MathAction SandBoxObserverAsIdempotent2

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- SandBox
  - SandBoxObserverAsIdempotent
    - SandBoxObserverAsIdempotent2 <-- You are here.

SandBoxAlgebraOfObservers2

Obs(2) is a 4 dimensional Frobenius Algebra

Generators of Obs(2)

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)set output abbreviate on

V := OrderedVariableList [p,q]

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

(1)

Type: TYPE

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M := FreeMonoid V

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

(2)

Type: TYPE

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gens:List M := enumerate()$V

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

(3)

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

Representation

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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

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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

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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

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K := FRAC POLY INT

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

(5)

Type: TYPE

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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

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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

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

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

Type: VOID

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

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

Type: VOID

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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

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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

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

Type: VOID

Reduction

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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(monomial(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]))
      ) -> 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

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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

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--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(2), gens(1))=γ(gens(1), gens(2))]), monomial(mod(i*j)))$MK for j in basis] for i in basis]

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

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

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

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

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

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

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

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

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

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

(8)

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

Structure Constants

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

$\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ m [ p ] , m [ q ] , �� [ pp ] , �� [ pq ] , �� [ qp ] , �� [ qq ] ] , \hbox{\axiomType{INT}\ }))$

(9)

Type: TYPE

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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([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT)))) has been added to
      workspace.

Type: VOID

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ss:=map(mat3, basis)

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Compiling function mat3 with type FMONOID(OVAR([p,q])) -> LIST(LIST(
      FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT))))

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

(10)

Type: LIST(LIST(LIST(FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT)))))

Algebra

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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

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

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

$\label{eq11}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ m [ p ] , m [ q ] , �� [ pp ] , �� [ pq ] , �� [ qp ] , �� [ qq ] ] , \hbox{\axiomType{INT}\ })) , 4, [ p , q , pq , qp ] , [ [ [ m [ p ] , 0, 0, m [ p ] * m [ q ] * �� [ pq ]^2 ] , [ 0, 0, 0, 0 ] , [ m [ p ] * m [ q ] * �� [ pq ]^2, 0, 0, m [ p ] * m [ q ]^2 * �� [ pq ]^2 ] , [ 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0 ] , [ 0, m [ q ] , m [ p ] * m [ q ] * �� [ pq ]^2, 0 ] , [ 0, 0, 0, 0 ] , [ 0, m [ p ] * m [ q ] * �� [ pq ]^2, m [ p ]^2 * m [ q ] * �� [ pq ]^2, 0 ] ] , [ [ 0, 1, m [ p ] , 0 ] , [ 0, 0, 0, 0 ] , [ 0, m [ q ] , m [ p ] * m [ q ] * �� [ pq ]^2, 0 ] , [ 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0 ] , [ 1, 0, 0, m [ q ] ] , [ 0, 0, 0, 0 ] , [ m [ p ] , 0, 0, m [ p ] * m [ q ] * �� [ pq ]^2 ] ] ])$

(11)

Type: TYPE

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alternative?()$A

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

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

(12)

Type: BOOLEAN

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antiAssociative?()$A

   algebra is not anti-associative

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

(13)

Type: BOOLEAN

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antiCommutative?()$A

   algebra is not anti-commutative

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

(14)

Type: BOOLEAN

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associative?()$A

   algebra is associative

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

(15)

Type: BOOLEAN

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commutative?()$A

   algebra is not commutative

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

(16)

Type: BOOLEAN

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flexible?()$A

   algebra is flexible

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

(17)

Type: BOOLEAN

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jacobiIdentity?()$A

   Jacobi identity does not hold

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

(18)

Type: BOOLEAN

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jordanAdmissible?()$A

   algebra is not Jordan admissible

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

(19)

Type: BOOLEAN

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jordanAlgebra?()$A

   algebra is not commutative
   this is not a Jordan algebra

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

(20)

Type: BOOLEAN

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leftAlternative?()$A

   algebra is left alternative

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

(21)

Type: BOOLEAN

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lieAdmissible?()$A

   algebra is Lie admissible

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

(22)

Type: BOOLEAN

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lieAlgebra?()$A

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

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

(23)

Type: BOOLEAN

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--powerAssociative?()$A
rightAlternative?()$A

   algebra is right alternative

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

(24)

Type: BOOLEAN

Check Multiplication

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AB := entries basis()$A

$\label{eq25}\left[ p , \: q , \: pq , \: qp \right]$

(25)

Type: LIST(ALGSC(FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT)),4,[p,q,pq,qp],[[[m[p],0,0,m[p]*m[q]*γ[pq]^2],[0,0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,m[p]*m[q]^2*γ[pq]^2],[0,0,0,0]],[[0,0,0,0],[0,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0],[0,m[p]*m[q]*γ[pq]^2,m[p]^2*m[q]*γ[pq]^2,0]],[[0,1,m[p],0],[0,0,0,0],[0,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,m[q]],[0,0,0,0],[m[p],0,0,m[p]*m[q]*γ[pq]^2]]]))

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

   Function declaration A2MK : ALGSC(FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[
      qp],γ[qq]],INT)),4,[p,q,pq,qp],[[[m[p],0,0,m[p]*m[q]*γ[pq]^2],[0,
      0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,m[p]*m[q]^2*γ[pq]^2],[0,0,0,0]],[[0
      ,0,0,0],[0,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0],[0,m[p]*m[q]*γ[pq]
      ^2,m[p]^2*m[q]*γ[pq]^2,0]],[[0,1,m[p],0],[0,0,0,0],[0,m[q],m[p]*m
      [q]*γ[pq]^2,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,m[q]],[0,0,0,0],[m[p]
      ,0,0,m[p]*m[q]*γ[pq]^2]]]) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,
      q]))) has been added to workspace.

Type: VOID

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

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Compiling function A2MK with type ALGSC(FRAC(DMP([m[p],m[q],γ[pp],γ[
      pq],γ[qp],γ[qq]],INT)),4,[p,q,pq,qp],[[[m[p],0,0,m[p]*m[q]*γ[pq]^
      2],[0,0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,m[p]*m[q]^2*γ[pq]^2],[0,0,0,0
      ]],[[0,0,0,0],[0,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0],[0,m[p]*m[q]
      *γ[pq]^2,m[p]^2*m[q]*γ[pq]^2,0]],[[0,1,m[p],0],[0,0,0,0],[0,m[q],
      m[p]*m[q]*γ[pq]^2,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,m[q]],[0,0,0,0]
      ,[m[p],0,0,m[p]*m[q]*γ[pq]^2]]]) -> FM(FRAC(POLY(INT)),FMONOID(
      OVAR([p,q])))

$\label{eq26} \mbox{\rm true}$

(26)

Type: BOOLEAN

Trace

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

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

(27)

Type: LIST(FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT)))

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

$\label{eq28}\left[{2 \ {m_{p}}}, \:{2 \ {m_{q}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}\right]$

(28)

Type: LIST(FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT)))

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

Type: VOID

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[trace(i) for i in AB]

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Compiling function trace with type ALGSC(FRAC(DMP([m[p],m[q],γ[pp],γ
      [pq],γ[qp],γ[qq]],INT)),4,[p,q,pq,qp],[[[m[p],0,0,m[p]*m[q]*γ[pq]
      ^2],[0,0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,m[p]*m[q]^2*γ[pq]^2],[0,0,0,
      0]],[[0,0,0,0],[0,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0],[0,m[p]*m[q
      ]*γ[pq]^2,m[p]^2*m[q]*γ[pq]^2,0]],[[0,1,m[p],0],[0,0,0,0],[0,m[q]
      ,m[p]*m[q]*γ[pq]^2,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,m[q]],[0,0,0,0
      ],[m[p],0,0,m[p]*m[q]*γ[pq]^2]]]) -> FRAC(DMP([m[p],m[q],γ[pp],γ[
      pq],γ[qp],γ[qq]],INT))

$\label{eq29}\left[{m_{p}}, \:{m_{q}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}\right]$

(29)

Type: LIST(FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT)))

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p:=AB(1); q:=AB(2);

Type: ALGSC(FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT)),4,[p,q,pq,qp],[[[m[p],0,0,m[p]*m[q]*γ[pq]^2],[0,0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,m[p]*m[q]^2*γ[pq]^2],[0,0,0,0]],[[0,0,0,0],[0,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0],[0,m[p]*m[q]*γ[pq]^2,m[p]^2*m[q]*γ[pq]^2,0]],[[0,1,m[p],0],[0,0,0,0],[0,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,m[q]],[0,0,0,0],[m[p],0,0,m[p]*m[q]*γ[pq]^2]]])

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test(p*p=trace(p)*p)

$\label{eq30} \mbox{\rm true}$

(30)

Type: BOOLEAN

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test(q*q=trace(q)*q)

$\label{eq31} \mbox{\rm true}$

(31)

Type: BOOLEAN

Center

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

$\label{eq32}1$

(32)

Type: PI

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c:=C(1)

$\label{eq33}qp + pq -{{m_{p}}\ q}-{{m_{q}}\ p}$

(33)

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

$\label{eq34}\left[ 0, \: 0, \: 0, \: 0 \right]$

(34)

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c*c

$\label{eq35}\begin{array}{@{}l} \displaystyle {{\left({{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}-{{m_{p}}\ {m_{q}}}\right)}\ qp}+{{\left({{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}-{{m_{p}}\ {m_{q}}}\right)}\ pq}+ \ \ \displaystyle {{\left(-{{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}+{{{m_{p}}^{2}}\ {m_{q}}}\right)}\ q}+ \ \ \displaystyle {{\left(-{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}+{{m_{p}}\ {{m_{q}}^{2}}}\right)}\ p}$

(35)

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test(c*c=c)

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

(36)

Type: BOOLEAN

Unit

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n := #gens/trace(c) * c

$\label{eq37}\begin{array}{@{}l} \displaystyle {{1 \over{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}-{{m_{p}}\ {m_{q}}}}}\ qp}+{{1 \over{{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}-{{m_{p}}\ {m_{q}}}}}\ pq}- \ \ \displaystyle {{1 \over{{{m_{q}}\ {{��_{pq}}^{2}}}-{m_{q}}}}\ q}-{{1 \over{{{m_{p}}\ {{��_{pq}}^{2}}}-{m_{p}}}}\ p}$

(37)

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trace(n)

$\label{eq38}2$

(38)

Type: FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT))

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test(n*n=n)

$\label{eq39} \mbox{\rm true}$

(39)

Type: BOOLEAN

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test(n=unit()$A)

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

(40)

Type: BOOLEAN

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

$\label{eq41}{{��_{pq}}^{2}}- 1$

(41)

Type: DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT)

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

$\label{eq42}\left[ \begin{array}{cc} 1 &{��_{pq}} \ {��_{pq}}& 1$

(42)

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

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test(f = - determinant(ff))

$\label{eq43} \mbox{\rm true}$

(43)

Type: BOOLEAN

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

$\label{eq44}{{{1 \over{{m_{p}}\ {m_{q}}}}\ qp}+{{1 \over{{m_{p}}\ {m_{q}}}}\ pq}-{{1 \over{m_{q}}}\ q}-{{1 \over{m_{p}}}\ p}}\over{{{��_{pq}}^{2}}- 1}$

(44)

Type: OUTFORM

Orthogonal Observers

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dual(p) == trace(p)*n - p

Type: VOID

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--dual(p) == n - (1/trace(p))*p
p' := dual p

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Compiling function dual with type ALGSC(FRAC(DMP([m[p],m[q],γ[pp],γ[
      pq],γ[qp],γ[qq]],INT)),4,[p,q,pq,qp],[[[m[p],0,0,m[p]*m[q]*γ[pq]^
      2],[0,0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,m[p]*m[q]^2*γ[pq]^2],[0,0,0,0
      ]],[[0,0,0,0],[0,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0],[0,m[p]*m[q]
      *γ[pq]^2,m[p]^2*m[q]*γ[pq]^2,0]],[[0,1,m[p],0],[0,0,0,0],[0,m[q],
      m[p]*m[q]*γ[pq]^2,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,m[q]],[0,0,0,0]
      ,[m[p],0,0,m[p]*m[q]*γ[pq]^2]]]) -> ALGSC(FRAC(DMP([m[p],m[q],γ[
      pp],γ[pq],γ[qp],γ[qq]],INT)),4,[p,q,pq,qp],[[[m[p],0,0,m[p]*m[q]*
      γ[pq]^2],[0,0,0,0],[m[p]*m[q]*γ[pq]^2,0,0,m[p]*m[q]^2*γ[pq]^2],[0
      ,0,0,0]],[[0,0,0,0],[0,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0],[0,m[p
      ]*m[q]*γ[pq]^2,m[p]^2*m[q]*γ[pq]^2,0]],[[0,1,m[p],0],[0,0,0,0],[0
      ,m[q],m[p]*m[q]*γ[pq]^2,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,m[q]],[0,
      0,0,0],[m[p],0,0,m[p]*m[q]*γ[pq]^2]]])

$\label{eq45}\begin{array}{@{}l} \displaystyle {{1 \over{{{m_{q}}\ {{��_{pq}}^{2}}}-{m_{q}}}}\ qp}+{{1 \over{{{m_{q}}\ {{��_{pq}}^{2}}}-{m_{q}}}}\ pq}-{{{m_{p}}\over{{{m_{q}}\ {{��_{pq}}^{2}}}-{m_{q}}}}\ q}- \ \ \displaystyle {{{{��_{pq}}^{2}}\over{{{��_{pq}}^{2}}- 1}}\ p}$

(45)

fricas

trace p'

$\label{eq46}m_{p}$

(46)

Type: FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT))

fricas

p'' := dual p'

$\label{eq47}p$

(47)

fricas

trace p''

$\label{eq48}m_{p}$

(48)

Type: FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT))

fricas

test(p' * p' = trace(p')*p')

$\label{eq49} \mbox{\rm true}$

(49)

Type: BOOLEAN

fricas

p * p'

$\label{eq50}0$

(50)

fricas

p' * p

$\label{eq51}0$

(51)

fricas

q' := dual q

$\label{eq52}\begin{array}{@{}l} \displaystyle {{1 \over{{{m_{p}}\ {{��_{pq}}^{2}}}-{m_{p}}}}\ qp}+{{1 \over{{{m_{p}}\ {{��_{pq}}^{2}}}-{m_{p}}}}\ pq}-{{{{��_{pq}}^{2}}\over{{{��_{pq}}^{2}}- 1}}\ q}- \ \ \displaystyle {{{m_{q}}\over{{{m_{p}}\ {{��_{pq}}^{2}}}-{m_{p}}}}\ p}$

(52)

fricas

trace(q')

$\label{eq53}m_{q}$

(53)

Type: FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT))

fricas

test(q' * q' = trace(q')*q')

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

(54)

Type: BOOLEAN

fricas

q * q'

$\label{eq55}0$

(55)

fricas

q' * q

$\label{eq56}0$

(56)

fricas

p' * q'

$\label{eq57}{{1 \over{{{��_{pq}}^{2}}- 1}}\ qp}+{{{{��_{pq}}^{2}}\over{{{��_{pq}}^{2}}- 1}}\ pq}-{{{{m_{p}}\ {{��_{pq}}^{2}}}\over{{{��_{pq}}^{2}}- 1}}\ q}-{{{{m_{q}}\ {{��_{pq}}^{2}}}\over{{{��_{pq}}^{2}}- 1}}\ p}$

(57)

fricas

q' * p'

$\label{eq58}{{{{��_{pq}}^{2}}\over{{{��_{pq}}^{2}}- 1}}\ qp}+{{1 \over{{{��_{pq}}^{2}}- 1}}\ pq}-{{{{m_{p}}\ {{��_{pq}}^{2}}}\over{{{��_{pq}}^{2}}- 1}}\ q}-{{{{m_{q}}\ {{��_{pq}}^{2}}}\over{{{��_{pq}}^{2}}- 1}}\ p}$

(58)

fricas

p' * q

$\label{eq59}- pq +{{m_{p}}\ q}$

(59)

fricas

q * p'

$\label{eq60}- qp +{{m_{p}}\ q}$

(60)

fricas

p * q'

$\label{eq61}- pq +{{m_{q}}\ p}$

(61)

fricas

q' * p

$\label{eq62}- qp +{{m_{q}}\ p}$

(62)

Orthogonal Observers are Derivations if there are only two observers

fricas

test(p'*(p*q) = (p'*p)*q + p*(p'*q))

$\label{eq63} \mbox{\rm true}$

(63)

Type: BOOLEAN

fricas

test(q'*(p*q) = (q'*p)*q + p*(q'*q))

$\label{eq64} \mbox{\rm true}$

(64)

Type: BOOLEAN

fricas

test((p*q)*p' = (p*p')*q + p*(q*p'))

$\label{eq65} \mbox{\rm true}$

(65)

Type: BOOLEAN

fricas

test((p*q)*q' = (p*q')*q + p*(q*q'))

$\label{eq66} \mbox{\rm true}$

(66)

Type: BOOLEAN

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])

$\label{eq67}{{1 \over{��_{qp}}}\ qp}+{{1 \over{��_{pq}}}\ pq}+{{{m_{q}}\over{��_{qq}}}\ q}+{{{m_{p}}\over{��_{pp}}}\ p}$

(67)

fricas

trace(P)

$\label{eq68}{\left( \begin{array}{@{}l} \displaystyle {{{m_{p}}^{2}}\ {��_{qp}}\ {��_{qq}}}+{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}}+{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}+ \ \ \displaystyle {{{m_{q}}^{2}}\ {��_{pp}}\ {��_{qp}}}$

(68)

Type: FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT))

fricas

c:=1/trace(P)*P

$\label{eq69}\begin{array}{@{}l} \displaystyle {{{{��_{pp}}\ {��_{qq}}}\over{\left( \begin{array}{@{}l} \displaystyle {{{m_{p}}^{2}}\ {��_{qp}}\ {��_{qq}}}+{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}}+ \ \ \displaystyle {{m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}+{{{m_{q}}^{2}}\ {��_{pp}}\ {��_{qp}}}$

(69)

fricas

c*c-c

$\label{eq70}\begin{array}{@{}l} \displaystyle {{{-{{m_{p}}\ {m_{q}}\ {{��_{pp}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {{��_{qq}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {{��_{qp}}^{2}}\ {��_{qq}}}}\over{\left( \begin{array}{@{}l} \displaystyle {{{m_{p}}^{4}}\ {{��_{qp}}^{2}}\ {{��_{qq}}^{2}}}+{2 \ {{m_{p}}^{3}}\ {m_{q}}\ {��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qp}}\ {{��_{qq}}^{2}}}+ \ \ \displaystyle {2 \ {{m_{p}}^{3}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {{��_{qp}}^{2}}\ {{��_{qq}}^{2}}}+{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pp}}^{2}}\ {{��_{pq}}^{4}}\ {{��_{qq}}^{2}}}+ \ \ \displaystyle {2 \ {{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pp}}^{2}}\ {{��_{pq}}^{3}}\ {��_{qp}}\ {{��_{qq}}^{2}}}+ \ \ \displaystyle {{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pp}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}\ {{��_{qq}}^{2}}}+ \ \ \displaystyle {2 \ {{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {{��_{qp}}^{2}}\ {��_{qq}}}+{2 \ {m_{p}}\ {{m_{q}}^{3}}\ {{��_{pp}}^{2}}\ {{��_{pq}}^{2}}\ {��_{qp}}\ {��_{qq}}}+ \ \ \displaystyle {2 \ {m_{p}}\ {{m_{q}}^{3}}\ {{��_{pp}}^{2}}\ {��_{pq}}\ {{��_{qp}}^{2}}\ {��_{qq}}}+{{{m_{q}}^{4}}\ {{��_{pp}}^{2}}\ {{��_{qp}}^{2}}}$

(70)

fricas

trace(c)

$\label{eq71}1$

(71)

Type: FRAC(DMP([m[p],m[q],γ[pp],γ[pq],γ[qp],γ[qq]],INT))