoTime of the rupture process under creepconditionsrdquo IzvestiyaAkademii Nauk SSSR Otdelenie Tekhnich-eskikh Nauk vol 8 pp 26ndash31 1958

[20] J Isometsa and S-G Sjolind ldquoA continuum damage mechanicsmodel for fiber reinforced compositesrdquo International Journal ofDamage Mechanics vol 8 no 1 pp 2ndash17 1999

[21] YWeitsman ldquoContinuum damagemodel for viscoelastic mate-rialsrdquo Journal of Applied Mechanics vol 55 no 4 pp 773ndash7801988

[22] G Z Voyiadjis and P I Kattan ldquoDamage mechanics with fabrictensorsrdquo Mechanics of Advanced Materials and Structures vol13 no 4 pp 285ndash301 2006

[23] Y Qiang L Zhongkui and L G Tham ldquoAn explicit expressionof second-order fabric-tensor dependent elastic compliancetensorrdquoMechanics Research Communications vol 28 no 3 pp255ndash260 2001

[24] A J M Spencer ldquoTheory of fabric-reinforced viscous fluidsrdquoComposites Part A Applied Science and Manufacturing vol 31no 12 pp 1311ndash1321 2000

[25] A J Spencer ldquoA theory of viscoplasticity for fabric-reinforcedcompositesrdquo Journal of the Mechanics and Physics of Solids vol49 no 11 pp 2667ndash2687 2001

[26] S C Cowin ldquoThe relationship between the elasticity tensor andthe fabric tensorrdquoMechanics of Materials vol 4 no 2 pp 137ndash147 1985

[27] K-D Leng and Q Yang ldquoFabric tensor characterization oftensor-valued directional data solution accuracy and sym-metrizationrdquo Journal of Applied Mathematics vol 2012 ArticleID 516060 22 pages 2012

[28] S Jemioło and J J Telega ldquoFabric tensor and constitutive equa-tions for a class of plastic and locking orthotropic materialsrdquoArchives of Mechanics vol 49 no 6 pp 1041ndash1067 1997

[29] B Elmabrouk and J R Berger ldquoBoundary element analysis foreffective stiffness tensors effect of fabric tensor determinationmethodrdquo Computational Mechanics vol 51 no 4 pp 391ndash3982013

[30] S C Cowin ldquoAnisotropic poroelasticity fabric tensor formula-tionrdquoMechanics of Materials vol 36 no 8 pp 665ndash677 2004

[31] K I Kanatani ldquoDistribution of directional data and fabrictensorsrdquo International Journal of Engineering Science vol 22 no2 pp 149ndash164 1984

[32] E S Suhubi Continuum MechanicsmdashIntroduction ITU TheFaculty of Arts and Sciences Istanbul Turkey 1994

[33] M Usal M R Usal and A H Ercelik ldquoA constitutive model forarbitrary fiber-reinforced composite materials having micro-cracks based on continuum damage mechanicsrdquo Journal ofApplied Engineering vol 2 no 5 pp 63ndash81 2014

[34] Y Weitsman ldquoDamage coupled with heat conduction in uni-axially reinforced compositesrdquo Transactions ASME Journal ofApplied Mechanics vol 55 no 3 pp 641ndash647 1988

[35] A J M Spencer Deformations of Fibre-Reinforced MaterialsClarendon Press Oxford UK 1972

[36] A J M Spencer Continuum Theory of the Mechanics ofFibre-Reinforced Composites Springer International Centre for

Mechanical Sciences Course and LecturesNewYorkNYUSA1984

[37] A C Eringen Mechanics of Continua Robert E KriegerPublishing Hungtington NY USA 1980

[38] A C Eringen Continuum PhysicsmdashMathematics vol 1 Aca-demic Press New York NY USA 1971

[39] Q-S Zheng and A J Spencer ldquoTensors which characterizeanisotropiesrdquo International Journal of Engineering Science vol31 no 5 pp 679ndash693 1993

[40] Q-S Zheng ldquoOn transversely isotropic orthotropic and rela-tive isotropic functions of symmetric tensors skew-symmetrictensors and vectors Part V the irreducibility of the represen-tations for three dimensional orthotropic functions and thesummaryrdquo International Journal of Engineering Science vol 31no 10 pp 1445ndash1453 1993

[41] A J M Spencer ldquoTheory of invariantsrdquo in Continuum PhysicsA C Eringen Ed vol 1 pp 239ndash353 Academic Press NewYork NY USA 1971

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


120590y

120591

120590x

120591

A1

AK

Ak

A2

Alowast

2

Alowast

k

(a)

A Alowast

(b)

Figure 1 (a) A sketch of a representative volume element containing119873microcracks [21] (b) Open and closed surfaces of a microcrack

phenomenon in materials that tend to self-heal This frame-work extends the well-known Kachanovrsquos [19] effective con-figuration and the concept of the effective stress space to self-healing materials by introducing the healing natural con-figuration in order to incorporate the microdamage healingeffects [18]

Continuum damage mechanics (CDM) has been provento be a valuable tool for the determination of properties dete-rioration due to microcracks of composites In continuumdamage mechanics (CDM) models families of damage arerepresented as continuous internal state variables and con-cepts of irreversible thermodynamics are used in derivingconstitutive equations [20] A number of models of damagebehavior of fiber reinforced composites are based on themethods of continuum damage mechanics (CDM) For thecase of damage in viscoelastic materials two sets of internalvariables are needed Weitsman [21] presented a formulationto treat this case In this formulation damage in the formof microcracks is represented by two symmetric secondrank tensors related to the total area of active and passivemicrocracks in a representative volume element (RVE) of thematerial and viscoelasticity is introduced through 119899 scalarvalued internal state variables (ISVs) Since the scale ofmolecular rearrangement is much smaller than the scale ofmicrocracks the effect of damage on viscoelastic retardationtime is ignoredThe formulation uses Taylor series expansionof Gibbs free energy (Figure 1) [2]

Voyiadjis and Kattan [22] have presented a new formu-lation to link continuum damage mechanics with the con-cept of fabric tensors within the framework of classicalelasticity assuming small strains The new theory was calleddamagemechanicswith fabric tensors A fourth-rankdamagetensor was used and its exact relationship to the fabric ten-sors was illustrated A [23] model of damage mechanics fordirectional data was formulated using fabric tensors [22]Qiang et al have described accurately by a fourth-order fabric

tensor of the cracks within the framework of noninteractionapproximation the compliance increment due to the presenceof generally nonuniformly distributed cracks They haveexpressed that the compliance increment can be approxi-mated by the corresponding second-order fabric tensor [24]Spencer has formulated constitutive equations for flow offabric-reinforced composite materials which show viscousand viscoplastic response at the forming temperature [24 25]

The fabric tensor is a symmetric second rank tensorwhichcharacterizes the geometric arrangement of the porous mate-rial microstructure In developing this result it is assumedthat the matrix material of the porous elastic solid is isotropicand thus that the anisotropy of the porous elastic solid isdetermined by the fabric tensor [26] Fabric tensor has provedto be an effective tool statistically characterizing directionaldata in a smooth and frame-indifferent formDirectional dataarising from microscopic physics and mechanics can besummed up as tensor-valued orientation distribution func-tions [27] The subject of fabric tensor has been studied by anumber of different researchers [28ndash31]There are studies thatestablish a relationship between the fabric tensor with dam-age tensor However such studies are based on experimentaldata In these studies statistical distribution along a certainplane of the microcracks in the material has been consid-ered and an average density of microcracks has been usedOur study is not based on experimental data and is phe-nomenological

In this study in the scope of continuum damage me-chanics (CDM) considering representative volume ele-ment (RVE) the mechanical representation of damage wasexpressed with two interior conditional variables showingthe properties of a second degree symmetrical tensor It isassumed that an element from two different continuous fiberfamilies is placed on each point of the composite materialIn the scope of continuum mechanics balance equationshave been summarized and the combined form was stated











119867(119896)

119894119895 = 119860(119896)

119894 119860(119896)

119895 119867lowast(119896)

119894119895 = 119860lowast(119896)

119894 119860lowast(119896)

119895

(1)



H =

119873

sum

119896=1

H(119896) =119873

sum

119896=1

A(119896) otimes A(119896)

Hlowast =119873

sum

119896=1

Hlowast(119896) =119873

sum

119896=1


(2)




119867119870119871 equiv 119860119870119860119871 997904rArr 119870119871 equiv 119870119860119871 + 119860119870119871

(3)





119887119896 = 120582minus1

119887 119909119896119870119861119870 120582119887 equiv (119889119897

119889119871)

119887

1205822

119887 = 119862119870119871119861119870119861119871

119911119896 = 120582minus1

119911 119909119896119870119885119870 120582119911 equiv (119889119897

119889119871)

119911

1205822

119911 = 119862119870119871119885119870119885119871

(4)




120588 + 120588V119896119896 = 0 (5)


120588 (x 119905) =1205880 (X)119869 (x 119905)

(6)


120588V119901 = 120588119891119901 + 119905119903119901119903 (7)


120576119896119903119901119905119903119901 = 0 119905119903119901 = 119905119901119903 (8)


120588 120576 = 119905119896119897119889119896119897 + 119902119896119896 + 120588ℎ (9)







minus (Σ + 1205880120579120578) +

1

2119879119870119871119870119871 +

1

120579120579119870119876119870 ge 0 (11)


Σ equiv 1205880120595

119870119871 = 2119889119896119897119909119896119870119909119897119871 997904rArr 119889119896119897 =1

2119870119871119883119870119896119883119871119897

119879119870119871 equiv 119869119883119870119896119883119871119897119905119896119897 997904rArr 119905119896119897 = 119869minus1119909119896119870119909119897119871119879119870119871

119876119870 equiv 119869119883119870119896119902119896 997904rArr 119902119896 = 119869minus1119909119896119870119876119870

120579119870 = 119909119896119870120579119896 997904rArr 120579119896 = 119883119870119896120579119870

(12)



Σ = Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579) (13)

120578 = minus1

1205880

120597Σ

120597120579 (14)

120576 =1

1205880

(Σ minus120597Σ

120597120579120579) (15)

119876119870 = 0 (16)

119864119879119870119871 equiv 2120597Σ

120597119862119870119871

(17)


119884119872119873 equiv minus120597Σ

120597119867119872119873


120597119867119870119871

(18)

119863119879119872119873 = 119863119879119872119873 (119862119872119873 119872119873 119867119872119873 119872119873 119861119870 119885119870 120579)

(19)

1

2119863119879119872119873119872119873 minus 119884119872119873119872119873 ge 0 (20)

119863119879119870119871 (119862119872119873 0 119867119872119873 0 119861119870 119885119870 120579) = 0 (21)

119879119870119871 equiv 119864119879119870119871 + 119863119879119870119871 (22)




minus1= [119878119870119871]



1198831015840

119870 = 119878119870119871119883119871 119883119871 = 119878119879

1198711198701198831015840

119870 [119878119870119871]minus1

= [119878119870119871]119879

(23)


Σ (119878119870119875119878119871119877119862119875119877 119878119870119875119878119871119877119867119875119877 119878119870119871119861119871 119878119870119871119885119871 120579)

= Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

(24)

119863119879119870119871 (119878119869119875119878119873119877119862119875119877 119878119869119875119878119873119877119875119877 119878119869119875119878119873119877119867119875119877

119878119869119875119878119873119877119875119877 119878119869119875119861119875 119878119869119875119885119875 120579)

= 119878119870119875119878119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(25)



119869 = 1 or detC = 119868119868119868 = 1

1205822

119887 = 119862119870119871119861119870119861119871 = 1 1205822

119911 = 119862119870119871119885119870119885119871 = 1

(26)


119864119879119870119871 = minus119901119862minus1

119870119871 + Γ119887119861119870119861119871 + Γ119911119885119870119885119871 + 2120597Σ

120597119862119870119871

(27)




Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

= Σ (119872119870119875119872119871119877119862119875119877119872119870119875119872119871119877119867119875119877119872119870119871119861119871119872119870119871119885119871 120579)

(28)


minus1= [119872119870119871]



1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 = 119867119870119870

1198685 equiv 119861119870119861119870 1198686 equiv 119885119870119885119870

1198687 equiv 119861119870119885119870 1198688 equiv 119861119870119862119870119871119861119871 = 1205822

119887

1198689 equiv 119861119870119862119870119871119862119871119872119861119872 11986810 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986811 equiv 119885119870119862119870119871119862119871119872119885119872 11986812 equiv 119861119870119862119870119871119885119871

11986813 equiv 119861119870119862119870119871119862119871119872119885119872 11986814 equiv 119861119870119867119870119871119861119871

11986815 equiv 119861119870119862119870119871119867119871119872119861119872 11986816 equiv 119885119870119867119870119871119885119871

11986817 equiv 119885119870119862119870119871119867119871119872119885119872 11986818 equiv 119861119870119867119870119871119885119871

11986819 equiv 119861119870119862119870119871119867119871119872119885119872 11986820 equiv 119862119870119871119867119871119870

11986821 equiv 119862119870119871119862119871119872119867119872119870

(29)



119868 = 1198681 119868119868 =1

2(1198682

1 minus 1198682)

119868119868119868 =1

6(1198683

1 minus 311986811198682 + 21198683) = detC(30)



Σ = Σ (119868 119868119868 1198684 1198687 1198689 119868119898 120579) (119898 = 11 21) (31)



119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868

120597119868

120597119862119875119877

+120597Σ

120597119868119868

120597119868119868

120597119862119875119877

+120597Σ

120597119868119894

120597119868119894

120597119862119875119877

)

(119894 = 9 11 12 13 15 17 19 20 21)

119884119875119877 =120597Σ

120597119868119898

120597119868119898

120597119867119875119877

(119898 = 4 14 15 21)

(32)


119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868+

120597Σ

120597119868119868119862119871119871)120575119875119877 minus

120597Σ

120597119868119868119862119875119877

+120597Σ

1205971198689

(119861119875119861119871119862119871119877 + 119862119875119871119861119871119861119877)

+120597Σ

12059711986811

(119885119875119885119871119862119871119877 + 119862119875119871119885119871119885119877)

+120597Σ

12059711986812

119861119875119885119877

+120597Σ

12059711986813

(119861119875119885119871119862119871119877 + 119862119875119871119861119871119885119877)

+120597Σ

12059711986815

(119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+120597Σ

12059711986817

(119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+120597Σ

12059711986819

(119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+120597Σ

12059711986820

119867119875119877 +120597Σ

12059711986821

(119862119875119871119867119871119877 + 119867119875119871119862119871119877)

(33)


119884119875119877 =120597Σ

1205971198684

120575119875119877 +120597Σ

12059711986814

119861119875119861119877 +120597Σ

12059711986815

119862119875119870119861119870119861119877

+120597Σ

12059711986816

119885119875119885119877 +120597Σ

12059711986817

119862119875119871119885119871119885119877 +120597Σ

12059711986818

119861119875119885119877

+120597Σ

12059711986819

119862119875119871119861119871119885119877 +120597Σ

12059711986820

119862119875119877 +120597Σ

12059711986821

119862119875119871119862119871119877

(34)





Σ = Σ119894119895119886119894119895119868119894119868119895 (119894 119895 = 1 2 4 7 9 11 21) 119886119894119895 = 119886119895119894

(35)


120597Σ

120597119868= 2 (11988611119868 + 11988612119868119868 + 1198861119896119868119896)

120597Σ

120597119868119868= 2 (11988612119868 + 11988622119868119868 + 1198862119896119868119896)

120597Σ

120597119868119898

= 2 (1198861198981119868 + 1198861198982119868119868 + 119886119898119896119868119896)

(119898 = 4 9 11 21) (119896 = 4 7 9 11 21)

(36)




119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877) + 12057222119867119875119877

(37)


119884119875119877 = 1205742119864119870119870120575119875119877 + 1205743119867119870119870120575119875119877 + 1205744119861119870119867119870119873119861119873120575119875119877

+ 1205745119885119870119867119870119873119885119873120575119875119877 + 1205746119861119875119861119877 + 1205747119864119870119870119861119875119861119877

+ 1205748119867119870119870119861119875119861119877 + 1205749119885119870119867119870119873119885119873119861119875119861119877 + 12057410119864119875119873119861119873119861119877

+ 12057411119885119875119885119877 + 12057412119864119870119870119885119875119885119877 + 12057413119867119870119870119885119875119885119877

+ 12057414119861119870119867119870119873119861119873119885119875119885119877 + 12057415119864119875119873119885119873119885119877

+ 12057416119861119870119885119870119861119875119885119877 + 12057417119861119870119864119870119873119885119873119861119875119885119877

+ 12057418119861119870119867119870119873119885119873119861119875119885119877 + 12057419119861119870119885119870119864119875119873119861119873119885119877 + 12057420119864119875119877

(38)








119863119879119870119871 (119872119869119875119872119873119877119862119875119877119872119869119875119872119873119877119875119877119872119869119875119872119873119877119867119875119877

119872119869119875119872119873119877119875119877119872119869119875119861119875119872119869119875119885119875 120579)

= 119872119870119875119872119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(39)


Consider

R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

equiv 119881119870119881119871 119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

(40)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

(41)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816119881119870=0

(42)



Ξ equiv Ξ (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870) (43)


1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 equiv 119870119870

1198685 equiv 119870119871119871119870 1198686 equiv 119870119871119871119872119872119870

1198687 equiv 119867119870119870 1198688 equiv 119870119870


1198689 equiv 119862119870119871119871119870 11986810 equiv 119862119870119871119867119871119870

11986811 equiv 119862119870119871119871119870 11986812 equiv 119870119871119867119871119870

11986813 equiv 119870119871119871119870 11986814 equiv 119867119870119871119871119870

11986815 equiv 119862119870119871119862119871119872119872119870 11986816 equiv 119862119870119871119862119871119872119867119872119870

11986817 equiv 119862119870119871119862119871119872119872119870 11986818 equiv 119862119870119871119871119872119872119870

11986819 equiv 119870119871119871119872119867119872119870 11986820 equiv 119870119871119871119872119872119870

11986821 equiv 119862119870119871119871119872119867119872119870 11986822 equiv 119862119870119871119871119872119872119870

11986823 equiv 119862119870119871119867119871119872119872119870 11986824 equiv 119862119870119871119871119872119872119870

11986825 equiv 119862119870119871119862119871119872119872119873119867119873119870 11986826 equiv 119862119870119871119862119871119872119872119873119873119870

11986827 equiv 119862119870119871119862119871119872119867119872119873119873119870 11986828 equiv 119870119871119871119872119862119872119873119867119873119870

11986829 equiv 119870119871119871119872119862119872119873119873119870 11986830 equiv 119870119871119871119872119867119872119873119873119870

11986831 equiv 119862119870119871119862119871119872119872119873119873119870 11986832 equiv 119862119870119871119871119872119867119872119873119873119870

11986833 equiv 119861119870119861119870 11986834 equiv 119885119870119885119870

11986835 equiv 119861119870119885119870 11986836 equiv 119861119870119862119870119871119861119871 = 1205822

119887

11986837 equiv 119861119870119862119870119871119862119871119872119861119872 11986838 equiv 119861119870119870119871119861119871

11986839 equiv 119861119870119870119871119871119872119861119872 11986840 equiv 119861119870119867119870119871119861119871

11986841 equiv 119861119870119870119871119861119871 11986842 equiv 119861119870119862119870119871119871119872119861119872

11986843 equiv 119861119870119862119870119871119867119871119872119861119872 11986844 equiv 119861119870119862119870119871119871119872119861119872

11986845 equiv 119861119870119870119871119867119871119872119861119872 11986846 equiv 119861119870119870119871119871119872119861119872

11986847 equiv 119861119870119867119870119871119871119872119861119872

11986848 equiv 119861119870119862119870119871119871119872119867119872119873119861119873

11986849 equiv 119861119870119862119870119871119871119872119872119873119861119873

11986850 equiv 119861119870119862119870119871119867119871119872119872119873119861119873

11986851 equiv 119861119870119870119871119867119871119872119872119873119861119873

11986852 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11986853 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986854 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119861119878

11986855 equiv 119861119870119870119871119871119872119867119872119873119873119878119861119878

11986856 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119861119878

11986857 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986858 equiv 119885119870119862119870119871119862119871119872119885119872

11986859 equiv 119885119870119870119871119885119871 11986860 equiv 119885119870119870119871119871119872119885119872

11986861 equiv 119885119870119867119870119871119885119871 11986862 equiv 119885119870119870119871119885119871

11986863 equiv 119885119870119862119870119871119871119872119885119872 11986864 equiv 119885119870119862119870119871119867119871119872119885119872

11986865 equiv 119885119870119862119870119871119871119872119885119872 11986866 equiv 119885119870119870119871119867119871119872119885119872

11986867 equiv 119885119870119870119871119871119872119885119872 11986868 equiv 119885119870119867119870119871119871119872119885119872

11986869 equiv 119885119870119862119870119871119871119872119867119872119873119885119873

11986870 equiv 119885119870119862119870119871119871119872119872119873119885119873

11986871 equiv 119885119870119862119870119871119867119871119872119872119873119885119873

11986872 equiv 119885119870119870119871119867119871119872119872119873119885119873

11986873 equiv 119885119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986874 equiv 119885119870119862119870119871119862119871119872119872119873119873119878119885119878

11986875 equiv 119885119870119870119871119871119872119867119872119873119862119873119878119885119878

11986876 equiv 119885119870119870119871119871119872119867119872119873119873119878119885119878

11986877 equiv 119885119870119862119870119871119871119872119867119872119873119873119878119885119878

11986878 equiv 119861119870119862119870119871119885119871

11986879 equiv 119861119870119862119870119871119862119871119872119885119872 11986880 equiv 119861119870119870119871119885119871

11986881 equiv 119861119870119870119871119871119872119885119872 11986882 equiv 119861119870119867119870119871119885119871

11986883 equiv 119861119870119870119871119885119871 11986884 equiv 119861119870119862119870119871119871119872119885119872

11986885 equiv 119861119870119862119870119871119867119871119872119885119872 11986886 equiv 119861119870119862119870119871119871119872119885119872

11986887 equiv 119861119870119870119871119867119871119872119885119872 11986888 equiv 119861119870119870119871119871119872119885119872

11986889 equiv 119861119870119867119870119871119871119872119885119872

11986890 equiv 119861119870119862119870119871119871119872119867119872119873119885119873

11986891 equiv 119861119870119862119870119871119871119872119872119873119885119873

11986892 equiv 119861119870119862119870119871119867119871119872119872119873119885119873

11986893 equiv 119861119870119870119871119867119871119872119872119873119885119873

11986894 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986895 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986896 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119885119878

11986897 equiv 119861119870119870119871119871119872119867119872119873119873119878119885119878

11986898 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119885119878

(44)




1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)



R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)




119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















119867(119896)

119894119895 = 119860(119896)

119894 119860(119896)

119895 119867lowast(119896)

119894119895 = 119860lowast(119896)

119894 119860lowast(119896)

119895

(1)



H =

119873

sum

119896=1

H(119896) =119873

sum

119896=1

A(119896) otimes A(119896)

Hlowast =119873

sum

119896=1

Hlowast(119896) =119873

sum

119896=1


(2)




119867119870119871 equiv 119860119870119860119871 997904rArr 119870119871 equiv 119870119860119871 + 119860119870119871

(3)





119887119896 = 120582minus1

119887 119909119896119870119861119870 120582119887 equiv (119889119897

119889119871)

119887

1205822

119887 = 119862119870119871119861119870119861119871

119911119896 = 120582minus1

119911 119909119896119870119885119870 120582119911 equiv (119889119897

119889119871)

119911

1205822

119911 = 119862119870119871119885119870119885119871

(4)




120588 + 120588V119896119896 = 0 (5)


120588 (x 119905) =1205880 (X)119869 (x 119905)

(6)


120588V119901 = 120588119891119901 + 119905119903119901119903 (7)


120576119896119903119901119905119903119901 = 0 119905119903119901 = 119905119901119903 (8)


120588 120576 = 119905119896119897119889119896119897 + 119902119896119896 + 120588ℎ (9)







minus (Σ + 1205880120579120578) +

1

2119879119870119871119870119871 +

1

120579120579119870119876119870 ge 0 (11)


Σ equiv 1205880120595

119870119871 = 2119889119896119897119909119896119870119909119897119871 997904rArr 119889119896119897 =1

2119870119871119883119870119896119883119871119897

119879119870119871 equiv 119869119883119870119896119883119871119897119905119896119897 997904rArr 119905119896119897 = 119869minus1119909119896119870119909119897119871119879119870119871

119876119870 equiv 119869119883119870119896119902119896 997904rArr 119902119896 = 119869minus1119909119896119870119876119870

120579119870 = 119909119896119870120579119896 997904rArr 120579119896 = 119883119870119896120579119870

(12)



Σ = Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579) (13)

120578 = minus1

1205880

120597Σ

120597120579 (14)

120576 =1

1205880

(Σ minus120597Σ

120597120579120579) (15)

119876119870 = 0 (16)

119864119879119870119871 equiv 2120597Σ

120597119862119870119871

(17)


119884119872119873 equiv minus120597Σ

120597119867119872119873


120597119867119870119871

(18)

119863119879119872119873 = 119863119879119872119873 (119862119872119873 119872119873 119867119872119873 119872119873 119861119870 119885119870 120579)

(19)

1

2119863119879119872119873119872119873 minus 119884119872119873119872119873 ge 0 (20)

119863119879119870119871 (119862119872119873 0 119867119872119873 0 119861119870 119885119870 120579) = 0 (21)

119879119870119871 equiv 119864119879119870119871 + 119863119879119870119871 (22)




minus1= [119878119870119871]



1198831015840

119870 = 119878119870119871119883119871 119883119871 = 119878119879

1198711198701198831015840

119870 [119878119870119871]minus1

= [119878119870119871]119879

(23)


Σ (119878119870119875119878119871119877119862119875119877 119878119870119875119878119871119877119867119875119877 119878119870119871119861119871 119878119870119871119885119871 120579)

= Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

(24)

119863119879119870119871 (119878119869119875119878119873119877119862119875119877 119878119869119875119878119873119877119875119877 119878119869119875119878119873119877119867119875119877

119878119869119875119878119873119877119875119877 119878119869119875119861119875 119878119869119875119885119875 120579)

= 119878119870119875119878119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(25)



119869 = 1 or detC = 119868119868119868 = 1

1205822

119887 = 119862119870119871119861119870119861119871 = 1 1205822

119911 = 119862119870119871119885119870119885119871 = 1

(26)


119864119879119870119871 = minus119901119862minus1

119870119871 + Γ119887119861119870119861119871 + Γ119911119885119870119885119871 + 2120597Σ

120597119862119870119871

(27)




Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

= Σ (119872119870119875119872119871119877119862119875119877119872119870119875119872119871119877119867119875119877119872119870119871119861119871119872119870119871119885119871 120579)

(28)


minus1= [119872119870119871]



1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 = 119867119870119870

1198685 equiv 119861119870119861119870 1198686 equiv 119885119870119885119870

1198687 equiv 119861119870119885119870 1198688 equiv 119861119870119862119870119871119861119871 = 1205822

119887

1198689 equiv 119861119870119862119870119871119862119871119872119861119872 11986810 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986811 equiv 119885119870119862119870119871119862119871119872119885119872 11986812 equiv 119861119870119862119870119871119885119871

11986813 equiv 119861119870119862119870119871119862119871119872119885119872 11986814 equiv 119861119870119867119870119871119861119871

11986815 equiv 119861119870119862119870119871119867119871119872119861119872 11986816 equiv 119885119870119867119870119871119885119871

11986817 equiv 119885119870119862119870119871119867119871119872119885119872 11986818 equiv 119861119870119867119870119871119885119871

11986819 equiv 119861119870119862119870119871119867119871119872119885119872 11986820 equiv 119862119870119871119867119871119870

11986821 equiv 119862119870119871119862119871119872119867119872119870

(29)



119868 = 1198681 119868119868 =1

2(1198682

1 minus 1198682)

119868119868119868 =1

6(1198683

1 minus 311986811198682 + 21198683) = detC(30)



Σ = Σ (119868 119868119868 1198684 1198687 1198689 119868119898 120579) (119898 = 11 21) (31)



119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868

120597119868

120597119862119875119877

+120597Σ

120597119868119868

120597119868119868

120597119862119875119877

+120597Σ

120597119868119894

120597119868119894

120597119862119875119877

)

(119894 = 9 11 12 13 15 17 19 20 21)

119884119875119877 =120597Σ

120597119868119898

120597119868119898

120597119867119875119877

(119898 = 4 14 15 21)

(32)


119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868+

120597Σ

120597119868119868119862119871119871)120575119875119877 minus

120597Σ

120597119868119868119862119875119877

+120597Σ

1205971198689

(119861119875119861119871119862119871119877 + 119862119875119871119861119871119861119877)

+120597Σ

12059711986811

(119885119875119885119871119862119871119877 + 119862119875119871119885119871119885119877)

+120597Σ

12059711986812

119861119875119885119877

+120597Σ

12059711986813

(119861119875119885119871119862119871119877 + 119862119875119871119861119871119885119877)

+120597Σ

12059711986815

(119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+120597Σ

12059711986817

(119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+120597Σ

12059711986819

(119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+120597Σ

12059711986820

119867119875119877 +120597Σ

12059711986821

(119862119875119871119867119871119877 + 119867119875119871119862119871119877)

(33)


119884119875119877 =120597Σ

1205971198684

120575119875119877 +120597Σ

12059711986814

119861119875119861119877 +120597Σ

12059711986815

119862119875119870119861119870119861119877

+120597Σ

12059711986816

119885119875119885119877 +120597Σ

12059711986817

119862119875119871119885119871119885119877 +120597Σ

12059711986818

119861119875119885119877

+120597Σ

12059711986819

119862119875119871119861119871119885119877 +120597Σ

12059711986820

119862119875119877 +120597Σ

12059711986821

119862119875119871119862119871119877

(34)





Σ = Σ119894119895119886119894119895119868119894119868119895 (119894 119895 = 1 2 4 7 9 11 21) 119886119894119895 = 119886119895119894

(35)


120597Σ

120597119868= 2 (11988611119868 + 11988612119868119868 + 1198861119896119868119896)

120597Σ

120597119868119868= 2 (11988612119868 + 11988622119868119868 + 1198862119896119868119896)

120597Σ

120597119868119898

= 2 (1198861198981119868 + 1198861198982119868119868 + 119886119898119896119868119896)

(119898 = 4 9 11 21) (119896 = 4 7 9 11 21)

(36)




119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877) + 12057222119867119875119877

(37)


119884119875119877 = 1205742119864119870119870120575119875119877 + 1205743119867119870119870120575119875119877 + 1205744119861119870119867119870119873119861119873120575119875119877

+ 1205745119885119870119867119870119873119885119873120575119875119877 + 1205746119861119875119861119877 + 1205747119864119870119870119861119875119861119877

+ 1205748119867119870119870119861119875119861119877 + 1205749119885119870119867119870119873119885119873119861119875119861119877 + 12057410119864119875119873119861119873119861119877

+ 12057411119885119875119885119877 + 12057412119864119870119870119885119875119885119877 + 12057413119867119870119870119885119875119885119877

+ 12057414119861119870119867119870119873119861119873119885119875119885119877 + 12057415119864119875119873119885119873119885119877

+ 12057416119861119870119885119870119861119875119885119877 + 12057417119861119870119864119870119873119885119873119861119875119885119877

+ 12057418119861119870119867119870119873119885119873119861119875119885119877 + 12057419119861119870119885119870119864119875119873119861119873119885119877 + 12057420119864119875119877

(38)








119863119879119870119871 (119872119869119875119872119873119877119862119875119877119872119869119875119872119873119877119875119877119872119869119875119872119873119877119867119875119877

119872119869119875119872119873119877119875119877119872119869119875119861119875119872119869119875119885119875 120579)

= 119872119870119875119872119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(39)


Consider

R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

equiv 119881119870119881119871 119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

(40)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

(41)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816119881119870=0

(42)



Ξ equiv Ξ (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870) (43)


1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 equiv 119870119870

1198685 equiv 119870119871119871119870 1198686 equiv 119870119871119871119872119872119870

1198687 equiv 119867119870119870 1198688 equiv 119870119870


1198689 equiv 119862119870119871119871119870 11986810 equiv 119862119870119871119867119871119870

11986811 equiv 119862119870119871119871119870 11986812 equiv 119870119871119867119871119870

11986813 equiv 119870119871119871119870 11986814 equiv 119867119870119871119871119870

11986815 equiv 119862119870119871119862119871119872119872119870 11986816 equiv 119862119870119871119862119871119872119867119872119870

11986817 equiv 119862119870119871119862119871119872119872119870 11986818 equiv 119862119870119871119871119872119872119870

11986819 equiv 119870119871119871119872119867119872119870 11986820 equiv 119870119871119871119872119872119870

11986821 equiv 119862119870119871119871119872119867119872119870 11986822 equiv 119862119870119871119871119872119872119870

11986823 equiv 119862119870119871119867119871119872119872119870 11986824 equiv 119862119870119871119871119872119872119870

11986825 equiv 119862119870119871119862119871119872119872119873119867119873119870 11986826 equiv 119862119870119871119862119871119872119872119873119873119870

11986827 equiv 119862119870119871119862119871119872119867119872119873119873119870 11986828 equiv 119870119871119871119872119862119872119873119867119873119870

11986829 equiv 119870119871119871119872119862119872119873119873119870 11986830 equiv 119870119871119871119872119867119872119873119873119870

11986831 equiv 119862119870119871119862119871119872119872119873119873119870 11986832 equiv 119862119870119871119871119872119867119872119873119873119870

11986833 equiv 119861119870119861119870 11986834 equiv 119885119870119885119870

11986835 equiv 119861119870119885119870 11986836 equiv 119861119870119862119870119871119861119871 = 1205822

119887

11986837 equiv 119861119870119862119870119871119862119871119872119861119872 11986838 equiv 119861119870119870119871119861119871

11986839 equiv 119861119870119870119871119871119872119861119872 11986840 equiv 119861119870119867119870119871119861119871

11986841 equiv 119861119870119870119871119861119871 11986842 equiv 119861119870119862119870119871119871119872119861119872

11986843 equiv 119861119870119862119870119871119867119871119872119861119872 11986844 equiv 119861119870119862119870119871119871119872119861119872

11986845 equiv 119861119870119870119871119867119871119872119861119872 11986846 equiv 119861119870119870119871119871119872119861119872

11986847 equiv 119861119870119867119870119871119871119872119861119872

11986848 equiv 119861119870119862119870119871119871119872119867119872119873119861119873

11986849 equiv 119861119870119862119870119871119871119872119872119873119861119873

11986850 equiv 119861119870119862119870119871119867119871119872119872119873119861119873

11986851 equiv 119861119870119870119871119867119871119872119872119873119861119873

11986852 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11986853 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986854 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119861119878

11986855 equiv 119861119870119870119871119871119872119867119872119873119873119878119861119878

11986856 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119861119878

11986857 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986858 equiv 119885119870119862119870119871119862119871119872119885119872

11986859 equiv 119885119870119870119871119885119871 11986860 equiv 119885119870119870119871119871119872119885119872

11986861 equiv 119885119870119867119870119871119885119871 11986862 equiv 119885119870119870119871119885119871

11986863 equiv 119885119870119862119870119871119871119872119885119872 11986864 equiv 119885119870119862119870119871119867119871119872119885119872

11986865 equiv 119885119870119862119870119871119871119872119885119872 11986866 equiv 119885119870119870119871119867119871119872119885119872

11986867 equiv 119885119870119870119871119871119872119885119872 11986868 equiv 119885119870119867119870119871119871119872119885119872

11986869 equiv 119885119870119862119870119871119871119872119867119872119873119885119873

11986870 equiv 119885119870119862119870119871119871119872119872119873119885119873

11986871 equiv 119885119870119862119870119871119867119871119872119872119873119885119873

11986872 equiv 119885119870119870119871119867119871119872119872119873119885119873

11986873 equiv 119885119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986874 equiv 119885119870119862119870119871119862119871119872119872119873119873119878119885119878

11986875 equiv 119885119870119870119871119871119872119867119872119873119862119873119878119885119878

11986876 equiv 119885119870119870119871119871119872119867119872119873119873119878119885119878

11986877 equiv 119885119870119862119870119871119871119872119867119872119873119873119878119885119878

11986878 equiv 119861119870119862119870119871119885119871

11986879 equiv 119861119870119862119870119871119862119871119872119885119872 11986880 equiv 119861119870119870119871119885119871

11986881 equiv 119861119870119870119871119871119872119885119872 11986882 equiv 119861119870119867119870119871119885119871

11986883 equiv 119861119870119870119871119885119871 11986884 equiv 119861119870119862119870119871119871119872119885119872

11986885 equiv 119861119870119862119870119871119867119871119872119885119872 11986886 equiv 119861119870119862119870119871119871119872119885119872

11986887 equiv 119861119870119870119871119867119871119872119885119872 11986888 equiv 119861119870119870119871119871119872119885119872

11986889 equiv 119861119870119867119870119871119871119872119885119872

11986890 equiv 119861119870119862119870119871119871119872119867119872119873119885119873

11986891 equiv 119861119870119862119870119871119871119872119872119873119885119873

11986892 equiv 119861119870119862119870119871119867119871119872119872119873119885119873

11986893 equiv 119861119870119870119871119867119871119872119872119873119885119873

11986894 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986895 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986896 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119885119878

11986897 equiv 119861119870119870119871119871119872119867119872119873119873119878119885119878

11986898 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119885119878

(44)




1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)



R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)




119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119887119896 = 120582minus1

119887 119909119896119870119861119870 120582119887 equiv (119889119897

119889119871)

119887

1205822

119887 = 119862119870119871119861119870119861119871

119911119896 = 120582minus1

119911 119909119896119870119885119870 120582119911 equiv (119889119897

119889119871)

119911

1205822

119911 = 119862119870119871119885119870119885119871

(4)




120588 + 120588V119896119896 = 0 (5)


120588 (x 119905) =1205880 (X)119869 (x 119905)

(6)


120588V119901 = 120588119891119901 + 119905119903119901119903 (7)


120576119896119903119901119905119903119901 = 0 119905119903119901 = 119905119901119903 (8)


120588 120576 = 119905119896119897119889119896119897 + 119902119896119896 + 120588ℎ (9)







minus (Σ + 1205880120579120578) +

1

2119879119870119871119870119871 +

1

120579120579119870119876119870 ge 0 (11)


Σ equiv 1205880120595

119870119871 = 2119889119896119897119909119896119870119909119897119871 997904rArr 119889119896119897 =1

2119870119871119883119870119896119883119871119897

119879119870119871 equiv 119869119883119870119896119883119871119897119905119896119897 997904rArr 119905119896119897 = 119869minus1119909119896119870119909119897119871119879119870119871

119876119870 equiv 119869119883119870119896119902119896 997904rArr 119902119896 = 119869minus1119909119896119870119876119870

120579119870 = 119909119896119870120579119896 997904rArr 120579119896 = 119883119870119896120579119870

(12)



Σ = Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579) (13)

120578 = minus1

1205880

120597Σ

120597120579 (14)

120576 =1

1205880

(Σ minus120597Σ

120597120579120579) (15)

119876119870 = 0 (16)

119864119879119870119871 equiv 2120597Σ

120597119862119870119871

(17)


119884119872119873 equiv minus120597Σ

120597119867119872119873


120597119867119870119871

(18)

119863119879119872119873 = 119863119879119872119873 (119862119872119873 119872119873 119867119872119873 119872119873 119861119870 119885119870 120579)

(19)

1

2119863119879119872119873119872119873 minus 119884119872119873119872119873 ge 0 (20)

119863119879119870119871 (119862119872119873 0 119867119872119873 0 119861119870 119885119870 120579) = 0 (21)

119879119870119871 equiv 119864119879119870119871 + 119863119879119870119871 (22)




minus1= [119878119870119871]



1198831015840

119870 = 119878119870119871119883119871 119883119871 = 119878119879

1198711198701198831015840

119870 [119878119870119871]minus1

= [119878119870119871]119879

(23)


Σ (119878119870119875119878119871119877119862119875119877 119878119870119875119878119871119877119867119875119877 119878119870119871119861119871 119878119870119871119885119871 120579)

= Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

(24)

119863119879119870119871 (119878119869119875119878119873119877119862119875119877 119878119869119875119878119873119877119875119877 119878119869119875119878119873119877119867119875119877

119878119869119875119878119873119877119875119877 119878119869119875119861119875 119878119869119875119885119875 120579)

= 119878119870119875119878119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(25)



119869 = 1 or detC = 119868119868119868 = 1

1205822

119887 = 119862119870119871119861119870119861119871 = 1 1205822

119911 = 119862119870119871119885119870119885119871 = 1

(26)


119864119879119870119871 = minus119901119862minus1

119870119871 + Γ119887119861119870119861119871 + Γ119911119885119870119885119871 + 2120597Σ

120597119862119870119871

(27)




Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

= Σ (119872119870119875119872119871119877119862119875119877119872119870119875119872119871119877119867119875119877119872119870119871119861119871119872119870119871119885119871 120579)

(28)


minus1= [119872119870119871]



1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 = 119867119870119870

1198685 equiv 119861119870119861119870 1198686 equiv 119885119870119885119870

1198687 equiv 119861119870119885119870 1198688 equiv 119861119870119862119870119871119861119871 = 1205822

119887

1198689 equiv 119861119870119862119870119871119862119871119872119861119872 11986810 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986811 equiv 119885119870119862119870119871119862119871119872119885119872 11986812 equiv 119861119870119862119870119871119885119871

11986813 equiv 119861119870119862119870119871119862119871119872119885119872 11986814 equiv 119861119870119867119870119871119861119871

11986815 equiv 119861119870119862119870119871119867119871119872119861119872 11986816 equiv 119885119870119867119870119871119885119871

11986817 equiv 119885119870119862119870119871119867119871119872119885119872 11986818 equiv 119861119870119867119870119871119885119871

11986819 equiv 119861119870119862119870119871119867119871119872119885119872 11986820 equiv 119862119870119871119867119871119870

11986821 equiv 119862119870119871119862119871119872119867119872119870

(29)



119868 = 1198681 119868119868 =1

2(1198682

1 minus 1198682)

119868119868119868 =1

6(1198683

1 minus 311986811198682 + 21198683) = detC(30)



Σ = Σ (119868 119868119868 1198684 1198687 1198689 119868119898 120579) (119898 = 11 21) (31)



119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868

120597119868

120597119862119875119877

+120597Σ

120597119868119868

120597119868119868

120597119862119875119877

+120597Σ

120597119868119894

120597119868119894

120597119862119875119877

)

(119894 = 9 11 12 13 15 17 19 20 21)

119884119875119877 =120597Σ

120597119868119898

120597119868119898

120597119867119875119877

(119898 = 4 14 15 21)

(32)


119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868+

120597Σ

120597119868119868119862119871119871)120575119875119877 minus

120597Σ

120597119868119868119862119875119877

+120597Σ

1205971198689

(119861119875119861119871119862119871119877 + 119862119875119871119861119871119861119877)

+120597Σ

12059711986811

(119885119875119885119871119862119871119877 + 119862119875119871119885119871119885119877)

+120597Σ

12059711986812

119861119875119885119877

+120597Σ

12059711986813

(119861119875119885119871119862119871119877 + 119862119875119871119861119871119885119877)

+120597Σ

12059711986815

(119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+120597Σ

12059711986817

(119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+120597Σ

12059711986819

(119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+120597Σ

12059711986820

119867119875119877 +120597Σ

12059711986821

(119862119875119871119867119871119877 + 119867119875119871119862119871119877)

(33)


119884119875119877 =120597Σ

1205971198684

120575119875119877 +120597Σ

12059711986814

119861119875119861119877 +120597Σ

12059711986815

119862119875119870119861119870119861119877

+120597Σ

12059711986816

119885119875119885119877 +120597Σ

12059711986817

119862119875119871119885119871119885119877 +120597Σ

12059711986818

119861119875119885119877

+120597Σ

12059711986819

119862119875119871119861119871119885119877 +120597Σ

12059711986820

119862119875119877 +120597Σ

12059711986821

119862119875119871119862119871119877

(34)





Σ = Σ119894119895119886119894119895119868119894119868119895 (119894 119895 = 1 2 4 7 9 11 21) 119886119894119895 = 119886119895119894

(35)


120597Σ

120597119868= 2 (11988611119868 + 11988612119868119868 + 1198861119896119868119896)

120597Σ

120597119868119868= 2 (11988612119868 + 11988622119868119868 + 1198862119896119868119896)

120597Σ

120597119868119898

= 2 (1198861198981119868 + 1198861198982119868119868 + 119886119898119896119868119896)

(119898 = 4 9 11 21) (119896 = 4 7 9 11 21)

(36)




119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877) + 12057222119867119875119877

(37)


119884119875119877 = 1205742119864119870119870120575119875119877 + 1205743119867119870119870120575119875119877 + 1205744119861119870119867119870119873119861119873120575119875119877

+ 1205745119885119870119867119870119873119885119873120575119875119877 + 1205746119861119875119861119877 + 1205747119864119870119870119861119875119861119877

+ 1205748119867119870119870119861119875119861119877 + 1205749119885119870119867119870119873119885119873119861119875119861119877 + 12057410119864119875119873119861119873119861119877

+ 12057411119885119875119885119877 + 12057412119864119870119870119885119875119885119877 + 12057413119867119870119870119885119875119885119877

+ 12057414119861119870119867119870119873119861119873119885119875119885119877 + 12057415119864119875119873119885119873119885119877

+ 12057416119861119870119885119870119861119875119885119877 + 12057417119861119870119864119870119873119885119873119861119875119885119877

+ 12057418119861119870119867119870119873119885119873119861119875119885119877 + 12057419119861119870119885119870119864119875119873119861119873119885119877 + 12057420119864119875119877

(38)








119863119879119870119871 (119872119869119875119872119873119877119862119875119877119872119869119875119872119873119877119875119877119872119869119875119872119873119877119867119875119877

119872119869119875119872119873119877119875119877119872119869119875119861119875119872119869119875119885119875 120579)

= 119872119870119875119872119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(39)


Consider

R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

equiv 119881119870119881119871 119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

(40)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

(41)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816119881119870=0

(42)



Ξ equiv Ξ (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870) (43)


1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 equiv 119870119870

1198685 equiv 119870119871119871119870 1198686 equiv 119870119871119871119872119872119870

1198687 equiv 119867119870119870 1198688 equiv 119870119870


1198689 equiv 119862119870119871119871119870 11986810 equiv 119862119870119871119867119871119870

11986811 equiv 119862119870119871119871119870 11986812 equiv 119870119871119867119871119870

11986813 equiv 119870119871119871119870 11986814 equiv 119867119870119871119871119870

11986815 equiv 119862119870119871119862119871119872119872119870 11986816 equiv 119862119870119871119862119871119872119867119872119870

11986817 equiv 119862119870119871119862119871119872119872119870 11986818 equiv 119862119870119871119871119872119872119870

11986819 equiv 119870119871119871119872119867119872119870 11986820 equiv 119870119871119871119872119872119870

11986821 equiv 119862119870119871119871119872119867119872119870 11986822 equiv 119862119870119871119871119872119872119870

11986823 equiv 119862119870119871119867119871119872119872119870 11986824 equiv 119862119870119871119871119872119872119870

11986825 equiv 119862119870119871119862119871119872119872119873119867119873119870 11986826 equiv 119862119870119871119862119871119872119872119873119873119870

11986827 equiv 119862119870119871119862119871119872119867119872119873119873119870 11986828 equiv 119870119871119871119872119862119872119873119867119873119870

11986829 equiv 119870119871119871119872119862119872119873119873119870 11986830 equiv 119870119871119871119872119867119872119873119873119870

11986831 equiv 119862119870119871119862119871119872119872119873119873119870 11986832 equiv 119862119870119871119871119872119867119872119873119873119870

11986833 equiv 119861119870119861119870 11986834 equiv 119885119870119885119870

11986835 equiv 119861119870119885119870 11986836 equiv 119861119870119862119870119871119861119871 = 1205822

119887

11986837 equiv 119861119870119862119870119871119862119871119872119861119872 11986838 equiv 119861119870119870119871119861119871

11986839 equiv 119861119870119870119871119871119872119861119872 11986840 equiv 119861119870119867119870119871119861119871

11986841 equiv 119861119870119870119871119861119871 11986842 equiv 119861119870119862119870119871119871119872119861119872

11986843 equiv 119861119870119862119870119871119867119871119872119861119872 11986844 equiv 119861119870119862119870119871119871119872119861119872

11986845 equiv 119861119870119870119871119867119871119872119861119872 11986846 equiv 119861119870119870119871119871119872119861119872

11986847 equiv 119861119870119867119870119871119871119872119861119872

11986848 equiv 119861119870119862119870119871119871119872119867119872119873119861119873

11986849 equiv 119861119870119862119870119871119871119872119872119873119861119873

11986850 equiv 119861119870119862119870119871119867119871119872119872119873119861119873

11986851 equiv 119861119870119870119871119867119871119872119872119873119861119873

11986852 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11986853 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986854 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119861119878

11986855 equiv 119861119870119870119871119871119872119867119872119873119873119878119861119878

11986856 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119861119878

11986857 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986858 equiv 119885119870119862119870119871119862119871119872119885119872

11986859 equiv 119885119870119870119871119885119871 11986860 equiv 119885119870119870119871119871119872119885119872

11986861 equiv 119885119870119867119870119871119885119871 11986862 equiv 119885119870119870119871119885119871

11986863 equiv 119885119870119862119870119871119871119872119885119872 11986864 equiv 119885119870119862119870119871119867119871119872119885119872

11986865 equiv 119885119870119862119870119871119871119872119885119872 11986866 equiv 119885119870119870119871119867119871119872119885119872

11986867 equiv 119885119870119870119871119871119872119885119872 11986868 equiv 119885119870119867119870119871119871119872119885119872

11986869 equiv 119885119870119862119870119871119871119872119867119872119873119885119873

11986870 equiv 119885119870119862119870119871119871119872119872119873119885119873

11986871 equiv 119885119870119862119870119871119867119871119872119872119873119885119873

11986872 equiv 119885119870119870119871119867119871119872119872119873119885119873

11986873 equiv 119885119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986874 equiv 119885119870119862119870119871119862119871119872119872119873119873119878119885119878

11986875 equiv 119885119870119870119871119871119872119867119872119873119862119873119878119885119878

11986876 equiv 119885119870119870119871119871119872119867119872119873119873119878119885119878

11986877 equiv 119885119870119862119870119871119871119872119867119872119873119873119878119885119878

11986878 equiv 119861119870119862119870119871119885119871

11986879 equiv 119861119870119862119870119871119862119871119872119885119872 11986880 equiv 119861119870119870119871119885119871

11986881 equiv 119861119870119870119871119871119872119885119872 11986882 equiv 119861119870119867119870119871119885119871

11986883 equiv 119861119870119870119871119885119871 11986884 equiv 119861119870119862119870119871119871119872119885119872

11986885 equiv 119861119870119862119870119871119867119871119872119885119872 11986886 equiv 119861119870119862119870119871119871119872119885119872

11986887 equiv 119861119870119870119871119867119871119872119885119872 11986888 equiv 119861119870119870119871119871119872119885119872

11986889 equiv 119861119870119867119870119871119871119872119885119872

11986890 equiv 119861119870119862119870119871119871119872119867119872119873119885119873

11986891 equiv 119861119870119862119870119871119871119872119872119873119885119873

11986892 equiv 119861119870119862119870119871119867119871119872119872119873119885119873

11986893 equiv 119861119870119870119871119867119871119872119872119873119885119873

11986894 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986895 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986896 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119885119878

11986897 equiv 119861119870119870119871119871119872119867119872119873119873119878119885119878

11986898 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119885119878

(44)




1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)



R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)




119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119884119872119873 equiv minus120597Σ

120597119867119872119873


120597119867119870119871

(18)

119863119879119872119873 = 119863119879119872119873 (119862119872119873 119872119873 119867119872119873 119872119873 119861119870 119885119870 120579)

(19)

1

2119863119879119872119873119872119873 minus 119884119872119873119872119873 ge 0 (20)

119863119879119870119871 (119862119872119873 0 119867119872119873 0 119861119870 119885119870 120579) = 0 (21)

119879119870119871 equiv 119864119879119870119871 + 119863119879119870119871 (22)




minus1= [119878119870119871]



1198831015840

119870 = 119878119870119871119883119871 119883119871 = 119878119879

1198711198701198831015840

119870 [119878119870119871]minus1

= [119878119870119871]119879

(23)


Σ (119878119870119875119878119871119877119862119875119877 119878119870119875119878119871119877119867119875119877 119878119870119871119861119871 119878119870119871119885119871 120579)

= Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

(24)

119863119879119870119871 (119878119869119875119878119873119877119862119875119877 119878119869119875119878119873119877119875119877 119878119869119875119878119873119877119867119875119877

119878119869119875119878119873119877119875119877 119878119869119875119861119875 119878119869119875119885119875 120579)

= 119878119870119875119878119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(25)



119869 = 1 or detC = 119868119868119868 = 1

1205822

119887 = 119862119870119871119861119870119861119871 = 1 1205822

119911 = 119862119870119871119885119870119885119871 = 1

(26)


119864119879119870119871 = minus119901119862minus1

119870119871 + Γ119887119861119870119861119871 + Γ119911119885119870119885119871 + 2120597Σ

120597119862119870119871

(27)




Σ (119862119870119871 119867119870119871 119861119870 119885119870 120579)

= Σ (119872119870119875119872119871119877119862119875119877119872119870119875119872119871119877119867119875119877119872119870119871119861119871119872119870119871119885119871 120579)

(28)


minus1= [119872119870119871]



1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 = 119867119870119870

1198685 equiv 119861119870119861119870 1198686 equiv 119885119870119885119870

1198687 equiv 119861119870119885119870 1198688 equiv 119861119870119862119870119871119861119871 = 1205822

119887

1198689 equiv 119861119870119862119870119871119862119871119872119861119872 11986810 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986811 equiv 119885119870119862119870119871119862119871119872119885119872 11986812 equiv 119861119870119862119870119871119885119871

11986813 equiv 119861119870119862119870119871119862119871119872119885119872 11986814 equiv 119861119870119867119870119871119861119871

11986815 equiv 119861119870119862119870119871119867119871119872119861119872 11986816 equiv 119885119870119867119870119871119885119871

11986817 equiv 119885119870119862119870119871119867119871119872119885119872 11986818 equiv 119861119870119867119870119871119885119871

11986819 equiv 119861119870119862119870119871119867119871119872119885119872 11986820 equiv 119862119870119871119867119871119870

11986821 equiv 119862119870119871119862119871119872119867119872119870

(29)



119868 = 1198681 119868119868 =1

2(1198682

1 minus 1198682)

119868119868119868 =1

6(1198683

1 minus 311986811198682 + 21198683) = detC(30)



Σ = Σ (119868 119868119868 1198684 1198687 1198689 119868119898 120579) (119898 = 11 21) (31)



119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868

120597119868

120597119862119875119877

+120597Σ

120597119868119868

120597119868119868

120597119862119875119877

+120597Σ

120597119868119894

120597119868119894

120597119862119875119877

)

(119894 = 9 11 12 13 15 17 19 20 21)

119884119875119877 =120597Σ

120597119868119898

120597119868119898

120597119867119875119877

(119898 = 4 14 15 21)

(32)


119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868+

120597Σ

120597119868119868119862119871119871)120575119875119877 minus

120597Σ

120597119868119868119862119875119877

+120597Σ

1205971198689

(119861119875119861119871119862119871119877 + 119862119875119871119861119871119861119877)

+120597Σ

12059711986811

(119885119875119885119871119862119871119877 + 119862119875119871119885119871119885119877)

+120597Σ

12059711986812

119861119875119885119877

+120597Σ

12059711986813

(119861119875119885119871119862119871119877 + 119862119875119871119861119871119885119877)

+120597Σ

12059711986815

(119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+120597Σ

12059711986817

(119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+120597Σ

12059711986819

(119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+120597Σ

12059711986820

119867119875119877 +120597Σ

12059711986821

(119862119875119871119867119871119877 + 119867119875119871119862119871119877)

(33)


119884119875119877 =120597Σ

1205971198684

120575119875119877 +120597Σ

12059711986814

119861119875119861119877 +120597Σ

12059711986815

119862119875119870119861119870119861119877

+120597Σ

12059711986816

119885119875119885119877 +120597Σ

12059711986817

119862119875119871119885119871119885119877 +120597Σ

12059711986818

119861119875119885119877

+120597Σ

12059711986819

119862119875119871119861119871119885119877 +120597Σ

12059711986820

119862119875119877 +120597Σ

12059711986821

119862119875119871119862119871119877

(34)





Σ = Σ119894119895119886119894119895119868119894119868119895 (119894 119895 = 1 2 4 7 9 11 21) 119886119894119895 = 119886119895119894

(35)


120597Σ

120597119868= 2 (11988611119868 + 11988612119868119868 + 1198861119896119868119896)

120597Σ

120597119868119868= 2 (11988612119868 + 11988622119868119868 + 1198862119896119868119896)

120597Σ

120597119868119898

= 2 (1198861198981119868 + 1198861198982119868119868 + 119886119898119896119868119896)

(119898 = 4 9 11 21) (119896 = 4 7 9 11 21)

(36)




119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877) + 12057222119867119875119877

(37)


119884119875119877 = 1205742119864119870119870120575119875119877 + 1205743119867119870119870120575119875119877 + 1205744119861119870119867119870119873119861119873120575119875119877

+ 1205745119885119870119867119870119873119885119873120575119875119877 + 1205746119861119875119861119877 + 1205747119864119870119870119861119875119861119877

+ 1205748119867119870119870119861119875119861119877 + 1205749119885119870119867119870119873119885119873119861119875119861119877 + 12057410119864119875119873119861119873119861119877

+ 12057411119885119875119885119877 + 12057412119864119870119870119885119875119885119877 + 12057413119867119870119870119885119875119885119877

+ 12057414119861119870119867119870119873119861119873119885119875119885119877 + 12057415119864119875119873119885119873119885119877

+ 12057416119861119870119885119870119861119875119885119877 + 12057417119861119870119864119870119873119885119873119861119875119885119877

+ 12057418119861119870119867119870119873119885119873119861119875119885119877 + 12057419119861119870119885119870119864119875119873119861119873119885119877 + 12057420119864119875119877

(38)








119863119879119870119871 (119872119869119875119872119873119877119862119875119877119872119869119875119872119873119877119875119877119872119869119875119872119873119877119867119875119877

119872119869119875119872119873119877119875119877119872119869119875119861119875119872119869119875119885119875 120579)

= 119872119870119875119872119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(39)


Consider

R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

equiv 119881119870119881119871 119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

(40)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

(41)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816119881119870=0

(42)



Ξ equiv Ξ (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870) (43)


1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 equiv 119870119870

1198685 equiv 119870119871119871119870 1198686 equiv 119870119871119871119872119872119870

1198687 equiv 119867119870119870 1198688 equiv 119870119870


1198689 equiv 119862119870119871119871119870 11986810 equiv 119862119870119871119867119871119870

11986811 equiv 119862119870119871119871119870 11986812 equiv 119870119871119867119871119870

11986813 equiv 119870119871119871119870 11986814 equiv 119867119870119871119871119870

11986815 equiv 119862119870119871119862119871119872119872119870 11986816 equiv 119862119870119871119862119871119872119867119872119870

11986817 equiv 119862119870119871119862119871119872119872119870 11986818 equiv 119862119870119871119871119872119872119870

11986819 equiv 119870119871119871119872119867119872119870 11986820 equiv 119870119871119871119872119872119870

11986821 equiv 119862119870119871119871119872119867119872119870 11986822 equiv 119862119870119871119871119872119872119870

11986823 equiv 119862119870119871119867119871119872119872119870 11986824 equiv 119862119870119871119871119872119872119870

11986825 equiv 119862119870119871119862119871119872119872119873119867119873119870 11986826 equiv 119862119870119871119862119871119872119872119873119873119870

11986827 equiv 119862119870119871119862119871119872119867119872119873119873119870 11986828 equiv 119870119871119871119872119862119872119873119867119873119870

11986829 equiv 119870119871119871119872119862119872119873119873119870 11986830 equiv 119870119871119871119872119867119872119873119873119870

11986831 equiv 119862119870119871119862119871119872119872119873119873119870 11986832 equiv 119862119870119871119871119872119867119872119873119873119870

11986833 equiv 119861119870119861119870 11986834 equiv 119885119870119885119870

11986835 equiv 119861119870119885119870 11986836 equiv 119861119870119862119870119871119861119871 = 1205822

119887

11986837 equiv 119861119870119862119870119871119862119871119872119861119872 11986838 equiv 119861119870119870119871119861119871

11986839 equiv 119861119870119870119871119871119872119861119872 11986840 equiv 119861119870119867119870119871119861119871

11986841 equiv 119861119870119870119871119861119871 11986842 equiv 119861119870119862119870119871119871119872119861119872

11986843 equiv 119861119870119862119870119871119867119871119872119861119872 11986844 equiv 119861119870119862119870119871119871119872119861119872

11986845 equiv 119861119870119870119871119867119871119872119861119872 11986846 equiv 119861119870119870119871119871119872119861119872

11986847 equiv 119861119870119867119870119871119871119872119861119872

11986848 equiv 119861119870119862119870119871119871119872119867119872119873119861119873

11986849 equiv 119861119870119862119870119871119871119872119872119873119861119873

11986850 equiv 119861119870119862119870119871119867119871119872119872119873119861119873

11986851 equiv 119861119870119870119871119867119871119872119872119873119861119873

11986852 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11986853 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986854 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119861119878

11986855 equiv 119861119870119870119871119871119872119867119872119873119873119878119861119878

11986856 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119861119878

11986857 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986858 equiv 119885119870119862119870119871119862119871119872119885119872

11986859 equiv 119885119870119870119871119885119871 11986860 equiv 119885119870119870119871119871119872119885119872

11986861 equiv 119885119870119867119870119871119885119871 11986862 equiv 119885119870119870119871119885119871

11986863 equiv 119885119870119862119870119871119871119872119885119872 11986864 equiv 119885119870119862119870119871119867119871119872119885119872

11986865 equiv 119885119870119862119870119871119871119872119885119872 11986866 equiv 119885119870119870119871119867119871119872119885119872

11986867 equiv 119885119870119870119871119871119872119885119872 11986868 equiv 119885119870119867119870119871119871119872119885119872

11986869 equiv 119885119870119862119870119871119871119872119867119872119873119885119873

11986870 equiv 119885119870119862119870119871119871119872119872119873119885119873

11986871 equiv 119885119870119862119870119871119867119871119872119872119873119885119873

11986872 equiv 119885119870119870119871119867119871119872119872119873119885119873

11986873 equiv 119885119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986874 equiv 119885119870119862119870119871119862119871119872119872119873119873119878119885119878

11986875 equiv 119885119870119870119871119871119872119867119872119873119862119873119878119885119878

11986876 equiv 119885119870119870119871119871119872119867119872119873119873119878119885119878

11986877 equiv 119885119870119862119870119871119871119872119867119872119873119873119878119885119878

11986878 equiv 119861119870119862119870119871119885119871

11986879 equiv 119861119870119862119870119871119862119871119872119885119872 11986880 equiv 119861119870119870119871119885119871

11986881 equiv 119861119870119870119871119871119872119885119872 11986882 equiv 119861119870119867119870119871119885119871

11986883 equiv 119861119870119870119871119885119871 11986884 equiv 119861119870119862119870119871119871119872119885119872

11986885 equiv 119861119870119862119870119871119867119871119872119885119872 11986886 equiv 119861119870119862119870119871119871119872119885119872

11986887 equiv 119861119870119870119871119867119871119872119885119872 11986888 equiv 119861119870119870119871119871119872119885119872

11986889 equiv 119861119870119867119870119871119871119872119885119872

11986890 equiv 119861119870119862119870119871119871119872119867119872119873119885119873

11986891 equiv 119861119870119862119870119871119871119872119872119873119885119873

11986892 equiv 119861119870119862119870119871119867119871119872119872119873119885119873

11986893 equiv 119861119870119870119871119867119871119872119872119873119885119873

11986894 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986895 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986896 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119885119878

11986897 equiv 119861119870119870119871119871119872119867119872119873119873119878119885119878

11986898 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119885119878

(44)




1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)



R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)




119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119868 = 1198681 119868119868 =1

2(1198682

1 minus 1198682)

119868119868119868 =1

6(1198683

1 minus 311986811198682 + 21198683) = detC(30)



Σ = Σ (119868 119868119868 1198684 1198687 1198689 119868119898 120579) (119898 = 11 21) (31)



119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868

120597119868

120597119862119875119877

+120597Σ

120597119868119868

120597119868119868

120597119862119875119877

+120597Σ

120597119868119894

120597119868119894

120597119862119875119877

)

(119894 = 9 11 12 13 15 17 19 20 21)

119884119875119877 =120597Σ

120597119868119898

120597119868119898

120597119867119875119877

(119898 = 4 14 15 21)

(32)


119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877

+ 2(120597Σ

120597119868+

120597Σ

120597119868119868119862119871119871)120575119875119877 minus

120597Σ

120597119868119868119862119875119877

+120597Σ

1205971198689

(119861119875119861119871119862119871119877 + 119862119875119871119861119871119861119877)

+120597Σ

12059711986811

(119885119875119885119871119862119871119877 + 119862119875119871119885119871119885119877)

+120597Σ

12059711986812

119861119875119885119877

+120597Σ

12059711986813

(119861119875119885119871119862119871119877 + 119862119875119871119861119871119885119877)

+120597Σ

12059711986815

(119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+120597Σ

12059711986817

(119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+120597Σ

12059711986819

(119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+120597Σ

12059711986820

119867119875119877 +120597Σ

12059711986821

(119862119875119871119867119871119877 + 119867119875119871119862119871119877)

(33)


119884119875119877 =120597Σ

1205971198684

120575119875119877 +120597Σ

12059711986814

119861119875119861119877 +120597Σ

12059711986815

119862119875119870119861119870119861119877

+120597Σ

12059711986816

119885119875119885119877 +120597Σ

12059711986817

119862119875119871119885119871119885119877 +120597Σ

12059711986818

119861119875119885119877

+120597Σ

12059711986819

119862119875119871119861119871119885119877 +120597Σ

12059711986820

119862119875119877 +120597Σ

12059711986821

119862119875119871119862119871119877

(34)





Σ = Σ119894119895119886119894119895119868119894119868119895 (119894 119895 = 1 2 4 7 9 11 21) 119886119894119895 = 119886119895119894

(35)


120597Σ

120597119868= 2 (11988611119868 + 11988612119868119868 + 1198861119896119868119896)

120597Σ

120597119868119868= 2 (11988612119868 + 11988622119868119868 + 1198862119896119868119896)

120597Σ

120597119868119898

= 2 (1198861198981119868 + 1198861198982119868119868 + 119886119898119896119868119896)

(119898 = 4 9 11 21) (119896 = 4 7 9 11 21)

(36)




119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877) + 12057222119867119875119877

(37)


119884119875119877 = 1205742119864119870119870120575119875119877 + 1205743119867119870119870120575119875119877 + 1205744119861119870119867119870119873119861119873120575119875119877

+ 1205745119885119870119867119870119873119885119873120575119875119877 + 1205746119861119875119861119877 + 1205747119864119870119870119861119875119861119877

+ 1205748119867119870119870119861119875119861119877 + 1205749119885119870119867119870119873119885119873119861119875119861119877 + 12057410119864119875119873119861119873119861119877

+ 12057411119885119875119885119877 + 12057412119864119870119870119885119875119885119877 + 12057413119867119870119870119885119875119885119877

+ 12057414119861119870119867119870119873119861119873119885119875119885119877 + 12057415119864119875119873119885119873119885119877

+ 12057416119861119870119885119870119861119875119885119877 + 12057417119861119870119864119870119873119885119873119861119875119885119877

+ 12057418119861119870119867119870119873119885119873119861119875119885119877 + 12057419119861119870119885119870119864119875119873119861119873119885119877 + 12057420119864119875119877

(38)








119863119879119870119871 (119872119869119875119872119873119877119862119875119877119872119869119875119872119873119877119875119877119872119869119875119872119873119877119867119875119877

119872119869119875119872119873119877119875119877119872119869119875119861119875119872119869119875119885119875 120579)

= 119872119870119875119872119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(39)


Consider

R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

equiv 119881119870119881119871 119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

(40)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

(41)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816119881119870=0

(42)



Ξ equiv Ξ (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870) (43)


1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 equiv 119870119870

1198685 equiv 119870119871119871119870 1198686 equiv 119870119871119871119872119872119870

1198687 equiv 119867119870119870 1198688 equiv 119870119870


1198689 equiv 119862119870119871119871119870 11986810 equiv 119862119870119871119867119871119870

11986811 equiv 119862119870119871119871119870 11986812 equiv 119870119871119867119871119870

11986813 equiv 119870119871119871119870 11986814 equiv 119867119870119871119871119870

11986815 equiv 119862119870119871119862119871119872119872119870 11986816 equiv 119862119870119871119862119871119872119867119872119870

11986817 equiv 119862119870119871119862119871119872119872119870 11986818 equiv 119862119870119871119871119872119872119870

11986819 equiv 119870119871119871119872119867119872119870 11986820 equiv 119870119871119871119872119872119870

11986821 equiv 119862119870119871119871119872119867119872119870 11986822 equiv 119862119870119871119871119872119872119870

11986823 equiv 119862119870119871119867119871119872119872119870 11986824 equiv 119862119870119871119871119872119872119870

11986825 equiv 119862119870119871119862119871119872119872119873119867119873119870 11986826 equiv 119862119870119871119862119871119872119872119873119873119870

11986827 equiv 119862119870119871119862119871119872119867119872119873119873119870 11986828 equiv 119870119871119871119872119862119872119873119867119873119870

11986829 equiv 119870119871119871119872119862119872119873119873119870 11986830 equiv 119870119871119871119872119867119872119873119873119870

11986831 equiv 119862119870119871119862119871119872119872119873119873119870 11986832 equiv 119862119870119871119871119872119867119872119873119873119870

11986833 equiv 119861119870119861119870 11986834 equiv 119885119870119885119870

11986835 equiv 119861119870119885119870 11986836 equiv 119861119870119862119870119871119861119871 = 1205822

119887

11986837 equiv 119861119870119862119870119871119862119871119872119861119872 11986838 equiv 119861119870119870119871119861119871

11986839 equiv 119861119870119870119871119871119872119861119872 11986840 equiv 119861119870119867119870119871119861119871

11986841 equiv 119861119870119870119871119861119871 11986842 equiv 119861119870119862119870119871119871119872119861119872

11986843 equiv 119861119870119862119870119871119867119871119872119861119872 11986844 equiv 119861119870119862119870119871119871119872119861119872

11986845 equiv 119861119870119870119871119867119871119872119861119872 11986846 equiv 119861119870119870119871119871119872119861119872

11986847 equiv 119861119870119867119870119871119871119872119861119872

11986848 equiv 119861119870119862119870119871119871119872119867119872119873119861119873

11986849 equiv 119861119870119862119870119871119871119872119872119873119861119873

11986850 equiv 119861119870119862119870119871119867119871119872119872119873119861119873

11986851 equiv 119861119870119870119871119867119871119872119872119873119861119873

11986852 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11986853 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986854 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119861119878

11986855 equiv 119861119870119870119871119871119872119867119872119873119873119878119861119878

11986856 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119861119878

11986857 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986858 equiv 119885119870119862119870119871119862119871119872119885119872

11986859 equiv 119885119870119870119871119885119871 11986860 equiv 119885119870119870119871119871119872119885119872

11986861 equiv 119885119870119867119870119871119885119871 11986862 equiv 119885119870119870119871119885119871

11986863 equiv 119885119870119862119870119871119871119872119885119872 11986864 equiv 119885119870119862119870119871119867119871119872119885119872

11986865 equiv 119885119870119862119870119871119871119872119885119872 11986866 equiv 119885119870119870119871119867119871119872119885119872

11986867 equiv 119885119870119870119871119871119872119885119872 11986868 equiv 119885119870119867119870119871119871119872119885119872

11986869 equiv 119885119870119862119870119871119871119872119867119872119873119885119873

11986870 equiv 119885119870119862119870119871119871119872119872119873119885119873

11986871 equiv 119885119870119862119870119871119867119871119872119872119873119885119873

11986872 equiv 119885119870119870119871119867119871119872119872119873119885119873

11986873 equiv 119885119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986874 equiv 119885119870119862119870119871119862119871119872119872119873119873119878119885119878

11986875 equiv 119885119870119870119871119871119872119867119872119873119862119873119878119885119878

11986876 equiv 119885119870119870119871119871119872119867119872119873119873119878119885119878

11986877 equiv 119885119870119862119870119871119871119872119867119872119873119873119878119885119878

11986878 equiv 119861119870119862119870119871119885119871

11986879 equiv 119861119870119862119870119871119862119871119872119885119872 11986880 equiv 119861119870119870119871119885119871

11986881 equiv 119861119870119870119871119871119872119885119872 11986882 equiv 119861119870119867119870119871119885119871

11986883 equiv 119861119870119870119871119885119871 11986884 equiv 119861119870119862119870119871119871119872119885119872

11986885 equiv 119861119870119862119870119871119867119871119872119885119872 11986886 equiv 119861119870119862119870119871119871119872119885119872

11986887 equiv 119861119870119870119871119867119871119872119885119872 11986888 equiv 119861119870119870119871119871119872119885119872

11986889 equiv 119861119870119867119870119871119871119872119885119872

11986890 equiv 119861119870119862119870119871119871119872119867119872119873119885119873

11986891 equiv 119861119870119862119870119871119871119872119872119873119885119873

11986892 equiv 119861119870119862119870119871119867119871119872119872119873119885119873

11986893 equiv 119861119870119870119871119867119871119872119872119873119885119873

11986894 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986895 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986896 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119885119878

11986897 equiv 119861119870119870119871119871119872119867119872119873119873119878119885119878

11986898 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119885119878

(44)




1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)



R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)




119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Σ = Σ119894119895119886119894119895119868119894119868119895 (119894 119895 = 1 2 4 7 9 11 21) 119886119894119895 = 119886119895119894

(35)


120597Σ

120597119868= 2 (11988611119868 + 11988612119868119868 + 1198861119896119868119896)

120597Σ

120597119868119868= 2 (11988612119868 + 11988622119868119868 + 1198862119896119868119896)

120597Σ

120597119868119898

= 2 (1198861198981119868 + 1198861198982119868119868 + 119886119898119896119868119896)

(119898 = 4 9 11 21) (119896 = 4 7 9 11 21)

(36)




119864119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877) + 12057222119867119875119877

(37)


119884119875119877 = 1205742119864119870119870120575119875119877 + 1205743119867119870119870120575119875119877 + 1205744119861119870119867119870119873119861119873120575119875119877

+ 1205745119885119870119867119870119873119885119873120575119875119877 + 1205746119861119875119861119877 + 1205747119864119870119870119861119875119861119877

+ 1205748119867119870119870119861119875119861119877 + 1205749119885119870119867119870119873119885119873119861119875119861119877 + 12057410119864119875119873119861119873119861119877

+ 12057411119885119875119885119877 + 12057412119864119870119870119885119875119885119877 + 12057413119867119870119870119885119875119885119877

+ 12057414119861119870119867119870119873119861119873119885119875119885119877 + 12057415119864119875119873119885119873119885119877

+ 12057416119861119870119885119870119861119875119885119877 + 12057417119861119870119864119870119873119885119873119861119875119885119877

+ 12057418119861119870119867119870119873119885119873119861119875119885119877 + 12057419119861119870119885119870119864119875119873119861119873119885119877 + 12057420119864119875119877

(38)








119863119879119870119871 (119872119869119875119872119873119877119862119875119877119872119869119875119872119873119877119875119877119872119869119875119872119873119877119867119875119877

119872119869119875119872119873119877119875119877119872119869119875119861119875119872119869119875119885119875 120579)

= 119872119870119875119872119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(39)


Consider

R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

equiv 119881119870119881119871 119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

(40)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

(41)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816119881119870=0

(42)



Ξ equiv Ξ (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870) (43)


1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 equiv 119870119870

1198685 equiv 119870119871119871119870 1198686 equiv 119870119871119871119872119872119870

1198687 equiv 119867119870119870 1198688 equiv 119870119870


1198689 equiv 119862119870119871119871119870 11986810 equiv 119862119870119871119867119871119870

11986811 equiv 119862119870119871119871119870 11986812 equiv 119870119871119867119871119870

11986813 equiv 119870119871119871119870 11986814 equiv 119867119870119871119871119870

11986815 equiv 119862119870119871119862119871119872119872119870 11986816 equiv 119862119870119871119862119871119872119867119872119870

11986817 equiv 119862119870119871119862119871119872119872119870 11986818 equiv 119862119870119871119871119872119872119870

11986819 equiv 119870119871119871119872119867119872119870 11986820 equiv 119870119871119871119872119872119870

11986821 equiv 119862119870119871119871119872119867119872119870 11986822 equiv 119862119870119871119871119872119872119870

11986823 equiv 119862119870119871119867119871119872119872119870 11986824 equiv 119862119870119871119871119872119872119870

11986825 equiv 119862119870119871119862119871119872119872119873119867119873119870 11986826 equiv 119862119870119871119862119871119872119872119873119873119870

11986827 equiv 119862119870119871119862119871119872119867119872119873119873119870 11986828 equiv 119870119871119871119872119862119872119873119867119873119870

11986829 equiv 119870119871119871119872119862119872119873119873119870 11986830 equiv 119870119871119871119872119867119872119873119873119870

11986831 equiv 119862119870119871119862119871119872119872119873119873119870 11986832 equiv 119862119870119871119871119872119867119872119873119873119870

11986833 equiv 119861119870119861119870 11986834 equiv 119885119870119885119870

11986835 equiv 119861119870119885119870 11986836 equiv 119861119870119862119870119871119861119871 = 1205822

119887

11986837 equiv 119861119870119862119870119871119862119871119872119861119872 11986838 equiv 119861119870119870119871119861119871

11986839 equiv 119861119870119870119871119871119872119861119872 11986840 equiv 119861119870119867119870119871119861119871

11986841 equiv 119861119870119870119871119861119871 11986842 equiv 119861119870119862119870119871119871119872119861119872

11986843 equiv 119861119870119862119870119871119867119871119872119861119872 11986844 equiv 119861119870119862119870119871119871119872119861119872

11986845 equiv 119861119870119870119871119867119871119872119861119872 11986846 equiv 119861119870119870119871119871119872119861119872

11986847 equiv 119861119870119867119870119871119871119872119861119872

11986848 equiv 119861119870119862119870119871119871119872119867119872119873119861119873

11986849 equiv 119861119870119862119870119871119871119872119872119873119861119873

11986850 equiv 119861119870119862119870119871119867119871119872119872119873119861119873

11986851 equiv 119861119870119870119871119867119871119872119872119873119861119873

11986852 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11986853 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986854 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119861119878

11986855 equiv 119861119870119870119871119871119872119867119872119873119873119878119861119878

11986856 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119861119878

11986857 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986858 equiv 119885119870119862119870119871119862119871119872119885119872

11986859 equiv 119885119870119870119871119885119871 11986860 equiv 119885119870119870119871119871119872119885119872

11986861 equiv 119885119870119867119870119871119885119871 11986862 equiv 119885119870119870119871119885119871

11986863 equiv 119885119870119862119870119871119871119872119885119872 11986864 equiv 119885119870119862119870119871119867119871119872119885119872

11986865 equiv 119885119870119862119870119871119871119872119885119872 11986866 equiv 119885119870119870119871119867119871119872119885119872

11986867 equiv 119885119870119870119871119871119872119885119872 11986868 equiv 119885119870119867119870119871119871119872119885119872

11986869 equiv 119885119870119862119870119871119871119872119867119872119873119885119873

11986870 equiv 119885119870119862119870119871119871119872119872119873119885119873

11986871 equiv 119885119870119862119870119871119867119871119872119872119873119885119873

11986872 equiv 119885119870119870119871119867119871119872119872119873119885119873

11986873 equiv 119885119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986874 equiv 119885119870119862119870119871119862119871119872119872119873119873119878119885119878

11986875 equiv 119885119870119870119871119871119872119867119872119873119862119873119878119885119878

11986876 equiv 119885119870119870119871119871119872119867119872119873119873119878119885119878

11986877 equiv 119885119870119862119870119871119871119872119867119872119873119873119878119885119878

11986878 equiv 119861119870119862119870119871119885119871

11986879 equiv 119861119870119862119870119871119862119871119872119885119872 11986880 equiv 119861119870119870119871119885119871

11986881 equiv 119861119870119870119871119871119872119885119872 11986882 equiv 119861119870119867119870119871119885119871

11986883 equiv 119861119870119870119871119885119871 11986884 equiv 119861119870119862119870119871119871119872119885119872

11986885 equiv 119861119870119862119870119871119867119871119872119885119872 11986886 equiv 119861119870119862119870119871119871119872119885119872

11986887 equiv 119861119870119870119871119867119871119872119885119872 11986888 equiv 119861119870119870119871119871119872119885119872

11986889 equiv 119861119870119867119870119871119871119872119885119872

11986890 equiv 119861119870119862119870119871119871119872119867119872119873119885119873

11986891 equiv 119861119870119862119870119871119871119872119872119873119885119873

11986892 equiv 119861119870119862119870119871119867119871119872119872119873119885119873

11986893 equiv 119861119870119870119871119867119871119872119872119873119885119873

11986894 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986895 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986896 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119885119878

11986897 equiv 119861119870119870119871119871119872119867119872119873119873119878119885119878

11986898 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119885119878

(44)




1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)



R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)




119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119863119879119870119871 (119872119869119875119872119873119877119862119875119877119872119869119875119872119873119877119875119877119872119869119875119872119873119877119867119875119877

119872119869119875119872119873119877119875119877119872119869119875119861119875119872119869119875119885119875 120579)

= 119872119870119875119872119871119877 119863119879119875119877 (119862119869119873 119869119873 119867119869119873 119869119873 119861119869 119885119869 120579)

(39)


Consider

R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

equiv 119881119870119881119871 119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

(40)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

(41)


119863119879119870119871 (119862119875119877 119875119877 119867119875119877 119875119877 119861119875 119885119875)

=1

2

1205972R (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870 119881119870)

120597119881119870120597119881119871

10038161003816100381610038161003816100381610038161003816100381610038161003816119881119870=0

(42)



Ξ equiv Ξ (119862119870119871 119870119871 119867119870119871 119870119871 119861119870 119885119870) (43)


1198681 equiv 119862119870119870 1198682 equiv 119862119870119871119862119871119870

1198683 equiv 119862119870119871119862119871119872119862119872119870 1198684 equiv 119870119870

1198685 equiv 119870119871119871119870 1198686 equiv 119870119871119871119872119872119870

1198687 equiv 119867119870119870 1198688 equiv 119870119870


1198689 equiv 119862119870119871119871119870 11986810 equiv 119862119870119871119867119871119870

11986811 equiv 119862119870119871119871119870 11986812 equiv 119870119871119867119871119870

11986813 equiv 119870119871119871119870 11986814 equiv 119867119870119871119871119870

11986815 equiv 119862119870119871119862119871119872119872119870 11986816 equiv 119862119870119871119862119871119872119867119872119870

11986817 equiv 119862119870119871119862119871119872119872119870 11986818 equiv 119862119870119871119871119872119872119870

11986819 equiv 119870119871119871119872119867119872119870 11986820 equiv 119870119871119871119872119872119870

11986821 equiv 119862119870119871119871119872119867119872119870 11986822 equiv 119862119870119871119871119872119872119870

11986823 equiv 119862119870119871119867119871119872119872119870 11986824 equiv 119862119870119871119871119872119872119870

11986825 equiv 119862119870119871119862119871119872119872119873119867119873119870 11986826 equiv 119862119870119871119862119871119872119872119873119873119870

11986827 equiv 119862119870119871119862119871119872119867119872119873119873119870 11986828 equiv 119870119871119871119872119862119872119873119867119873119870

11986829 equiv 119870119871119871119872119862119872119873119873119870 11986830 equiv 119870119871119871119872119867119872119873119873119870

11986831 equiv 119862119870119871119862119871119872119872119873119873119870 11986832 equiv 119862119870119871119871119872119867119872119873119873119870

11986833 equiv 119861119870119861119870 11986834 equiv 119885119870119885119870

11986835 equiv 119861119870119885119870 11986836 equiv 119861119870119862119870119871119861119871 = 1205822

119887

11986837 equiv 119861119870119862119870119871119862119871119872119861119872 11986838 equiv 119861119870119870119871119861119871

11986839 equiv 119861119870119870119871119871119872119861119872 11986840 equiv 119861119870119867119870119871119861119871

11986841 equiv 119861119870119870119871119861119871 11986842 equiv 119861119870119862119870119871119871119872119861119872

11986843 equiv 119861119870119862119870119871119867119871119872119861119872 11986844 equiv 119861119870119862119870119871119871119872119861119872

11986845 equiv 119861119870119870119871119867119871119872119861119872 11986846 equiv 119861119870119870119871119871119872119861119872

11986847 equiv 119861119870119867119870119871119871119872119861119872

11986848 equiv 119861119870119862119870119871119871119872119867119872119873119861119873

11986849 equiv 119861119870119862119870119871119871119872119872119873119861119873

11986850 equiv 119861119870119862119870119871119867119871119872119872119873119861119873

11986851 equiv 119861119870119870119871119867119871119872119872119873119861119873

11986852 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11986853 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986854 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119861119878

11986855 equiv 119861119870119870119871119871119872119867119872119873119873119878119861119878

11986856 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119861119878

11986857 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986858 equiv 119885119870119862119870119871119862119871119872119885119872

11986859 equiv 119885119870119870119871119885119871 11986860 equiv 119885119870119870119871119871119872119885119872

11986861 equiv 119885119870119867119870119871119885119871 11986862 equiv 119885119870119870119871119885119871

11986863 equiv 119885119870119862119870119871119871119872119885119872 11986864 equiv 119885119870119862119870119871119867119871119872119885119872

11986865 equiv 119885119870119862119870119871119871119872119885119872 11986866 equiv 119885119870119870119871119867119871119872119885119872

11986867 equiv 119885119870119870119871119871119872119885119872 11986868 equiv 119885119870119867119870119871119871119872119885119872

11986869 equiv 119885119870119862119870119871119871119872119867119872119873119885119873

11986870 equiv 119885119870119862119870119871119871119872119872119873119885119873

11986871 equiv 119885119870119862119870119871119867119871119872119872119873119885119873

11986872 equiv 119885119870119870119871119867119871119872119872119873119885119873

11986873 equiv 119885119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986874 equiv 119885119870119862119870119871119862119871119872119872119873119873119878119885119878

11986875 equiv 119885119870119870119871119871119872119867119872119873119862119873119878119885119878

11986876 equiv 119885119870119870119871119871119872119867119872119873119873119878119885119878

11986877 equiv 119885119870119862119870119871119871119872119867119872119873119873119878119885119878

11986878 equiv 119861119870119862119870119871119885119871

11986879 equiv 119861119870119862119870119871119862119871119872119885119872 11986880 equiv 119861119870119870119871119885119871

11986881 equiv 119861119870119870119871119871119872119885119872 11986882 equiv 119861119870119867119870119871119885119871

11986883 equiv 119861119870119870119871119885119871 11986884 equiv 119861119870119862119870119871119871119872119885119872

11986885 equiv 119861119870119862119870119871119867119871119872119885119872 11986886 equiv 119861119870119862119870119871119871119872119885119872

11986887 equiv 119861119870119870119871119867119871119872119885119872 11986888 equiv 119861119870119870119871119871119872119885119872

11986889 equiv 119861119870119867119870119871119871119872119885119872

11986890 equiv 119861119870119862119870119871119871119872119867119872119873119885119873

11986891 equiv 119861119870119862119870119871119871119872119872119873119885119873

11986892 equiv 119861119870119862119870119871119867119871119872119872119873119885119873

11986893 equiv 119861119870119870119871119867119871119872119872119873119885119873

11986894 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986895 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986896 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119885119878

11986897 equiv 119861119870119870119871119871119872119867119872119873119873119878119885119878

11986898 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119885119878

(44)




1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)



R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)




119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198689 equiv 119862119870119871119871119870 11986810 equiv 119862119870119871119867119871119870

11986811 equiv 119862119870119871119871119870 11986812 equiv 119870119871119867119871119870

11986813 equiv 119870119871119871119870 11986814 equiv 119867119870119871119871119870

11986815 equiv 119862119870119871119862119871119872119872119870 11986816 equiv 119862119870119871119862119871119872119867119872119870

11986817 equiv 119862119870119871119862119871119872119872119870 11986818 equiv 119862119870119871119871119872119872119870

11986819 equiv 119870119871119871119872119867119872119870 11986820 equiv 119870119871119871119872119872119870

11986821 equiv 119862119870119871119871119872119867119872119870 11986822 equiv 119862119870119871119871119872119872119870

11986823 equiv 119862119870119871119867119871119872119872119870 11986824 equiv 119862119870119871119871119872119872119870

11986825 equiv 119862119870119871119862119871119872119872119873119867119873119870 11986826 equiv 119862119870119871119862119871119872119872119873119873119870

11986827 equiv 119862119870119871119862119871119872119867119872119873119873119870 11986828 equiv 119870119871119871119872119862119872119873119867119873119870

11986829 equiv 119870119871119871119872119862119872119873119873119870 11986830 equiv 119870119871119871119872119867119872119873119873119870

11986831 equiv 119862119870119871119862119871119872119872119873119873119870 11986832 equiv 119862119870119871119871119872119867119872119873119873119870

11986833 equiv 119861119870119861119870 11986834 equiv 119885119870119885119870

11986835 equiv 119861119870119885119870 11986836 equiv 119861119870119862119870119871119861119871 = 1205822

119887

11986837 equiv 119861119870119862119870119871119862119871119872119861119872 11986838 equiv 119861119870119870119871119861119871

11986839 equiv 119861119870119870119871119871119872119861119872 11986840 equiv 119861119870119867119870119871119861119871

11986841 equiv 119861119870119870119871119861119871 11986842 equiv 119861119870119862119870119871119871119872119861119872

11986843 equiv 119861119870119862119870119871119867119871119872119861119872 11986844 equiv 119861119870119862119870119871119871119872119861119872

11986845 equiv 119861119870119870119871119867119871119872119861119872 11986846 equiv 119861119870119870119871119871119872119861119872

11986847 equiv 119861119870119867119870119871119871119872119861119872

11986848 equiv 119861119870119862119870119871119871119872119867119872119873119861119873

11986849 equiv 119861119870119862119870119871119871119872119872119873119861119873

11986850 equiv 119861119870119862119870119871119867119871119872119872119873119861119873

11986851 equiv 119861119870119870119871119867119871119872119872119873119861119873

11986852 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11986853 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986854 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119861119878

11986855 equiv 119861119870119870119871119871119872119867119872119873119873119878119861119878

11986856 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119861119878

11986857 equiv 119885119870119862119870119871119885119871 = 1205822

119911

11986858 equiv 119885119870119862119870119871119862119871119872119885119872

11986859 equiv 119885119870119870119871119885119871 11986860 equiv 119885119870119870119871119871119872119885119872

11986861 equiv 119885119870119867119870119871119885119871 11986862 equiv 119885119870119870119871119885119871

11986863 equiv 119885119870119862119870119871119871119872119885119872 11986864 equiv 119885119870119862119870119871119867119871119872119885119872

11986865 equiv 119885119870119862119870119871119871119872119885119872 11986866 equiv 119885119870119870119871119867119871119872119885119872

11986867 equiv 119885119870119870119871119871119872119885119872 11986868 equiv 119885119870119867119870119871119871119872119885119872

11986869 equiv 119885119870119862119870119871119871119872119867119872119873119885119873

11986870 equiv 119885119870119862119870119871119871119872119872119873119885119873

11986871 equiv 119885119870119862119870119871119867119871119872119872119873119885119873

11986872 equiv 119885119870119870119871119867119871119872119872119873119885119873

11986873 equiv 119885119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986874 equiv 119885119870119862119870119871119862119871119872119872119873119873119878119885119878

11986875 equiv 119885119870119870119871119871119872119867119872119873119862119873119878119885119878

11986876 equiv 119885119870119870119871119871119872119867119872119873119873119878119885119878

11986877 equiv 119885119870119862119870119871119871119872119867119872119873119873119878119885119878

11986878 equiv 119861119870119862119870119871119885119871

11986879 equiv 119861119870119862119870119871119862119871119872119885119872 11986880 equiv 119861119870119870119871119885119871

11986881 equiv 119861119870119870119871119871119872119885119872 11986882 equiv 119861119870119867119870119871119885119871

11986883 equiv 119861119870119870119871119885119871 11986884 equiv 119861119870119862119870119871119871119872119885119872

11986885 equiv 119861119870119862119870119871119867119871119872119885119872 11986886 equiv 119861119870119862119870119871119871119872119885119872

11986887 equiv 119861119870119870119871119867119871119872119885119872 11986888 equiv 119861119870119870119871119871119872119885119872

11986889 equiv 119861119870119867119870119871119871119872119885119872

11986890 equiv 119861119870119862119870119871119871119872119867119872119873119885119873

11986891 equiv 119861119870119862119870119871119871119872119872119873119885119873

11986892 equiv 119861119870119862119870119871119867119871119872119872119873119885119873

11986893 equiv 119861119870119870119871119867119871119872119872119873119885119873

11986894 equiv 119861119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11986895 equiv 119861119870119870119871119871119872119867119872119873119862119873119878119861119878

11986896 equiv 119861119870119862119870119871119862119871119872119872119873119873119878119885119878

11986897 equiv 119861119870119870119871119871119872119867119872119873119873119878119885119878

11986898 equiv 119861119870119862119870119871119871119872119867119872119873119873119878119885119878

(44)




1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)



R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)




119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198701 equiv 119881119870119861119870 1198702 equiv 119881119870119885119870

1198703 equiv 119881119870119862119870119871119861119871 1198704 equiv 119881119870119862119870119871119885119871

1198705 equiv 119881119870119870119871119861119871 1198706 equiv 119881119870119870119871119885119871

1198707 equiv 119881119870119867119870119871119861119871 1198708 equiv 119881119870119867119870119871119885119871

1198709 equiv 119881119870119870119871119861119871 11987010 equiv 119881119870119870119871119885119871

11987011 equiv 119881119870119862119870119872119862119872119871119861119871 11987012 equiv 119881119870119862119870119872119862119872119871119885119871

11987013 equiv 119881119870119870119872119872119871119861119871 11987014 equiv 119881119870119870119872119872119871119885119871

11987015 equiv 119881119870119862119870119871119871119872119861119872 11987016 equiv 119881119870119862119870119871119871119872119885119872

11987017 equiv 119881119870119862119870119871119867119871119872119861119872 11987018 equiv 119881119870119862119870119871119867119871119872119885119872

11987019 equiv 119881119870119862119870119871119871119872119861119872 11987020 equiv 119881119870119862119870119871119871119872119885119872

11987021 equiv 119881119870119870119871119867119871119872119861119872

11987022 equiv 119881119870119870119871119867119871119872119885119872

11987023 equiv 119881119870119870119871119871119872119861119872

11987024 equiv 119881119870119870119871119871119872119885119872

11987025 equiv 119881119870119867119870119871119871119872119861119872

11987026 equiv 119881119870119867119870119871119871119872119885119872

11987027 equiv 119881119870119862119870119871119871119872119867119872119873119861119873

11987028 equiv 119881119870119862119870119871119871119872119867119872119873119885119873

11987029 equiv 119881119870119862119870119871119867119871119872119872119873119861119873

11987030 equiv 119881119870119862119870119871119867119871119872119872119873119885119873

11987031 equiv 119881119870119870119871119867119871119872119872119873119861119873

11987032 equiv 119881119870119870119871119867119871119872119872119873119885119873

11987033 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119861119878

11987034 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119885119878

11987035 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119861119878

11987036 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119885119878

11987037 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119861119878

11987038 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119885119878

11987039 equiv 119881119870119870119871119871119872119867119872119873119873119878119861119878

11987040 equiv 119881119870119870119871119871119872119867119872119873119873119878119885119878

11987041 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119861119878

11987042 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119885119878

11987043 equiv 119881119870119881119870 11987044 equiv 119881119870119862119870119871119881119871

11987045 equiv 119881119870119870119871119881119871 11987046 equiv 119881119870119867119870119871119881119871

11987047 equiv 119881119870119870119871119881119871 11987048 equiv 119881119870119862119870119871119862119871119872119881119872

11987049 equiv 119881119870119870119871119871119872119881119872 11987050 equiv 119881119870119862119870119871119871119872119881119872

11987051 equiv 119881119870119862119870119871119867119871119872119881119872 11987052 equiv 119881119870119862119870119871119871119872119881119872

11987053 equiv 119881119870119870119871119867119871119872119881119872

11987054 equiv 119881119870119870119871119871119872119881119872

11987055 equiv 119881119870119867119870119871119871119872119881119872

11987056 equiv 119881119870119862119870119871119871119872119867119872119873119881119873

11987057 equiv 119881119870119862119870119871119871119872119872119873119881119873

11987058 equiv 119881119870119862119870119871119867119871119872119872119873119881119873

11987059 equiv 119881119870119870119871119867119871119872119872119873119881119873

11987060 equiv 119881119870119862119870119871119862119871119872119872119873119867119873119878119881119878

11987061 equiv 119881119870119862119870119871119862119871119872119872119873119873119878119881119878

11987062 equiv 119881119870119870119871119871119872119867119872119873119862119873119878119881119878

11987063 equiv 119881119870119870119871119871119872119867119872119873119873119878119881119878

11987064 equiv 119881119870119862119870119871119871119872119867119872119873119873119878119881119878

(45)



R (119862 119860 119861 119881)

=

42

sum

120572=1

42

sum

120573=1

120582120572120573119870120572119870120573 + 120582011987043

+ 120582111987044 + sdot sdot sdot + 120582119898119870119899

119898 = 2 3 4 20 119899 = 45 46 47 64

(46)




119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119863119879119875119877 = 1205820120575119875119877 + 1205821119862119875119877 + 1205822119875119877 + 1205823119867119875119877

+ 1205824119875119877 + 12058211119861119875119861119877

+ 12058212 (119861119875119885119877 + 119885119875119861119877)

+ 12058213 (119861119875119861119871119862119871119877 + 119862119875119870119861119870119861119877)

+ 12058214 (119861119875119885119871119862119871119877 + 119862119875119870119885119870119861119877)

+ 12058215 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 12058216 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058217 (119861119875119861119871119867119871119877 + 119867119875119870119861119870119861119877)

+ 12058218 (119861119875119885119871119867119871119877 + 119867119875119870119885119870119861119877)

+ 12058219 (119861119875119861119871119871119877 + 119875119870119861119870119861119877)

+ 120582110 (119861119875119885119871119871119877 + 119875119870119885119870119861119877)

+ 12058222119885119875119885119877 + 12058223 (119885119875119861119871119862119871119877 + 119862119875119870119861119870119885119877)

+ 12058224 (119885119875119885119871119862119871119877 + 119862119875119870119885119870119885119877)

+ 12058225 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 12058226 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

+ 12058227 (119885119875119861119871119867119871119877 + 119867119875119870119861119870119885119877)

+ 12058228 (119885119875119885119871119867119871119877 + 119867119875119870119885119870119885119877)

+ 12058229 (119885119875119861119871119871119877 + 119875119870119861119870119885119877)

+ 120582210 (119885119875119885119871119871119877 + 119875119870119885119870119885119877)

(47)


1198691 = 3 + 2119864119870119871 1198692 = 2119870119870

1198693 = 119867119870119870 1198694 = 119870119870

1198695 = 119861119870119885119870 1198696 = 2119861119870119870119871119861119871

1198697 = 119861119870119867119870119871119861119871 1198698 = 119861119870119870119871119861119871

1198699 = 2119885119870119870119871119885119871 11986910 = 119885119870119867119870119871119885119871

11986911 = 119885119870119870119871119885119871 11986912 = 119861119870119885119870 + 2119861119870119864119870119871119861119871

11986913 = 2119861119870119870119871119885119871 11986914 = 119861119870119867119870119871119885119871

11986915 = 119861119870119870119871119885119871

(48)


120582120572 = 1205730 +

15

sum

119894=1

120573119894119869119894 0 le 120572 le 2 10 (49)




120573 = 119901 120572 = 19 rArr 120573 = 119903 120572 = 1 10 rArr 120573 = 119902 120572 = 22 rArr 120573 =

119900 120572 = 23 rArr 120573 = 119904 120572 = 24 rArr 120573 = s 120572 = 25 rArr 120573 = 119905 120572 =




119863119879119875119877 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ3119870119870120575119875119877

+ Γ4119867119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ13119875119877 + Γ14119867119875119877 + Γ15119875119877

+ Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ18119870119870119861119875119861119877 + Γ19119867119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ22119885119870119867119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)


+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ39119870119870119885119875119885119877 + Γ40119867119870119870119885119875119885119877

+ Γ41119870119870119885119875119885119877 + Γ42119861119870119870119871119861119871119885119875119885119877

+ Γ43119861119870119867119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(50)



0 = Γ1120575119875119877 + Γ2119864119870119870120575119875119877 + Γ4119867119870119870120575119875119877

+ Γ7119861119870119867119870119871119861119871120575119875119877 + Γ10119885119870119867119870119871119885119871120575119875119877 + Γ12119864119875119877

+ Γ14119867119875119877 + Γ16119861119875119861119877 + Γ17119864119870119870119861119875119861119877

+ Γ19119867119870119870119861119875119861119877 + Γ22119885119870119867119870119871119885119871119861119875119861119877

+ Γ24119861119870119885119870 (119861119875119885119877 + 119885119875119861119877)

+ Γ25119861119870119864119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ27119861119870119867119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ29 (119861119875119861119871119864119871119875 + 119864119875119871119861119871119861119877)

+ Γ30119861119870119885119870 (119861119875119885119871119864119871119875 + 119864119875119871119885119871119861119877)

+ Γ33 (119861119875119861119871119867119871119875 + 119867119875119871119861119871119861119877)

+ Γ34119861119870119885119870 (119861119875119885119871119867119871119875 + 119867119875119871119885119871119861119877)

+ Γ37119885119875119885119877 + Γ38119864119870119870119885119875119885119877

+ Γ40119867119870119870119885119875119885119877 + Γ43119861119870119867119870119871119861119871119885119875119885119877

+ Γ45119861119870119885119870 (119885119875119861119871119864119871119875 + 119864119875119871119861119871119885119877)

+ Γ46 (119885119875119885119871119864119871119875 + 119864119875119871119885119871119885119877)

+ Γ49119861119870119885119870 (119885119875119861119871119867119871119875 + 119867119875119871119861119871119885119877)

+ Γ50 (119885119875119885119871119867119871119875 + 119867119875119871119885119871119885119877)

(51)


1 2 4 7 10 12 14 16 17 19 22 24 25 27 29 30 33 34 37


119863119879119875119877 = Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877 + Γ6119861119870119870119871119861119871120575119875119877

+ Γ8119861119870119870119871119861119871120575119875119877 + Γ9119885119870119870119871119885119871120575119875119877

+ Γ11119885119870119870119871119885119871120575119875119877 + Γ13119875119877

+ Γ15119875119877 + Γ18119870119870119861119875119861119877 + Γ20119870119870119861119875119861119877

+ Γ21119885119870119870119871119885119871119861119875119861119877 + Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(52)




119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879119875119877 = minus119901119862minus1

119875119877 + Γ119887119861119875119861119877 + Γ119911119885119875119885119877 + 1205722119864119870119870120575119875119877

+ 1205723119867119870119870120575119875119877 + 1205724119861119870119867119870119873119861119873120575119875119877

+ 1205725119885119870119867119870119873119885119873120575119875119877 + 1205726119864119875119877 + 1205727119864119870119870119861119875119861119877

+ 1205728119867119870119870119861119875119861119877 + 1205729119885119870119867119870119873119885119873119861119875119861119877

+ 12057210 [119861119875119861119877 + (119861119875119861119871119864119871119877 + 119864119875119871119861119871119861119877)]

+ 12057211119864119870119870119885119875119885119877 + 12057212119867119870119870119885119875119885119877

+ 12057213119861119870119867119870119873119861119873119885119875119885119877

+ 12057214 [119885119875119885119877 + (119885119875119885119871119864119871119877 + 119864119875119871119885119871119885119877)]

+ 12057215119861119870119885119870119861119875119885119877 + 12057216119861119870119864119870119873119885119873119861119875119885119877

+ 12057217119861119870119867119870119873119885119873119861119875119885119877

+ 12057218119861119870119885119870 (119861119875119885119871119864119871119877 + 119864119875119871119861119871119885119877)

+ 12057219 (119861119875119861119871119867119871119877 + 119867119875119871119861119871119861119877)

+ 12057220 (119885119875119885119871119867119871119877 + 119867119875119871119885119871119885119877)

+ 12057221119861119870119885119870 (119861119875119885119871119867119871119877 + 119867119875119871119861119871119885119877)

+ 12057222119867119875119877 + Γ3119870119870120575119875119877 + Γ5119870119870120575119875119877

+ Γ6119861119870119870119871119861119871120575119875119877 + Γ8119861119870119870119871119861119871120575119875119877

+ Γ9119885119870119870119871119885119871120575119875119877 + Γ11119885119870119870119871119885119871120575119875119877

+ Γ13119875119877 + Γ15119875119877 + Γ18119870119870119861119875119861119877

+ Γ20119870119870119861119875119861119877 + Γ21119885119870119870119871119885119871119861119875119861119877

+ Γ23119885119870119870119871119885119871119861119875119861119877

+ Γ26119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ28119861119870119870119871119885119871 (119861119875119885119877 + 119885119875119861119877)

+ Γ31 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ32119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ35 (119861119875119861119871119871119875 + 119875119871119861119871119861119877)

+ Γ36119861119870119885119870 (119861119875119885119871119871119875 + 119875119871119885119871119861119877)

+ Γ39119870119870119885119875119885119877 + Γ41119870119870119885119875119885119877

+ Γ42119861119870119870119871119861119871119885119875119885119877 + Γ44119861119870119870119871119861119871119885119875119885119877

+ Γ47119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ48 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

+ Γ51119861119870119885119870 (119885119875119861119871119871119875 + 119875119871119861119871119885119877)

+ Γ52 (119885119875119885119871119871119875 + 119875119871119885119871119885119877)

(53)













References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article On Continuum Damage Modeling of Fiber