|
|
|
last edited 7 years ago by Bill Page |
Edit detail for SandBoxObserverAsIdempotent2 revision 7 of 18
| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ||
|
Editor: Bill Page
Time: 2014/03/14 03:34:52 GMT+0 |
||
| Note: Orthogonal Observers | ||
added: monomial(x:MK):M == leadingMonomial(x) 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] -- 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)), m(gens(1))=1, m(gens(2))=1]), monomial(mod(i*j)))$MK for j in basis] for i in basis] -- nilpotent --MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=0, γ(gens(2), gens(2))=0, γ(gens(2), gens(1))=γ(gens(1), gens(2)), m(gens(1))=1, m(gens(2))=1]), monomial(mod(i*j)))$MK for j in basis] for i in basis] added: Orthogonal Observers \begin{axiom} p:=AB(1) p*p-p q:=AB(2) q*q-q p*q p' := n-p trace(p') p' * p' - p' q' := n-q trace(q') q' * q' - q' p' * q' \end{axiom}
Obs(2) is a 4 dimensional Frobenius Algebra
This page uses FriCAS? as underlying system.
fricas
)version
>> System error: The variable *BUILD-VERSION* is unbound.
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 == leadingMonomial(x)
Function declaration monomial : FM(FRAC(POLY(INT)),FMONOID(OVAR([p, q ]))) -> FMONOID(OVAR([p, q])) has been added to workspace.
Type: VOID
fricas
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
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(base(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(base(ij),
return(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)), m(gens(1))=1, m(gens(2))=1]), monomial(mod(i*j)))$MK for j in basis] for i in basis]
fricas
Compiling function base with type FM(FRAC(POLY(INT)),FMONOID(OVAR([p , q]))) -> FMONOID(OVAR([p, q]))
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 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 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])))
fricas
Compiling function monomial with type FM(FRAC(POLY(INT)),FMONOID( OVAR([p, q]))) -> FMONOID(OVAR([p, q]))
![\label{eq8}\begin{array}{@{}l}
\displaystyle
\left[{\left[ p , \:{p \ q}, \:{p \ q}, \:{{{��_{pq}}^{2}}\ p}\right]}, \:{\left[{q \ p}, \: q , \:{{{��_{pq}}^{2}}\ q}, \:{q \ p}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{{{��_{pq}}^{2}}\ p}, \:{p \ q}, \:{{{��_{pq}}^{2}}\ p \ q}, \:{{{��_{pq}}^{2}}\ p}\right]}, \:{\left[{q \ p}, \:{{{��_{pq}}^{2}}\ q}, \:{{{��_{pq}}^{2}}\ q}, \:{{{��_{pq}}^{2}}\ q \ p}\right]}\right]](images/7446404652345645486-16.0px.png "\label{eq8}\begin{array}{@{}l}
\displaystyle
\left[{\left[ p , \:{p \ q}, \:{p \ q}, \:{{{��_{pq}}^{2}}\ p}\right]}, \:{\left[{q \ p}, \: q , \:{{{��_{pq}}^{2}}\ q}, \:{q \ p}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{{{��_{pq}}^{2}}\ p}, \:{p \ q}, \:{{{��_{pq}}^{2}}\ p \ q}, \:{{{��_{pq}}^{2}}\ p}\right]}, \:{\left[{q \ p}, \:{{{��_{pq}}^{2}}\ q}, \:{{{��_{pq}}^{2}}\ 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}\ } ([ <em> 01 mp , </em> 01 mq , * 01 �� pp , * 01 �� pq , * 01 �� qp , * 01 �� qq ] , \hbox{\axiomType{INT}\ }))](images/9024064143619987076-16.0px.png "\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
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([*01mp, *01mq, *01γpp, *01γpq, *01γqp, *01γqq], 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([*01mp, *01mq, *01γpp, *01γpq, *01γqp, *01γqq], INT))))
![\label{eq10}\begin{array}{@{}l}
\displaystyle
\left[{\left[{\left[ 1, \: 0, \: 0, \:{{��_{pq}}^{2}}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[{{��_{pq}}^{2}}, \: 0, \: 0, \:{{��_{pq}}^{2}}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 1, \:{{��_{pq}}^{2}}, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \:{{��_{pq}}^{2}}, \:{{��_{pq}}^{2}}, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 1, \: 1, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 1, \:{{��_{pq}}^{2}}, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 1, \: 0, \: 0, \: 1 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 1, \: 0, \: 0, \:{{��_{pq}}^{2}}\right]}\right]}\right]](images/695259835203015146-16.0px.png "\label{eq10}\begin{array}{@{}l}
\displaystyle
\left[{\left[{\left[ 1, \: 0, \: 0, \:{{��_{pq}}^{2}}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[{{��_{pq}}^{2}}, \: 0, \: 0, \:{{��_{pq}}^{2}}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 1, \:{{��_{pq}}^{2}}, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \:{{��_{pq}}^{2}}, \:{{��_{pq}}^{2}}, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 1, \: 1, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 1, \:{{��_{pq}}^{2}}, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 1, \: 0, \: 0, \: 1 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 1, \: 0, \: 0, \:{{��_{pq}}^{2}}\right]}\right]}\right]") | (10) |
Type: LIST(LIST(LIST(FRAC(DMP([*01mp,
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}\ } ([ <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}\ } ])](images/8680127660451763981-16.0px.png "\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
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?()$A
Internal Error The function powerAssociative? with signature hashcode is missing from domain AlgebraGivenByStructuralConstants (Fraction (DistributedMultivariatePolynomial ((*01m p) (*01m q) (*01γ pp) (*01γ pq) (*01γ qp) (*01γ qq)) (Integer))) 4(p q pq qp)UNPRINTABLE
Check Multiplication
fricas
AB := entries basis()$A
![\label{eq24}\left[ p , \: q , \: pq , \: qp \right]](images/3056819212626973450-16.0px.png "\label{eq24}\left[ p , \: q , \: pq , \: qp \right]") | (24) |
Type: LIST(ALGSC(FRAC(DMP([*01mp,
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, *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
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, *01γpp , *01γpq, *01γqp, *01γqq], INT)), 4, [p, q, pq, qp], [MATRIX, MATRIX, MATRIX, MATRIX]) -> FM(FRAC(POLY(INT)), FMONOID(OVAR([p, q])))
 | (25) |
Type: BOOLEAN
Trace
fricas
[rightTrace(i)$A for i in AB]
![\label{eq26}\left[ 2, \: 2, \:{2 \ {{��_{pq}}^{2}}}, \:{2 \ {{��_{pq}}^{2}}}\right]](images/3155229599196629679-16.0px.png "\label{eq26}\left[ 2, \: 2, \:{2 \ {{��_{pq}}^{2}}}, \:{2 \ {{��_{pq}}^{2}}}\right]") | (26) |
Type: LIST(FRAC(DMP([*01mp,
fricas
[leftTrace(i)$A for i in AB]
![\label{eq27}\left[ 2, \: 2, \:{2 \ {{��_{pq}}^{2}}}, \:{2 \ {{��_{pq}}^{2}}}\right]](images/3268604311637558992-16.0px.png "\label{eq27}\left[ 2, \: 2, \:{2 \ {{��_{pq}}^{2}}}, \:{2 \ {{��_{pq}}^{2}}}\right]") | (27) |
Type: LIST(FRAC(DMP([*01mp,
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, *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[ 1, \: 1, \:{{��_{pq}}^{2}}, \:{{��_{pq}}^{2}}\right]](images/2796986871387818131-16.0px.png "\label{eq28}\left[ 1, \: 1, \:{{��_{pq}}^{2}}, \:{{��_{pq}}^{2}}\right]") | (28) |
Type: LIST(FRAC(DMP([*01mp,
Center
fricas
C:=basisOfCenter()$AlgebraPackage(R,A); # C
 | (29) |
Type: PI
fricas
c:=C(1)
 | (30) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
[c*i-i*c for i in AB]
![\label{eq31}\left[ 0, \: 0, \: 0, \: 0 \right]](images/2555932230319657100-16.0px.png "\label{eq31}\left[ 0, \: 0, \: 0, \: 0 \right]") | (31) |
Type: LIST(ALGSC(FRAC(DMP([*01mp,
fricas
test(c*c=c)
 | (32) |
Type: BOOLEAN
Unit
fricas
rightTrace(c)
 | (33) |
Type: FRAC(DMP([*01mp,
fricas
n := #basis / rightTrace(c) * c
 | (34) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
trace(n)
 | (35) |
Type: FRAC(DMP([*01mp,
fricas
test(n*n=n)
 | (36) |
Type: BOOLEAN
fricas
test(n=unit()$A)
 | (37) |
Type: BOOLEAN
fricas
f:=gcd map(x+->denom x,coordinates(n))
 | (38) |
Type: DMP([*01mp,
fricas
ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]
![\label{eq39}\left[
\begin{array}{cc}
{��_{pp}}&{��_{pq}}
\
{��_{qp}}&{��_{qq}}](images/1526093921744674070-16.0px.png "\label{eq39}\left[
\begin{array}{cc}
{��_{pp}}&{��_{pq}}
\
{��_{qp}}&{��_{qq}}") | (39) |
Type: MATRIX(FRAC(DMP([*01mp,
fricas
test(f = determinant(ff))
 | (40) |
Type: BOOLEAN
fricas
(f*n)::OutputForm / f::OutputForm
 | (41) |
Type: OUTFORM
Orthogonal Observers
fricas
p:=AB(1)
 | (42) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
p*p-p
 | (43) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
q:=AB(2)
 | (44) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
q*q-q
 | (45) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
p*q
 | (46) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
p' := n-p
 | (47) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
trace(p')
 | (48) |
Type: FRAC(DMP([*01mp,
fricas
p' * p' - p'
 | (49) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
q' := n-q
 | (50) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
trace(q')
 | (51) |
Type: FRAC(DMP([*01mp,
fricas
q' * q' - q'
 | (52) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
p' * q'
 | (53) |
Type: ALGSC(FRAC(DMP([*01mp,
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{eq54}\left[{{��_{pp}}= 0}, \:{{��_{pq}}={��_{pq}}}, \:{{��_{qp}}= -{��_{pq}}}, \:{{��_{qq}}= 0}\right]](images/3430539307425571049-16.0px.png "\label{eq54}\left[{{��_{pp}}= 0}, \:{{��_{pq}}={��_{pq}}}, \:{{��_{qp}}= -{��_{pq}}}, \:{{��_{qq}}= 0}\right]") | (54) |
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, *01γpp, *01γpq, *01γ qp, *01γqq], INT)) -> FRAC(DMP([*01mp, *01mq, *01γpp, *01γpq, *01γqp, *01γqq], 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, *01γpp, *01γ pq, *01γqp, *01γqq], INT)) -> FRAC(DMP([*01mp, *01mq, *01γpp, *01γpq, *01γqp, *01γqq], INT))
 | (55) |
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])
 | (56) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
trace(P)
 | (57) |
Type: FRAC(DMP([*01mp,
fricas
c:=1/trace(P)*P
![\label{eq58}\begin{array}{@{}l}
\displaystyle
{{{{��_{pp}}\ {��_{qq}}}\over{{{��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}}+{{��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}+{{��_{pp}}\ {��_{qp}}}+{{��_{qp}}\ {��_{qq}}}}}\ qp}+
\
\
\displaystyle
{{{{��_{pp}}\ {��_{qp}}\ {��_{qq}}}\over{{{��_{pp}}\ {{��_{pq}}^{3}}\ {��_{qq}}}+{{��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qp}}\ {��_{qq}}}+{{��_{pp}}\ {��_{pq}}\ {��_{qp}}}+{{��_{pq}}\ {��_{qp}}\ {��_{qq}}}}}\ pq}+
\
\
\displaystyle
{{{{��_{pp}}\ {��_{qp}}}\over{{{��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}}+{{��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}+{{��_{pp}}\ {��_{qp}}}+{{��_{qp}}\ {��_{qq}}}}}\ q}+
\
\
\displaystyle
{{{{��_{qp}}\ {��_{qq}}}\over{{{��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}}+{{��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}+{{��_{pp}}\ {��_{qp}}}+{{��_{qp}}\ {��_{qq}}}}}\ p}](images/5354178195822688430-16.0px.png "\label{eq58}\begin{array}{@{}l}
\displaystyle
{{{{��_{pp}}\ {��_{qq}}}\over{{{��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}}+{{��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}+{{��_{pp}}\ {��_{qp}}}+{{��_{qp}}\ {��_{qq}}}}}\ qp}+
\
\
\displaystyle
{{{{��_{pp}}\ {��_{qp}}\ {��_{qq}}}\over{{{��_{pp}}\ {{��_{pq}}^{3}}\ {��_{qq}}}+{{��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qp}}\ {��_{qq}}}+{{��_{pp}}\ {��_{pq}}\ {��_{qp}}}+{{��_{pq}}\ {��_{qp}}\ {��_{qq}}}}}\ pq}+
\
\
\displaystyle
{{{{��_{pp}}\ {��_{qp}}}\over{{{��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}}+{{��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}+{{��_{pp}}\ {��_{qp}}}+{{��_{qp}}\ {��_{qq}}}}}\ q}+
\
\
\displaystyle
{{{{��_{qp}}\ {��_{qq}}}\over{{{��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qq}}}+{{��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}+{{��_{pp}}\ {��_{qp}}}+{{��_{qp}}\ {��_{qq}}}}}\ p}") | (58) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
c*c-c
![\label{eq59}\begin{array}{@{}l}
\displaystyle
{{{-{{{��_{pp}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {{��_{qq}}^{2}}}+{{��_{pp}}\ {{��_{qp}}^{2}}\ {��_{qq}}}}\over{\left(
\begin{array}{@{}l}
\displaystyle
{{{��_{pp}}^{2}}\ {{��_{pq}}^{4}}\ {{��_{qq}}^{2}}}+{2 \ {{��_{pp}}^{2}}\ {{��_{pq}}^{3}}\ {��_{qp}}\ {{��_{qq}}^{2}}}+
\
\
\displaystyle
{{{��_{pp}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}\ {{��_{qq}}^{2}}}+{2 \ {{��_{pp}}^{2}}\ {{��_{pq}}^{2}}\ {��_{qp}}\ {��_{qq}}}+
\
\
\displaystyle
{2 \ {{��_{pp}}^{2}}\ {��_{pq}}\ {{��_{qp}}^{2}}\ {��_{qq}}}+{{{��_{pp}}^{2}}\ {{��_{qp}}^{2}}}+{2 \ {��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qp}}\ {{��_{qq}}^{2}}}+
\
\
\displaystyle
{2 \ {��_{pp}}\ {��_{pq}}\ {{��_{qp}}^{2}}\ {{��_{qq}}^{2}}}+{2 \ {��_{pp}}\ {{��_{qp}}^{2}}\ {��_{qq}}}+{{{��_{qp}}^{2}}\ {{��_{qq}}^{2}}}](images/4228354792001030699-16.0px.png "\label{eq59}\begin{array}{@{}l}
\displaystyle
{{{-{{{��_{pp}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {{��_{qq}}^{2}}}+{{��_{pp}}\ {{��_{qp}}^{2}}\ {��_{qq}}}}\over{\left(
\begin{array}{@{}l}
\displaystyle
{{{��_{pp}}^{2}}\ {{��_{pq}}^{4}}\ {{��_{qq}}^{2}}}+{2 \ {{��_{pp}}^{2}}\ {{��_{pq}}^{3}}\ {��_{qp}}\ {{��_{qq}}^{2}}}+
\
\
\displaystyle
{{{��_{pp}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}\ {{��_{qq}}^{2}}}+{2 \ {{��_{pp}}^{2}}\ {{��_{pq}}^{2}}\ {��_{qp}}\ {��_{qq}}}+
\
\
\displaystyle
{2 \ {{��_{pp}}^{2}}\ {��_{pq}}\ {{��_{qp}}^{2}}\ {��_{qq}}}+{{{��_{pp}}^{2}}\ {{��_{qp}}^{2}}}+{2 \ {��_{pp}}\ {{��_{pq}}^{2}}\ {��_{qp}}\ {{��_{qq}}^{2}}}+
\
\
\displaystyle
{2 \ {��_{pp}}\ {��_{pq}}\ {{��_{qp}}^{2}}\ {{��_{qq}}^{2}}}+{2 \ {��_{pp}}\ {{��_{qp}}^{2}}\ {��_{qq}}}+{{{��_{qp}}^{2}}\ {{��_{qq}}^{2}}}") | (59) |
Type: ALGSC(FRAC(DMP([*01mp,
fricas
trace(c)
 | (60) |
Type: FRAC(DMP([*01mp,
Scalar Product
fricas
S := matrix [[trace(x*y) for y in AB] for x in AB]
![\label{eq61}\left[
\begin{array}{cccc}
1 &{{��_{pq}}^{2}}&{{��_{pq}}^{2}}&{{��_{pq}}^{2}}
\
{{��_{pq}}^{2}}& 1 &{{��_{pq}}^{2}}&{{��_{pq}}^{2}}
\
{{��_{pq}}^{2}}&{{��_{pq}}^{2}}&{{��_{pq}}^{4}}&{{��_{pq}}^{2}}
\
{{��_{pq}}^{2}}&{{��_{pq}}^{2}}&{{��_{pq}}^{2}}&{{��_{pq}}^{4}}](images/6367070167330943682-16.0px.png "\label{eq61}\left[
\begin{array}{cccc}
1 &{{��_{pq}}^{2}}&{{��_{pq}}^{2}}&{{��_{pq}}^{2}}
\
{{��_{pq}}^{2}}& 1 &{{��_{pq}}^{2}}&{{��_{pq}}^{2}}
\
{{��_{pq}}^{2}}&{{��_{pq}}^{2}}&{{��_{pq}}^{4}}&{{��_{pq}}^{2}}
\
{{��_{pq}}^{2}}&{{��_{pq}}^{2}}&{{��_{pq}}^{2}}&{{��_{pq}}^{4}}") | (61) |
Type: MATRIX(FRAC(DMP([*01mp,