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last edited 7 years ago by Bill Page |
Edit detail for SandBoxObserverAsIdempotent2 revision 17 of 18
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Editor: Bill Page
Time: 2017/04/18 21:25:28 GMT+0 |
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| Note: cleanup | ||
added:
Brackets
\begin{axiom}
p*q - q*p
p'*q - q'*p
p'*q' - q'*p'
\end{axiom}
Obs(2) is a 4 dimensional Frobenius Algebra
Generators of Obs(2)
fricas
)set output abbreviate on
V := OrderedVariableList [p,q]
![\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q ])](images/7148988253953892369-16.0px.png "\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q ])") | (1) |
Type: TYPE
fricas
M := FreeMonoid V
![\label{eq2}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ]))](images/6789939370406249697-16.0px.png "\label{eq2}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ]))") | (2) |
Type: TYPE
fricas
gens:List M := enumerate()$V
![\label{eq3}\left[ p , \: q \right]](images/8791685478538925680-16.0px.png "\label{eq3}\left[ p , \: q \right]") | (3) |
Type: LIST(FMONOID(OVAR([p,
Representation
fricas
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 ])))}](images/8129632100567326246-16.0px.png "\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
fricas
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
fricas
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
fricas
K := FRAC POLY 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 ])))](images/5915684182462122441-16.0px.png "\label{eq6}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ })) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ])))") | (6) |
Type: TYPE
fricas
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
fricas
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
fricas
m(x:M):Symbol == subscript('m,
Function declaration m : FMONOID(OVAR([p,
Type: VOID
fricas
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
fricas
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]](images/3713261661853592382-16.0px.png "\label{eq7}\left[ p , \: q , \:{p \ q}, \:{q \ p}\right]") | (7) |
Type: LIST(FMONOID(OVAR([p,
Idempotent
fricas
rule1(ij:MK):MK ==
for k in gens repeat
kk := divide(monomial(ij),
Function declaration rule1 : FM(FRAC(POLY(INT)),
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(monomial(ij),
Function declaration rule2 : FM(FRAC(POLY(INT)),
Type: VOID
An endomorphism on the K-Module is defined by the 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)),
Type: VOID
Matrix
Algebra is the free algebra product 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(2), gens(1))=γ(gens(1), gens(2))]), 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]))) -> FMONOID(OVAR([p, q]))
fricas
Compiling function coeff with type FM(FRAC(POLY(INT)),FMONOID(OVAR([ p, q]))) -> FRAC(POLY(INT))
fricas
Compiling function m with type FMONOID(OVAR([p,q])) -> SYMBOL
fricas
Compiling function γ with type (FMONOID(OVAR([p,q])), FMONOID(OVAR([p , q]))) -> SYMBOL
fricas
Compiling function leftDiv with type Union(Record(lm: FMONOID(OVAR([
p,fricas
Compiling function rightDiv with type Union(Record(lm: FMONOID(OVAR(
[p,fricas
Compiling function rule1 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([ p, q]))) -> FM(FRAC(POLY(INT)), FMONOID(OVAR([p, q])))
fricas
Compiling function rule2 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([ p, q]))) -> FM(FRAC(POLY(INT)), FMONOID(OVAR([p, q])))
fricas
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]](images/3386446109531113874-16.0px.png "\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)),
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}\ } ([ m [ p ] , m [ q ] , �� [ pp ] , �� [ pq ] , �� [ qp ] , �� [ qq ] ] , \hbox{\axiomType{INT}\ }))](images/5824725434220301926-16.0px.png "\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ m [ p ] , m [ q ] , �� [ pp ] , �� [ pq ] , �� [ qp ] , �� [ qq ] ] , \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])) -> LIST(LIST(FRAC( DMP([m[p], m[q], γ[pp], γ[pq], γ[qp], γ[qq]], INT)))) has been added to workspace.
Type: VOID
fricas
ss:=map(mat3,basis)
fricas
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]](images/5549623448471642623-16.0px.png "\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],
Algebra
fricas
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
fricas
A:=AlgebraGivenByStructuralConstants(R,#(basis)::PI, map(cats, basis), ss::Vector(Matrix R))
fricas
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 ] ] ])](images/4607386245546421185-16.0px.png "\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
fricas
alternative?()$A
algebra satisfies 2*associator(a,b, b) = 0 = 2*associator(a, a, b) = 0
 | (12) |
Type: BOOLEAN
fricas
antiAssociative?()$A
algebra is not anti-associative
 | (13) |
Type: BOOLEAN
fricas
antiCommutative?()$A
algebra is not anti-commutative
 | (14) |
Type: BOOLEAN
fricas
associative?()$A
algebra is associative
 | (15) |
Type: BOOLEAN
fricas
commutative?()$A
algebra is not commutative
 | (16) |
Type: BOOLEAN
fricas
flexible?()$A
algebra is flexible
 | (17) |
Type: BOOLEAN
fricas
jacobiIdentity?()$A
Jacobi identity does not hold
 | (18) |
Type: BOOLEAN
fricas
jordanAdmissible?()$A
algebra is not Jordan admissible
 | (19) |
Type: BOOLEAN
fricas
jordanAlgebra?()$A
algebra is not commutative this is not a Jordan algebra
 | (20) |
Type: BOOLEAN
fricas
leftAlternative?()$A
algebra is left alternative
 | (21) |
Type: BOOLEAN
fricas
lieAdmissible?()$A
algebra is Lie admissible
 | (22) |
Type: BOOLEAN
fricas
lieAlgebra?()$A
algebra is not anti-commutative this is not a Lie algebra
 | (23) |
Type: BOOLEAN
fricas
--powerAssociative?()$A rightAlternative?()$A
algebra is right alternative
 | (24) |
Type: BOOLEAN
Check Multiplication
fricas
AB := entries basis()$A
![\label{eq25}\left[ p , \: q , \: pq , \: qp \right]](images/4914048732794342905-16.0px.png "\label{eq25}\left[ p , \: q , \: pq , \: qp \right]") | (25) |
Type: LIST(ALGSC(FRAC(DMP([m[p],
fricas
A2MK(z:A):MK==reduce(+,map((x:R, y:M):MK+->(x::K)*y, coordinates(z), basis))
Function declaration A2MK : ALGSC(FRAC(DMP([m[p],m[q], γ[pp], γ[pq], γ[ qp], γ[qq]], INT)), 4, [p, q, pq, qp], [[[m[p], 0, 0, m[p]*m[q]*γ[pq]^2], [0, 0, 0, 0], [m[p]*m[q]*γ[pq]^2, 0, 0, m[p]*m[q]^2*γ[pq]^2], [0, 0, 0, 0]], [[0 , 0, 0, 0], [0, m[q], m[p]*m[q]*γ[pq]^2, 0], [0, 0, 0, 0], [0, m[p]*m[q]*γ[pq] ^2, m[p]^2*m[q]*γ[pq]^2, 0]], [[0, 1, m[p], 0], [0, 0, 0, 0], [0, m[q], m[p]*m [q]*γ[pq]^2, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [1, 0, 0, m[q]], [0, 0, 0, 0], [m[p] , 0, 0, m[p]*m[q]*γ[pq]^2]]]) -> FM(FRAC(POLY(INT)), FMONOID(OVAR([p, q]))) has been added to workspace.
Type: VOID
fricas
test(MT=map(x+->map(A2MK,x), [[i*j for j in AB] for i in AB]))
fricas
Compiling function A2MK with type ALGSC(FRAC(DMP([m[p],m[q], γ[pp], γ[ pq], γ[qp], γ[qq]], INT)), 4, [p, q, pq, qp], [[[m[p], 0, 0, m[p]*m[q]*γ[pq]^ 2], [0, 0, 0, 0], [m[p]*m[q]*γ[pq]^2, 0, 0, m[p]*m[q]^2*γ[pq]^2], [0, 0, 0, 0 ]], [[0, 0, 0, 0], [0, m[q], m[p]*m[q]*γ[pq]^2, 0], [0, 0, 0, 0], [0, m[p]*m[q] *γ[pq]^2, m[p]^2*m[q]*γ[pq]^2, 0]], [[0, 1, m[p], 0], [0, 0, 0, 0], [0, m[q], m[p]*m[q]*γ[pq]^2, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [1, 0, 0, m[q]], [0, 0, 0, 0] , [m[p], 0, 0, m[p]*m[q]*γ[pq]^2]]]) -> FM(FRAC(POLY(INT)), FMONOID( OVAR([p, q])))
 | (26) |
Type: BOOLEAN
Trace
fricas
[rightTrace(i)$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]](images/2226356419064648495-16.0px.png "\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],
fricas
[leftTrace(i)$A for i in AB]
![\label{eq28}\left[{2 \ {m_{p}}}, \:{2 \ {m_{q}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}\right]](images/6360403875611301140-16.0px.png "\label{eq28}\left[{2 \ {m_{p}}}, \:{2 \ {m_{q}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}\right]") | (28) |
Type: LIST(FRAC(DMP([m[p],
fricas
trace(i)==rightTrace(i) / #gens
Type: VOID
fricas
[trace(i) for i in AB]
fricas
Compiling function trace with type ALGSC(FRAC(DMP([m[p],m[q], γ[pp], γ [pq], γ[qp], γ[qq]], INT)), 4, [p, q, pq, qp], [[[m[p], 0, 0, m[p]*m[q]*γ[pq] ^2], [0, 0, 0, 0], [m[p]*m[q]*γ[pq]^2, 0, 0, m[p]*m[q]^2*γ[pq]^2], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, m[q], m[p]*m[q]*γ[pq]^2, 0], [0, 0, 0, 0], [0, m[p]*m[q ]*γ[pq]^2, m[p]^2*m[q]*γ[pq]^2, 0]], [[0, 1, m[p], 0], [0, 0, 0, 0], [0, m[q] , m[p]*m[q]*γ[pq]^2, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [1, 0, 0, m[q]], [0, 0, 0, 0 ], [m[p], 0, 0, m[p]*m[q]*γ[pq]^2]]]) -> FRAC(DMP([m[p], m[q], γ[pp], γ[ pq], γ[qp], γ[qq]], INT))
![\label{eq29}\left[{m_{p}}, \:{m_{q}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}\right]](images/4189989349366466049-16.0px.png "\label{eq29}\left[{m_{p}}, \:{m_{q}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}\right]") | (29) |
Type: LIST(FRAC(DMP([m[p],
fricas
p:=AB(1); q:=AB(2);
Type: ALGSC(FRAC(DMP([m[p],
fricas
test(p*p=trace(p)*p)
 | (30) |
Type: BOOLEAN
fricas
test(q*q=trace(q)*q)
 | (31) |
Type: BOOLEAN
Center
fricas
C:=basisOfCenter()$AlgebraPackage(R,A); # C
 | (32) |
Type: PI
fricas
c:=C(1)
 | (33) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
[c*i-i*c for i in AB]
![\label{eq34}\left[ 0, \: 0, \: 0, \: 0 \right]](images/3593100846523843307-16.0px.png "\label{eq34}\left[ 0, \: 0, \: 0, \: 0 \right]") | (34) |
Type: LIST(ALGSC(FRAC(DMP([m[p],
fricas
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}](images/1995256963800017520-16.0px.png "\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) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
test(c*c=c)
 | (36) |
Type: BOOLEAN
Unit
fricas
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}](images/8165389870526853569-16.0px.png "\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) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
trace(n)
 | (38) |
Type: FRAC(DMP([m[p],
fricas
test(n*n=n)
 | (39) |
Type: BOOLEAN
fricas
test(n=unit()$A)
 | (40) |
Type: BOOLEAN
fricas
f:=gcd map(x+->denom x,coordinates(n))
 | (41) |
Type: DMP([m[p],
fricas
--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](images/8693557477723880685-16.0px.png "\label{eq42}\left[
\begin{array}{cc}
1 &{��_{pq}}
\
{��_{pq}}& 1") | (42) |
Type: MATRIX(FRAC(POLY(INT)))
fricas
test(f = - determinant(ff))
 | (43) |
Type: BOOLEAN
fricas
(f*n)::OutputForm / f::OutputForm
 | (44) |
Type: OUTFORM
Lie bracket
fricas
p*q
 | (45) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
pq:=p*q-q*p
 | (46) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
trace(pq)
 | (47) |
Type: FRAC(DMP([m[p],
Orthogonal Observers
fricas
dual(p) == trace(p)*n - p
Type: VOID
fricas
--dual(p) == n - (1/trace(p))*p --dual(p) == n - p p' := dual p
fricas
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{eq48}\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}](images/1062555358702008566-16.0px.png "\label{eq48}\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}") | (48) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
trace p'
 | (49) |
Type: FRAC(DMP([m[p],
fricas
p'' := dual p'
 | (50) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
trace p''
 | (51) |
Type: FRAC(DMP([m[p],
fricas
test(p' * p' = trace(p')*p')
 | (52) |
Type: BOOLEAN
fricas
p * p'
 | (53) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
p' * p
 | (54) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
q' := dual q
![\label{eq55}\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}](images/6197194012320828539-16.0px.png "\label{eq55}\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}") | (55) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
trace(q')
 | (56) |
Type: FRAC(DMP([m[p],
fricas
test(q' * q' = trace(q')*q')
 | (57) |
Type: BOOLEAN
fricas
q * q'
 | (58) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
q' * q
 | (59) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
-- p' * q'
 | (60) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
p * q'
 | (61) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
p' * q
 | (62) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
-- q' * p'
 | (63) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
q * p'
 | (64) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
q' * p
 | (65) |
Type: ALGSC(FRAC(DMP([m[p],
Orthogonal Observers are Derivations
fricas
test(p'*(p*q) = (p'*p)*q + p*(p'*q))
 | (66) |
Type: BOOLEAN
fricas
test(q'*(p*q) = (q'*p)*q + p*(q'*q))
 | (67) |
Type: BOOLEAN
fricas
test((p*q)*p' = (p*p')*q + p*(q*p'))
 | (68) |
Type: BOOLEAN
fricas
test((p*q)*q' = (p*q')*q + p*(q*q'))
 | (69) |
Type: BOOLEAN
Brackets
fricas
p*q - q*p
 | (70) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
p'*q - q'*p
 | (71) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
p'*q' - q'*p'
 | (72) |
Type: ALGSC(FRAC(DMP([m[p],
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])
 | (73) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
trace(P)
![\label{eq74}{\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}}}](images/6576539199563318833-16.0px.png "\label{eq74}{\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}}}") | (74) |
Type: FRAC(DMP([m[p],
fricas
c:=1/trace(P)*P
![\label{eq75}\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}}}](images/4192891104519985914-16.0px.png "\label{eq75}\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}}}") | (75) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
c*c-c
![\label{eq76}\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}}}](images/5080657147482827202-16.0px.png "\label{eq76}\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}}}") | (76) |
Type: ALGSC(FRAC(DMP([m[p],
fricas
trace(c)
 | (77) |
Type: FRAC(DMP([m[p],