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SandBox Idempotent Observers ←	↑	→ SandBoxObserverAsIdempotent4

SandBoxObserverAsIdempotent2

Edit detail for SandBoxObserverAsIdempotent2 revision 2 of 18

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Editor: Bill Page Time: 2013/04/22 02:30:37 GMT+0
Note: display version

added:
)version

Obs(2) is a 4 dimensional Frobenius Algebra

axiom

)version

Value = "FriCAS 2012-10-18 compiled at Monday March 25, 2013 at 20:50:21 "

axiom

)set output abbreviate on

V := OrderedVariableList [p,q]

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

(1)

Type: TYPE

axiom

M := FreeMonoid V

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

(2)

Type: TYPE

axiom

gens:List M := enumerate()$V

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

(3)

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

axiom

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

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

(4)

Type: TYPE

axiom

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

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

Type: VOID

axiom

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

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

Type: VOID

axiom

K := FRAC POLY INT

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

(5)

Type: TYPE

axiom

MK := FreeModule(K,M)

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

(6)

Type: TYPE

axiom

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

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

Type: VOID

axiom

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

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

Type: VOID

axiom

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

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

Type: VOID

axiom

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

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

Type: VOID

Basis

axiom

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

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

(7)

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

Idempotent

axiom

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

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

Type: VOID

Reduction

axiom

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(base(ij),i*j*k)
        if ijk case divisible then
          ij := monom(leftDiv(ijk)*i*k*rightDiv(ijk), coeff(ij)*m(j)*γ(i,j)*γ(j,k)/γ(i,k))

  return(ij)

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

Type: VOID

An endomorphism on the K-Module is defined by the fixed point of applied rules

axiom

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

Type: VOID

Matrix

Algebra is the free algebra product modulo the fixed point

axiom

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

axiom

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

axiom

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

axiom

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

axiom

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

axiom

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

axiom

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

axiom

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

axiom

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

axiom

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

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[{\left[{{m_{p}}\ {��_{pp}}\ p}, \:{p \ q}, \:{{m_{p}}\ {��_{pp}}\ p \ q}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\ p}\right]}, \: \right. \ \ \displaystyle \left.{\left[{q \ p}, \:{{m_{q}}\ {��_{qq}}\ q}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\ q}, \:{{m_{q}}\ {��_{qq}}\ q \ p}\right]}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\ p}, \:{{m_{q}}\ {��_{qq}}\ p \ q}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\ p \ q}, \: \right. \ \ \displaystyle \left.{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}\ p}\right]$

(8)

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

Structure Constants

axiom

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

$\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ <em> 01 mp , </em> 01 mq , * 01 �� pp , * 01 �� pq , * 01 �� qp , * 01 �� qq ] , \hbox{\axiomType{INT}\ }))$

(9)

Type: TYPE

axiom

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])) -> LIST(LIST(FRAC(
      DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))) has been 
      added to workspace.

Type: VOID

axiom

ss:=map(mat3, basis)

axiom

Compiling function mat3 with type FMONOID(OVAR([p,q])) -> LIST(LIST(
      FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))))

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

(10)

Type: LIST(LIST(LIST(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))))

Algebra

axiom

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

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

Type: VOID

axiom

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

axiom

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

$\label{eq11}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ <em> 01 mp , </em> 01 mq , * 01 �� pp , * 01 �� pq , * 01 �� qp , * 01 �� qq ] , \hbox{\axiomType{INT}\ })) , 4, [ p , q , pq , qp ] , [ \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } ])$

(11)

Type: TYPE

axiom

alternative?()$A

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

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

(12)

Type: BOOLEAN

axiom

antiAssociative?()$A

   algebra is not anti-associative

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

(13)

Type: BOOLEAN

axiom

antiCommutative?()$A

   algebra is not anti-commutative

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

(14)

Type: BOOLEAN

axiom

associative?()$A

   algebra is associative

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

(15)

Type: BOOLEAN

axiom

commutative?()$A

   algebra is not commutative

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

(16)

Type: BOOLEAN

axiom

flexible?()$A

   algebra is flexible

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

(17)

Type: BOOLEAN

axiom

jacobiIdentity?()$A

   Jacobi identity does not hold

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

(18)

Type: BOOLEAN

axiom

jordanAdmissible?()$A

   algebra is not Jordan admissible

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

(19)

Type: BOOLEAN

axiom

jordanAlgebra?()$A

   algebra is not commutative
   this is not a Jordan algebra

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

(20)

Type: BOOLEAN

axiom

leftAlternative?()$A

   algebra is left alternative

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

(21)

Type: BOOLEAN

axiom

lieAdmissible?()$A

   algebra is Lie admissible

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

(22)

Type: BOOLEAN

axiom

lieAlgebra?()$A

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

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

(23)

Type: BOOLEAN

axiom

powerAssociative?()$A

   Internal Error
   The function powerAssociative? with signature hashcode is missing 
      from domain AlgebraGivenByStructuralConstants
      (Fraction (DistributedMultivariatePolynomial ((*01m p) (*01m q) (*01γ pp) (*01γ pq) (*01γ qp) (*01γ qq)) (Integer)))
      4(p q pq qp)UNPRINTABLE

Check Multiplication

axiom

AB := entries basis()$A

$\label{eq24}\left[ p , \: q , \: pq , \: qp \right]$

(24)

Type: LIST(ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX]))

axiom

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,*01γpp,*01γ
      pq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,
      MATRIX]) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q]))) has been 
      added to workspace.

Type: VOID

axiom

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

axiom

Compiling function A2MK with type ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp
      ,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,
      MATRIX]) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q])))

$\label{eq25} \mbox{\rm true}$

(25)

Type: BOOLEAN

Trace

axiom

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

$\label{eq26}\left[{2 \ {m_{p}}\ {��_{pp}}}, \:{2 \ {m_{q}}\ {��_{qq}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}\right]$

(26)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))

axiom

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

$\label{eq27}\left[{2 \ {m_{p}}\ {��_{pp}}}, \:{2 \ {m_{q}}\ {��_{qq}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}\right]$

(27)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))

axiom

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

Type: VOID

axiom

[trace(i) for i in AB]

axiom

Compiling function trace with type ALGSC(FRAC(DMP([*01mp,*01mq,*01γ
      pp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,
      MATRIX,MATRIX]) -> FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,
      *01γqq],INT))

$\label{eq28}\left[{{m_{p}}\ {��_{pp}}}, \:{{m_{q}}\ {��_{qq}}}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}\right]$

(28)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))

Center

axiom

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

$\label{eq29}1$

(29)

Type: PI

axiom

c:=C(1)

$\label{eq30}qp + pq -{{m_{p}}\ {��_{pp}}\ q}-{{m_{q}}\ {��_{qq}}\ p}$

(30)

Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])

axiom

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

$\label{eq31}\left[ 0, \: 0, \: 0, \: 0 \right]$

(31)

Type: LIST(ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX]))

axiom

test(c*c=c)

$\label{eq32} \mbox{\rm false}$

(32)

Type: BOOLEAN

Unit

axiom

rightTrace(c)

$\label{eq33}-{4 \ {m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{qq}}}+{4 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}$

(33)

Type: FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))

axiom

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

$\label{eq34}\begin{array}{@{}l} \displaystyle -{{1 \over{{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{qq}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}}}\ qp}- \ \ \displaystyle {{1 \over{{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{qq}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}}}\ pq}+{{{��_{pp}}\over{{{m_{q}}\ {��_{pp}}\ {��_{qq}}}-{{m_{q}}\ {��_{pq}}\ {��_{qp}}}}}\ q}+ \ \ \displaystyle {{{��_{qq}}\over{{{m_{p}}\ {��_{pp}}\ {��_{qq}}}-{{m_{p}}\ {��_{pq}}\ {��_{qp}}}}}\ p}$

(34)

Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])

axiom

trace(n)

$\label{eq35}2$

(35)

Type: FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))

axiom

test(n*n=n)

$\label{eq36} \mbox{\rm true}$

(36)

Type: BOOLEAN

axiom

test(n=unit()$A)

$\label{eq37} \mbox{\rm true}$

(37)

Type: BOOLEAN

axiom

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

$\label{eq38}{{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}$

(38)

Type: DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)

axiom

ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]

$\label{eq39}\left[ \begin{array}{cc} {��_{pp}}&{��_{pq}} \ {��_{qp}}&{��_{qq}}$

(39)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))

axiom

test(f = determinant(ff))

$\label{eq40} \mbox{\rm true}$

(40)

Type: BOOLEAN

axiom

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

$\label{eq41}{-{{1 \over{{m_{p}}\ {m_{q}}}}\ qp}-{{1 \over{{m_{p}}\ {m_{q}}}}\ pq}+{{{��_{pp}}\over{m_{q}}}\ q}+{{{��_{qq}}\over{m_{p}}}\ p}}\over{{{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}}$

(41)

Type: OUTFORM

Antisymmetric γ

axiom

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]

$\label{eq42}\left[{{��_{pp}}= 0}, \:{{��_{pq}}={��_{pq}}}, \:{{��_{qp}}= -{��_{pq}}}, \:{{��_{qq}}= 0}\right]$

(42)

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

axiom

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

   Function declaration anti : FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γ
      qp,*01γqq],INT)) -> FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,
      *01γqq],INT)) has been added to workspace.

Type: VOID

axiom

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

axiom

Compiling function anti with type FRAC(DMP([*01mp,*01mq,*01γpp,*01γ
      pq,*01γqp,*01γqq],INT)) -> FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,
      *01γqp,*01γqq],INT))

$\label{eq43}{-{{1 \over{{m_{p}}\ {m_{q}}}}\ qp}-{{1 \over{{m_{p}}\ {m_{q}}}}\ pq}}\over{{��_{pq}}^{2}}$

(43)

Type: OUTFORM

Momentum

axiom

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])

$\label{eq44}{{1 \over{��_{qp}}}\ qp}+{{1 \over{��_{pq}}}\ pq}+{{m_{q}}\ q}+{{m_{p}}\ p}$

(44)

Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])

axiom

trace(P)

$\label{eq45}{{{m_{p}}^{2}}\ {��_{pp}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}}+{{m_{p}}\ {m_{q}}\ {��_{qp}}}+{{{m_{q}}^{2}}\ {��_{qq}}}$

(45)

Type: FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))

axiom

c:=1/trace(P)*P

$\label{eq46}\begin{array}{@{}l} \displaystyle {{1 \over{{{{m_{p}}^{2}}\ {��_{pp}}\ {��_{qp}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}+{{m_{p}}\ {m_{q}}\ {{��_{qp}}^{2}}}+{{{m_{q}}^{2}}\ {��_{qp}}\ {��_{qq}}}}}\ qp}+ \ \ \displaystyle {{1 \over{{{{m_{p}}^{2}}\ {��_{pp}}\ {��_{pq}}}+{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}+{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{qq}}}}}\ pq}+ \ \ \displaystyle {{{m_{q}}\over{{{{m_{p}}^{2}}\ {��_{pp}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}}+{{m_{p}}\ {m_{q}}\ {��_{qp}}}+{{{m_{q}}^{2}}\ {��_{qq}}}}}\ q}+ \ \ \displaystyle {{{m_{p}}\over{{{{m_{p}}^{2}}\ {��_{pp}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}}+{{m_{p}}\ {m_{q}}\ {��_{qp}}}+{{{m_{q}}^{2}}\ {��_{qq}}}}}\ p}$

(46)

Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])

axiom

c*c-c

$\label{eq47}0$

(47)

Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])

axiom

trace(c)

$\label{eq48}1$

(48)

Type: FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))

Scalar Product

axiom

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

$\label{eq49}\left[ \begin{array}{cccc} {{{m_{p}}^{2}}\ {{��_{pp}}^{2}}}&{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}} \ {{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{q}}^{2}}\ {{��_{qq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}$

(49)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))