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SandBoxObserverAsIdempotent

Edit detail for SandBoxObserverAsIdempotent revision 4 of 21

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Editor: Bill Page Time: 2014/01/29 02:36:30 GMT+0
Note:

changed:
-          --if i=k then
-          --  ij := (coeff(ij)*m(i)*m(j)*γ(i,j)*γ(j,i) ) * _
-          --        (leftDiv(ijk)*i*rightDiv(ijk))
-          --else
          if i=k then
            ij := (coeff(ij)*m(i)*m(j)*γ(i,j)*γ(j,i) ) * _
                  (leftDiv(ijk)*i*rightDiv(ijk))
          else

changed:
-MT := [[mod(i*j) for j in basis] for i in basis]
--MT := [[mod(i*j) for j in basis] for i in basis]

changed:
---MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3)),m(gens(1))=1,m(gens(2))=1,m(gens(3))=1]),monomial(mod(i*j)))$MK for j in basis] for i in basis]
MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]),monomial(mod(i*j)))$MK for j in basis] for i in basis]

changed:
---MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=0,γ(gens(2),gens(2))=0,γ(gens(3),gens(3))=0,m(gens(1))=1,m(gens(2))=1,m(gens(3))=1]),monomial(mod(i*j)))$MK for j in basis] for i in basis]
--MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=0,γ(gens(2),gens(2))=0,γ(gens(3),gens(3))=0,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3)),m(gens(1))=1,m(gens(2))=1,m(gens(3))=1]),monomial(mod(i*j)))$MK for j in basis] for i in basis]

added:
rightAlternative?()$A

removed:
-rightAlternative?()$A

added:
p:=AB(1)
q:=AB(2)
r:=AB(3)

added:
p*p
p*q*r

changed:
-ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]
-test(f = determinant(ff))
--ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]
ff:= matrix [[eval(γ(i,j)::R,[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]) for j in gens] for i in gens]
--ff:= matrix [[eval(γ(i,j)::R,[γ(gens(1),gens(1))=0,γ(gens(2),gens(2))=0,γ(gens(3),gens(3))=0,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]) for j in gens] for i in gens]
-determinant(ff)
test(f = %)

Obs(3) is a 9 dimensional Frobenius Algrebra

fricas

)set output abbreviate on

V := OrderedVariableList [p,q,r]

$\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q , r ])$

(1)

Type: TYPE

fricas

M := FreeMonoid V

$\label{eq2}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ]))$

(2)

Type: TYPE

fricas

gens:List M := enumerate()$V

$\label{eq3}\left[ p , \: q , \: r \right]$

(3)

Type: LIST(FMONOID(OVAR([p,q,r])))

fricas

divisible := Record(lm: M,rm: M)

$\label{eq4}\mbox{\rm \hbox{\axiomType{Record}\ } (lm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])) , rm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])))}$

(4)

Type: TYPE

fricas

leftDiv(k:Union(divisible,"failed")):M == (k::divisible).lm

   Function declaration leftDiv : Union(Record(lm: FMONOID(OVAR([p,q,r]
      )),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p,q,r])
      ) has been added to workspace.

Type: VOID

fricas

rightDiv(k:Union(divisible,"failed")):M == (k::divisible).rm

   Function declaration rightDiv : Union(Record(lm: FMONOID(OVAR([p,q,r
      ])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p,q,r]
      )) has been added to workspace.

Type: VOID

fricas

K := FRAC POLY INT

$\label{eq5}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ }))$

(5)

Type: TYPE

fricas

MK := FreeModule(K,M)

$\label{eq6}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ })) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])))$

(6)

Type: TYPE

fricas

coeff(x:MK):K == leadingCoefficient(x)

   Function declaration coeff : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FRAC(POLY(INT)) has been added to workspace.

Type: VOID

fricas

monomial(x:MK):M == leadingMonomial(x)

   Function declaration monomial : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q
      ,r]))) -> FMONOID(OVAR([p,q,r])) has been added to workspace.

Type: VOID

fricas

m(x:M):K == subscript('m,[retract(x)::Symbol])

   Function declaration m : FMONOID(OVAR([p,q,r])) -> FRAC(POLY(INT)) 
      has been added to workspace.

Type: VOID

fricas

γ(x:M,y:M):K == subscript('γ,[concat(string retract x, string retract y)::Symbol])

   Function declaration γ : (FMONOID(OVAR([p,q,r])),FMONOID(OVAR([p,q,r
      ]))) -> FRAC(POLY(INT)) has been added to workspace.

Type: VOID

Basis

fricas

basis := concat(gens,concat [[j*i for i in gens | i~=j] for j in gens])

$\label{eq7}\left[ p , \: q , \: r , \:{p \ q}, \:{p \ r}, \:{q \ p}, \:{q \ r}, \:{r \ p}, \:{r \ q}\right]$

(7)

Type: LIST(FMONOID(OVAR([p,q,r])))

Idempotent

fricas

rule1(ij:MK):MK ==
  for k in gens repeat
    kk := divide(monomial(ij),k*k)
    if kk case divisible then
      ij:=(coeff(ij) * m(k)*γ(k,k)) * (leftDiv(kk) * k * rightDiv(kk))
  return(ij)

   Function declaration rule1 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added 
      to workspace.

Type: VOID

Reduction

fricas

rule2(ij:MK):MK ==
  for i in gens repeat
    for j in gens | j ~= i repeat
      for k in gens | k ~= j repeat
        ijk:=divide(leadingMonomial(ij),i*j*k)
        if ijk case divisible then
          if i=k then
            ij := (coeff(ij)*m(i)*m(j)*γ(i,j)*γ(j,i) ) * _
                  (leftDiv(ijk)*i*rightDiv(ijk))
          else
            ij := (coeff(ij)*m(j)*γ(i,j)*γ(j,k) / γ(i,k) ) * _
                  (leftDiv(ijk)*i*k*rightDiv(ijk))
  return(ij)

   Function declaration rule2 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added 
      to workspace.

Type: VOID

Modulo fixed point of applied rules

fricas

mod(ij:MK):MK ==
  ijFix:MK := 1
  while ijFix~=ij repeat
    ijFix := ij
    ij := rule1(ij)
    ij := rule2(ij)
  return(ij)

   Function declaration mod : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))
      ) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added to
      workspace.

Type: VOID

Matrix

Multiplication is monoidal concatenation modulo the fixed point

fricas

--MT := [[mod(i*j) for j in basis] for i in basis]
-- idempotent
MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]),monomial(mod(i*j)))$MK for j in basis] for i in basis]

fricas

Compiling function monomial with type FM(FRAC(POLY(INT)),FMONOID(
      OVAR([p,q,r]))) -> FMONOID(OVAR([p,q,r]))

fricas

Compiling function coeff with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FRAC(POLY(INT))

fricas

Compiling function m with type FMONOID(OVAR([p,q,r])) -> FRAC(POLY(
      INT))

fricas

Compiling function γ with type (FMONOID(OVAR([p,q,r])),FMONOID(OVAR(
      [p,q,r]))) -> FRAC(POLY(INT))

fricas

Compiling function leftDiv with type Union(Record(lm: FMONOID(OVAR([
      p,q,r])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p
      ,q,r]))

fricas

Compiling function rightDiv with type Union(Record(lm: FMONOID(OVAR(
      [p,q,r])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([
      p,q,r]))

fricas

Compiling function rule1 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

fricas

Compiling function rule2 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

fricas

Compiling function mod with type FM(FRAC(POLY(INT)),FMONOID(OVAR([p,
      q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ p}, \:{p \ q}, \:{p \ r}, \:{{m_{p}}\ p \ q}, \:{{m_{p}}\ p \ r}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p}, \:{{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ p \ r}, \: \right. \ \ \displaystyle \left.{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}\ p}, \:{{{{m_{r}}\ {��_{pr}}\ {��_{qr}}}\over{��_{pq}}}\ p \ q}\right]$

(8)

Type: LIST(LIST(FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))))

Structure Constants

fricas

R:=FRAC DMP(concat [[m(i) for i in gens],concat [[γ(j,i) for i in gens] for j in gens]], INT)

$\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ 01 mp , 01 mq , 01 mr , 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� rp , * 0 1 �� rq , * 01 �� rr ] , \hbox{\axiomType{INT}\ }))$

(9)

Type: TYPE

fricas

mat3(y:M):List List R == map(z+->map(x+->coefficient(x,y)::FRAC POLY INT,z),MT)

   Function declaration mat3 : FMONOID(OVAR([p,q,r])) -> LIST(LIST(FRAC
      (DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr
      ,*01γrp,*01γrq,*01γrr],INT)))) has been added to workspace.

Type: VOID

fricas

ss:=map(mat3, basis)

fricas

Compiling function mat3 with type FMONOID(OVAR([p,q,r])) -> LIST(
      LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γ
      qq,*01γqr,*01γrp,*01γrq,*01γrr],INT))))

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{\left[{m_{p}}, \: 0, \: 0, \: 0, \: 0, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \: 0, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \: 0, \: 0, \: 0, \: 0, \:{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}, \: 0, \:{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}, \: 0 \right]$

(10)

Type: LIST(LIST(LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))))

Algebra

fricas

cats(m:M):Symbol==concat(map(x+->string(x.gen::Symbol),factors m))::Symbol

   Function declaration cats : FMONOID(OVAR([p,q,r])) -> SYMBOL has 
      been added to workspace.

Type: VOID

fricas

A:=AlgebraGivenByStructuralConstants(R,#(basis)::PI,map(cats,basis),ss::Vector(Matrix R))

fricas

Compiling function cats with type FMONOID(OVAR([p,q,r])) -> SYMBOL

$\label{eq11}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ 01 mp , 01 mq , 01 mr , 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� rp , * 01 �� rq , * 01 �� rr ] , \hbox{\axiomType{INT}\ })) , 9, [ p , q , r , pq , pr , qp , qr , rp , rq ] , [ \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } ])$

(11)

Type: TYPE

fricas

alternative?()$A

   algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0

$\label{eq12} \mbox{\rm true}$

(12)

Type: BOOLEAN

fricas

antiAssociative?()$A

   algebra is not anti-associative

$\label{eq13} \mbox{\rm false}$

(13)

Type: BOOLEAN

fricas

antiCommutative?()$A

   algebra is not anti-commutative

$\label{eq14} \mbox{\rm false}$

(14)

Type: BOOLEAN

fricas

associative?()$A

   algebra is associative

$\label{eq15} \mbox{\rm true}$

(15)

Type: BOOLEAN

fricas

commutative?()$A

   algebra is not commutative

$\label{eq16} \mbox{\rm false}$

(16)

Type: BOOLEAN

fricas

flexible?()$A

   algebra is flexible

$\label{eq17} \mbox{\rm true}$

(17)

Type: BOOLEAN

fricas

jacobiIdentity?()$A

   Jacobi identity does not hold

$\label{eq18} \mbox{\rm false}$

(18)

Type: BOOLEAN

fricas

jordanAdmissible?()$A

   algebra is not Jordan admissible

$\label{eq19} \mbox{\rm false}$

(19)

Type: BOOLEAN

fricas

jordanAlgebra?()$A

   algebra is not commutative
   this is not a Jordan algebra

$\label{eq20} \mbox{\rm false}$

(20)

Type: BOOLEAN

fricas

leftAlternative?()$A

   algebra is left alternative

$\label{eq21} \mbox{\rm true}$

(21)

Type: BOOLEAN

fricas

rightAlternative?()$A

   algebra is right alternative

$\label{eq22} \mbox{\rm true}$

(22)

Type: BOOLEAN

fricas

lieAdmissible?()$A

   algebra is Lie admissible

$\label{eq23} \mbox{\rm true}$

(23)

Type: BOOLEAN

fricas

lieAlgebra?()$A

   algebra is not anti-commutative
   this is not a Lie algebra

$\label{eq24} \mbox{\rm false}$

(24)

Type: BOOLEAN

fricas

powerAssociative?()$A

   Internal Error
   The function powerAssociative? with signature hashcode is missing 
      from domain AlgebraGivenByStructuralConstants
      (Fraction (DistributedMultivariatePolynomial ((*01m p) (*01m q) (*01m r) (*01γ pp) (*01γ pq) (*01γ pr) (*01γ qp) (*01γ qq) (*01γ qr) (*01γ rp) (*01γ rq) (*01γ rr)) (Integer)))
      9(p q r pq pr qp qr rp rq)UNPRINTABLE

Check Multiplication

fricas

AB := entries basis()$A

$\label{eq25}\left[ p , \: q , \: r , \: pq , \: pr , \: qp , \: qr , \: rp , \: rq \right]$

(25)

Type: LIST(ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]))

fricas

p:=AB(1)

$\label{eq26}p$

(26)

Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX])

fricas

q:=AB(2)

$\label{eq27}q$

(27)

fricas

r:=AB(3)

$\label{eq28}r$

(28)

fricas

A2MK(z:A):MK==reduce(+,map((x:R,y:M):MK+->(x::K)*y,coordinates(z),basis))

   Function declaration A2MK : ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp
      ,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9
      ,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX]) -> FM(FRAC(POLY(INT)),FMONOID(OVAR(
      [p,q,r]))) has been added to workspace.

Type: VOID

fricas

test(MT=map(x+->map(A2MK,x),[[i*j for j in AB] for i in AB]))

fricas

Compiling function A2MK with type ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,
      *01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],
      INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> FM(FRAC(POLY(INT)),
      FMONOID(OVAR([p,q,r])))

$\label{eq29} \mbox{\rm true}$

(29)

Type: BOOLEAN

fricas

p*p

$\label{eq30}{m_{p}}\ p$

(30)

fricas

p*q*r

$\label{eq31}{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ pr$

(31)

Trace

fricas

[rightTrace(i)$A for i in AB]

$\label{eq32}\begin{array}{@{}l} \displaystyle \left[{3 \ {m_{p}}}, \:{3 \ {m_{q}}}, \:{3 \ {m_{r}}}, \:{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}\right]$

(32)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

[leftTrace(i)$A for i in AB]

$\label{eq33}\begin{array}{@{}l} \displaystyle \left[{3 \ {m_{p}}}, \:{3 \ {m_{q}}}, \:{3 \ {m_{r}}}, \:{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}\right]$

(33)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

trace(i)==rightTrace(i) / #gens

Type: VOID

fricas

[trace(i) for i in AB]

fricas

Compiling function trace with type ALGSC(FRAC(DMP([*01mp,*01mq,*01mr
      ,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],
      INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> FRAC(DMP([*01mp,*01mq,
      *01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,
      *01γrr],INT))

$\label{eq34}\begin{array}{@{}l} \displaystyle \left[{m_{p}}, \:{m_{q}}, \:{m_{r}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \: \right. \ \ \displaystyle \left.{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}\right]$

(34)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

Scalar Product

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.1,AB.2,AB.3]]

$\label{eq35}\left[ \begin{array}{ccc} {{m_{p}}^{2}}&{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{q}}^{2}}&{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{r}}^{2}}$

(35)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.4,AB.5,AB.6]]

$\label{eq36}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(36)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.7,AB.8,AB.9]]

$\label{eq37}\left[ \begin{array}{ccc} {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}$

(37)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.1,AB.2,AB.3]]

$\label{eq38}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(38)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.4,AB.5,AB.6]]

$\label{eq39}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{4}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{4}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{4}}}$

(39)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.7,AB.8,AB.9]]

$\label{eq40}\left[ \begin{array}{ccc} {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(40)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.1,AB.2,AB.3]]

$\label{eq41}\left[ \begin{array}{ccc} {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}} \ {{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}$

(41)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.4,AB.5,AB.6]]

$\label{eq42}\left[ \begin{array}{ccc} {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(42)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.7,AB.8,AB.9]]

$\label{eq43}\left[ \begin{array}{ccc} {{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{4}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{4}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{4}}}$

(43)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

Center

fricas

C:=basisOfCenter()$AlgebraPackage(R,A); # C

$\label{eq44}1$

(44)

Type: PI

fricas

c:=C(1)

$\label{eq45}\begin{array}{@{}l} \displaystyle rq +{{{{{m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {��_{pr}}\ {��_{qr}}}}\over{{{m_{p}}\ {��_{pq}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {��_{pr}}\ {��_{qr}}}}}\ rp}+ qr + \ \ \displaystyle {{{-{{m_{r}}\ {��_{pq}}\ {��_{qr}}}+{{m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}}\over{{{m_{p}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{{m_{p}}\ {��_{pq}}\ {��_{qr}}}}}\ qp}+{{{{{m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {��_{pr}}\ {��_{qr}}}}\over{{{m_{p}}\ {��_{pq}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {��_{pr}}\ {��_{qr}}}}}\ pr}+ \ \ \displaystyle {{{-{{m_{r}}\ {��_{pq}}\ {��_{qr}}}+{{m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}}\over{{{m_{p}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{{m_{p}}\ {��_{pq}}\ {��_{qr}}}}}\ pq}+{{{-{{m_{q}}\ {{��_{pq}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {��_{qr}}}}\over{{{��_{pq}}\ {��_{pr}}}-{��_{qr}}}}\ r}+ \ \ \displaystyle {{{-{{m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{r}}\ {��_{qr}}}}\over{{{��_{pq}}\ {��_{pr}}}-{��_{qr}}}}\ q}+{{{-{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}+{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\over{{{m_{p}}\ {��_{pq}}\ {��_{pr}}}-{{m_{p}}\ {��_{qr}}}}}\ p}$

(45)

fricas

[c*i-i*c for i in AB]

$\label{eq46}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(46)

fricas

test(c*c=c)

$\label{eq47} \mbox{\rm false}$

(47)

Type: BOOLEAN

Unit

fricas

rightTrace(c)

$\label{eq48}{\left( \begin{array}{@{}l} \displaystyle -{9 \ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}+{{18}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}- \ \ \displaystyle {9 \ {m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}-{9 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}+{9 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}$

(48)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

n := #basis / rightTrace(c) * c

$\label{eq49}\begin{array}{@{}l} \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{\left( \begin{array}{@{}l} \displaystyle {{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+ \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}$

(49)

fricas

trace(n)

$\label{eq50}3$

(50)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

test(n*n=n)

$\label{eq51} \mbox{\rm true}$

(51)

Type: BOOLEAN

fricas

[n*i-i for i in AB]

$\label{eq52}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(52)

fricas

[i*n-i for i in AB]

$\label{eq53}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(53)

fricas

test(n=unit()$A)

$\label{eq54} \mbox{\rm true}$

(54)

Type: BOOLEAN

fricas

f:=gcd map(x+->denom x,coordinates(n))

$\label{eq55}{{��_{pq}}^{2}}-{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{��_{pr}}^{2}}+{{��_{qr}}^{2}}- 1$

(55)

Type: DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)

fricas

--ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]
ff:= matrix [[eval(γ(i,j)::R,[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]) for j in gens] for i in gens]

$\label{eq56}\left[ \begin{array}{ccc} 1 &{��_{pq}}&{��_{pr}} \ {��_{pq}}& 1 &{��_{qr}} \ {��_{pr}}&{��_{qr}}& 1$

(56)

Type: MATRIX(FRAC(POLY(INT)))

fricas

--ff:= matrix [[eval(γ(i,j)::R,[γ(gens(1),gens(1))=0,γ(gens(2),gens(2))=0,γ(gens(3),gens(3))=0,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]) for j in gens] for i in gens]
-determinant(ff)

$\label{eq57}{{��_{qr}}^{2}}-{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{��_{pr}}^{2}}+{{��_{pq}}^{2}}- 1$

(57)

Type: FRAC(POLY(INT))

fricas

test(f = %)

$\label{eq58} \mbox{\rm true}$

(58)

Type: BOOLEAN

fricas

(f*n)::OutputForm / f::OutputForm

$\label{eq59}{\left( \begin{array}{@{}l} \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+ \ \ \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+{{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+ \ \ \displaystyle {{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+{{{{{��_{pq}}^{2}}- 1}\over{m_{r}}}\ r}+{{{{{��_{pr}}^{2}}- 1}\over{m_{q}}}\ q}+{{{{{��_{qr}}^{2}}- 1}\over{m_{p}}}\ p}$

(59)

Type: OUTFORM

Symmetric γ

fricas

eqSymm:List Equation EXPR INT := concat [[(i>j => γ(i,j)=γ(i,j);i=j =>γ(i,j)=γ(i,j);γ(i,j)=γ(j,i)) for j in gens] for i in gens]

$\label{eq60}\begin{array}{@{}l} \displaystyle \left[{{��_{pp}}={��_{pp}}}, \:{{��_{pq}}={��_{pq}}}, \:{{��_{pr}}={��_{pr}}}, \:{{��_{qp}}={��_{pq}}}, \:{{��_{qq}}={��_{qq}}}, \:{{��_{qr}}={��_{qr}}}, \:{{��_{rp}}={��_{pr}}}, \right. \ \ \displaystyle \left.\:{{��_{rq}}={��_{qr}}}, \:{{��_{rr}}={��_{rr}}}\right]$

(60)

Type: LIST(EQ(EXPR(INT)))

fricas

symm(x:R):R == subst(x::EXPR INT, eqSymm)::FRAC POLY INT

   Function declaration symm : FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γ
      pq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)) -> 
      FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,
      *01γqr,*01γrp,*01γrq,*01γrr],INT)) has been added to workspace.

Type: VOID

fricas

(symm(f)*map(symm ,coordinates(n))::A)::OutputForm / symm(f)::OutputForm

fricas

Compiling function symm with type FRAC(DMP([*01mp,*01mq,*01mr,*01γpp
      ,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))
       -> FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γ
      qq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

$\label{eq61}{\left( \begin{array}{@{}l} \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+ \ \ \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+{{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+ \ \ \displaystyle {{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+{{{{{��_{pq}}^{2}}- 1}\over{m_{r}}}\ r}+{{{{{��_{pr}}^{2}}- 1}\over{m_{q}}}\ q}+{{{{{��_{qr}}^{2}}- 1}\over{m_{p}}}\ p}$

(61)

Type: OUTFORM

Antisymmetric γ

fricas

eqAnti:List Equation EXPR INT := concat [[(i>j => γ(i,j)=γ(i,j);i=j =>γ(i,j)=0;γ(i,j)=-γ(j,i)) for j in gens] for i in gens]

$\label{eq62}\begin{array}{@{}l} \displaystyle \left[{{��_{pp}}= 0}, \:{{��_{pq}}={��_{pq}}}, \:{{��_{pr}}={��_{pr}}}, \:{{��_{qp}}= -{��_{pq}}}, \:{{��_{qq}}= 0}, \:{{��_{qr}}={��_{qr}}}, \: \right. \ \ \displaystyle \left.{{��_{rp}}= -{��_{pr}}}, \:{{��_{rq}}= -{��_{qr}}}, \:{{��_{rr}}= 0}\right]$

(62)

Type: LIST(EQ(EXPR(INT)))

fricas

anti(x:R):R == subst(x::EXPR INT, eqAnti)::FRAC POLY INT

   Function declaration anti : FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γ
      pq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)) -> 
      FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,
      *01γqr,*01γrp,*01γrq,*01γrr],INT)) has been added to workspace.

Type: VOID

fricas

(anti(f)*map(anti ,coordinates(n))::A)::OutputForm / anti(f)::OutputForm

fricas

Compiling function anti with type FRAC(DMP([*01mp,*01mq,*01mr,*01γpp
      ,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))
       -> FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γ
      qq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

$\label{eq63}{\left( \begin{array}{@{}l} \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+ \ \ \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+{{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+ \ \ \displaystyle {{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+{{{{{��_{pq}}^{2}}- 1}\over{m_{r}}}\ r}+{{{{{��_{pr}}^{2}}- 1}\over{m_{q}}}\ q}+{{{{{��_{qr}}^{2}}- 1}\over{m_{p}}}\ p}$

(63)

Type: OUTFORM

Momentum

fricas

P:=reduce(+,concat [[(1/γ(basis(i),basis(j)))::R*AB(i)*AB(j) for j in 1..size()$V] for i in 1..size()$V])

$\label{eq64}\begin{array}{@{}l} \displaystyle {{1 \over{��_{rq}}}\ rq}+{{1 \over{��_{rp}}}\ rp}+{{1 \over{��_{qr}}}\ qr}+{{1 \over{��_{qp}}}\ qp}+{{1 \over{��_{pr}}}\ pr}+{{1 \over{��_{pq}}}\ pq}+{{{m_{r}}\over{��_{rr}}}\ r}+ \ \ \displaystyle {{{m_{q}}\over{��_{qq}}}\ q}+{{{m_{p}}\over{��_{pp}}}\ p}$

(64)

fricas

trace(P)

$\label{eq65}{\left( \begin{array}{@{}l} \displaystyle {{{m_{p}}^{2}}\ {��_{qp}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+ \ \ \displaystyle {{m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+ \ \ \displaystyle {{m_{p}}\ {m_{r}}\ {��_{pp}}\ {{��_{pr}}^{2}}\ {��_{qp}}\ {��_{qq}}\ {��_{rq}}\ {��_{rr}}}+ \ \ \displaystyle {{m_{p}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{qp}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+{{{m_{q}}^{2}}\ {��_{pp}}\ {��_{qp}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+ \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{qp}}\ {��_{qq}}\ {{��_{qr}}^{2}}\ {��_{rp}}\ {��_{rr}}}+ \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{qp}}\ {��_{qq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+{{{m_{r}}^{2}}\ {��_{pp}}\ {��_{qp}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}}$

(65)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

c:=1/trace(P)*P

$\label{eq66}\begin{array}{@{}l} \displaystyle {{{{��_{pp}}\ {��_{qp}}\ {��_{qq}}\ {��_{rp}}\ {��_{rr}}}\over{\left( \begin{array}{@{}l} \displaystyle {{{m_{p}}^{2}}\ {��_{qp}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+ \ \ \displaystyle {{m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+ \ \ \displaystyle {{m_{p}}\ {m_{r}}\ {��_{pp}}\ {{��_{pr}}^{2}}\ {��_{qp}}\ {��_{qq}}\ {��_{rq}}\ {��_{rr}}}+ \ \ \displaystyle {{m_{p}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{qp}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+ \ \ \displaystyle {{{m_{q}}^{2}}\ {��_{pp}}\ {��_{qp}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+{{m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{qp}}\ {��_{qq}}\ {{��_{qr}}^{2}}\ {��_{rp}}\ {��_{rr}}}+ \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{qp}}\ {��_{qq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}\ {��_{rr}}}+ \ \ \displaystyle {{{m_{r}}^{2}}\ {��_{pp}}\ {��_{qp}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}}$

(66)

fricas

c*c-c

$\label{eq67}$

(67)

fricas

trace(c)

$\label{eq68}1$

(68)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

Scalar Product

fricas

S := matrix [[trace(x*y) for y in AB] for x in AB]

$\label{eq69}\left[ \begin{array}{ccccccccc} {{m_{p}}^{2}}&{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{q}}^{2}}&{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{r}}^{2}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{4}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{4}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{4}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{4}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{4}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{4}}}$

(69)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

determinant(S)/f^(2*#gens)

$\label{eq70}-{{{m_{p}}^{10}}\ {{m_{q}}^{10}}\ {{m_{r}}^{10}}\ {{��_{pq}}^{4}}\ {{��_{pr}}^{4}}\ {{��_{qr}}^{4}}}$

(70)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

C:=map(x+->factor(numer(x))/factor(denom(x)),inverse(S/f^2))

\label{eq71}\left[
\begin{array}{ccccccccc}
{{{{\left({��_{qr}}- 1 \right)}^{2}}\ {{\left({��_{qr}}+ 1 \right)}^{2}}}\over{{m_{p}}^{2}}}&{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}^{2}}\over{{m_{q}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}^{2}}\over{{m_{r}}\ {m_{p}}}}&{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}}\over{{��_{pq}}\ {m_{q}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {{m_{p}}^{2}}}}&{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}}\over{{��_{pq}}\ {m_{q}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}
\
{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}^{2}}\over{{m_{q}}\ {m_{p}}}}&{{{{\left({��_{pr}}- 1 \right)}^{2}}\ {{\left({��_{pr}}+ 1 \right)}^{2}}}\over{{m_{q}}^{2}}}&{{{\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}^{2}}\over{{m_{r}}\ {m_{q}}}}&{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}}\over{{��_{pq}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}&{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}}\over{{��_{pq}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {{m_{q}}^{2}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {{m_{q}}^{2}}}}
\
{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}^{2}}\over{{m_{r}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}^{2}}\over{{m_{r}}\ {m_{q}}}}&{{{{\left({��_{pq}}- 1 \right)}^{2}}\ {{\left({��_{pq}}+ 1 \right)}^{2}}}\over{{m_{r}}^{2}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pq}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {{m_{r}}^{2}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pq}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {{m_{r}}^{2}}\ {m_{q}}}}& -{{{\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {{m_{r}}^{2}}\ {m_{p}}}}& -{{{\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {{m_{r}}^{2}}\ {m_{q}}}}
\
{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}}\over{{��_{pq}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}}\over{{��_{pq}}\ {{m_{q}}^{2}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pq}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}&{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}^{2}}\over{{{��_{pq}}^{2}}\ {{m_{q}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}}\over{{{��_{pq}}^{2}}\ {{m_{q}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}
\
-{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {{m_{r}}^{2}}\ {m_{p}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}^{2}}\over{{{��_{pr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}}\over{{{��_{pr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{p}}^{2}}}}&{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}
\
{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}}\over{{��_{pq}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}}\over{{��_{pq}}\ {{m_{q}}^{2}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pq}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}&{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}}\over{{{��_{pq}}^{2}}\ {{m_{q}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}^{2}}\over{{{��_{pq}}^{2}}\ {{m_{q}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}
\
-{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {{m_{q}}^{2}}}}& -{{{\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {{m_{r}}^{2}}\ {m_{q}}}}& -{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}^{2}}\over{{{��_{qr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{q}}^{2}}}}&{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}}\over{{{��_{qr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{q}}^{2}}}}
\
-{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {{m_{r}}^{2}}\ {m_{p}}}}& -{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({��_{qr}}- 1 \right)}\ {\left({��_{qr}}+ 1 \right)}\ {\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}}\over{{{��_{pr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}^{2}}\over{{{��_{pr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}
\
-{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {{m_{q}}^{2}}}}& -{{{\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {{m_{r}}^{2}}\ {m_{q}}}}& -{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}&{{{\left({��_{pq}}-{{��_{pr}}\ {��_{qr}}}\right)}\ {\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}&{{{\left({��_{pr}}- 1 \right)}\ {\left({��_{pr}}+ 1 \right)}\ {\left({��_{pq}}- 1 \right)}\ {\left({��_{pq}}+ 1 \right)}}\over{{{��_{qr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{q}}^{2}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{��_{pr}}\right)}\ {\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{pr}}}-{��_{qr}}\right)}^{2}}\over{{{��_{qr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{q}}^{2}}}}

(71)

Type: MATRIX(FRAC(FR(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))))