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last edited 7 years ago by Bill Page |
Edit detail for SandBoxAlgebraOfObservers2 revision 1 of 5
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Editor: Bill Page
Time: 2017/04/26 22:43:06 GMT+0 |
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| Note: All idempotent operators | ||
changed: - Obs(2) is a 4 dimensional Frobenius Algebra Generators of Obs(2) \begin{axiom} )set output abbreviate on )set message type off V := OrderedVariableList [p,q] vars:List V := enumerate()$V \end{axiom} --Representation \begin{axiom} M := FreeMonoid V divisible := Record(lm: M,rm: M) gamma(i:Symbol,j:Symbol):Symbol == concat([string 'γ,string i,string j])::Symbol --subscript('γ,[concat(string i, string j)::Symbol]) mass(i:Symbol):Symbol == concat("m",string i)::Symbol -- subscript('m,[i]) B := OrderedVariableList(concat [ _ [mass i for i in vars], _ concat [[gamma(vars i ,vars j) for i in (j+1)..#vars] for j in 1..#vars] ]) K := FRAC SMP(Integer,B) --K := Expression Integer MK := FreeModule(K,M) m(x:V):K == mass(x::Symbol) m(vars 1) γ(x:V,y:V):K == if x<y then return variable(gamma(x::Symbol,y::Symbol))$B if x>y then return variable(gamma(y::Symbol,x::Symbol))$B return 1 γ(vars 2,vars 1) --Basis basis:List M := concat(vars,concat [[i::M*j::M for j in vars | i~=j] for i in vars]) \end{axiom} Idempotent: ii --> mᵢ γᵢᵢ i \begin{axiom} idem(p:MK):MK == -- p = c*q q := leadingSupport p c := leadingCoefficient p for i in vars::List M repeat f := divide(q, i*i) if f case divisible then -- q = f.lm * ii * f.rm return monomial(c * m i * γ(i,i), elt(f,lm) * i * elt(f,rm)) --return monomial(c * γ(i,i), elt(f,lm) * i * elt(f,rm)) return p idem(basis(1)*basis(1)) \end{axiom} Reductions: ijk --> mᵢmⱼ γᵢⱼγⱼₖ/γᵢₖ ik \begin{axiom} reduct(p:MK):MK == q := leadingSupport p c := leadingCoefficient p for i in vars repeat for j in vars::List M | j ~= i repeat for k in vars::List M | k ~= j repeat f:=divide(q, i*j*k) if f case divisible then return monomial(c * m j * γ(i,j) * γ(j,k) / γ(i,k), _ --return monomial(c * γ(i,j) * γ(j,k) / γ(i,k), _ elt(f,lm) * i * k * elt(f,rm)) return p reduct(basis(1)*basis(2)*basis(1)) \end{axiom} An endomorphism on the K-Module is defined by the fixed point of applied rules \begin{axiom} Y(p:MK):MK == repeat r := p; p := idem reduct r if r=p then return p Y(basis(1)*basis(2)) \end{axiom} Matrix Algebra is the free algebra product modulo the fixed point \begin{axiom} MT := [[Y(i*j) for j in basis] for i in basis] \end{axiom} Structure Constants \begin{axiom} mat3(y:M):List List K == map(z+->map(x+->coefficient(x,y),z),MT) ss:=map(mat3, basis) \end{axiom} Algebra \begin{axiom} cats(m:M):Symbol==concat(map(x+->string(x.gen::Symbol),factors m))::Symbol A:=AlgebraGivenByStructuralConstants(K,#(basis)::PI,map(cats,basis),ss::Vector(Matrix K)) alternative?()$A antiAssociative?()$A antiCommutative?()$A associative?()$A commutative?()$A flexible?()$A jacobiIdentity?()$A jordanAdmissible?()$A jordanAlgebra?()$A leftAlternative?()$A lieAdmissible?()$A lieAlgebra?()$A --powerAssociative?()$A rightAlternative?()$A \end{axiom} Check Multiplication \begin{axiom} AB := entries basis()$A A2MK(z:A):MK==reduce(+,map((x:K,y:M):MK+->(x::K)*y,coordinates(z),basis)) test(MT=map(x+->map(A2MK,x),[[i*j for j in AB] for i in AB])) \end{axiom} Trace \begin{axiom} [rightTrace(i)$A for i in AB] [leftTrace(i)$A for i in AB] trace(i)==rightTrace(i) / #vars [trace(i) for i in AB] \end{axiom} \begin{axiom} p:=AB(1); q:=AB(2); test(p*p=trace(p)*p) test(q*q=trace(q)*q) \end{axiom} Center \begin{axiom} C:=basisOfCenter()$AlgebraPackage(K,A); # C c:=C(1) [c*i-i*c for i in AB] c*c test(c*c=c) \end{axiom} Unit \begin{axiom} n := #vars/trace(c) * c test(n = unit()$A) trace(n) test(n*n=n) f:=gcd map(x+->denom x,coordinates(n)) --Silberstein symmetric matrix ff:= matrix [[(i=j => 1$K; γ(i,j)) for j in vars] for i in vars] test(f = - determinant(ff)) (f*n)::OutputForm / f::OutputForm \end{axiom} Orthogonal Observers \begin{axiom} dual(p) == trace(p)*n - p --dual(p) == n - (1/trace(p))*p p' := dual p trace p' p'' := dual p' trace p'' test(p' * p' = trace(p')*p') p * p' p' * p q' := dual q trace(q') test(q' * q' = trace(q')*q') q * q' q' * q p' * q' q' * p' p' * q q * p' p * q' q' * p \end{axiom} Orthogonal Observers are Derivations if there are only two observers \begin{axiom} test(p'*(p*q) = (p'*p)*q + p*(p'*q)) test(q'*(p*q) = (q'*p)*q + p*(q'*q)) test((p*q)*p' = (p*p')*q + p*(q*p')) test((p*q)*q' = (p*q')*q + p*(q*q')) \end{axiom} Momentum \begin{axiom} P:=reduce(+,concat [[1/γ(basis i,basis j)*AB(i)*AB(j) for j in 1..size()$V] for i in 1..size()$V]) trace(P) u:=1/trace(P)*P u*u-u trace(u) \end{axiom} All idempotents \begin{axiom} ideq:=conditionsForIdempotents()$GCNAALG(K,#(basis)::PI,map(cats,basis),ss::Vector(Matrix K)) gbs:=groebnerFactorize ideq; #gbs \end{axiom} \begin{axiom} )set output tex off )set output algebra on \end{axiom} \begin{axiom} s9:=solve(gbs.9) i9:=represents(reverse map(rhs,s9.1))$A test(i9=n) \end{axiom} \begin{axiom} s8:=solve(gbs.8) i8:=represents(reverse map(rhs,s8.1))$A test(i8=n-1/trace(p*q)*p*q) \end{axiom} \begin{axiom} s7:=solve(gbs.7) i7:=represents(reverse map(rhs,s7.1))$A test(i7=n-1/trace(q*p)*q*p) \end{axiom} \begin{axiom} s6:=solve(gbs.6) i6:=represents(reverse map(rhs,s6.1))$A test(i6=1/trace(q*p)*q*p) \end{axiom} \begin{axiom} gbs.5 s5:=solve(gbs.5) \end{axiom} \begin{axiom} s4:=solve(gbs.4) i4:=represents(reverse map(rhs,s4.1))$A \end{axiom} \begin{axiom} s3:=solve(gbs.3) i3:=represents(reverse map(rhs,s3.1))$A test(i3=1/trace(p*q)*p*q) \end{axiom} \begin{axiom} gbs.2 -- apparently we need to look for solutions in a larger ring ex2:=map(x+->interpret(x::InputForm)$InputFormFunctions1(FRAC POLY INT),concat(gbs.2,[%x3-%x4])) s2:=solve(ex2,[%x1,%x2,%x3,%x4]) #s2 -- need this to convert solution back to K (mp,mq,γqp):K i2:=represents(map(x+->interpret(rhs(x)::InputForm)$InputFormFunctions1(K),s2.1))$A i2':=represents(map(x+->interpret(rhs(x)::InputForm)$InputFormFunctions1(K),s2.2))$A test(n-i2=i2') expr2:=map(x+->interpret(x::InputForm)$InputFormFunctions1(EXPR INT)=0,concat(gbs.2,[])) solve(expr2,[%x1,%x2,%x3]) gbs.1 --s1:=solve(concat(gbs.1,[%x1-1,%x4-1])) s1:=solve(concat(gbs.1,[%x1-m('p)/trace(P),%x2-m('q)/trace(P)])) i1:=represents(reverse map(rhs,s1.1))$A test(i1=u) eval(gbs.1,[x=e for x in [%x1,%x2,%x3,%x4] for e in entries coordinates(u)]) eval(gbs.1,[x=e for x in [%x1,%x2,%x3,%x4] for e in entries coordinates(n-u)]) expr1a:=map(x+->interpret(x::InputForm)$InputFormFunctions1(EXPR INT)=0,concat(gbs.1,[])) solve(expr1a,[%x1,%x2]) expr1b:=map(x+->interpret(x::InputForm)$InputFormFunctions1(EXPR INT)=0,concat(gbs.1,[%x3-%x4])) solve(expr1b,[%x1,%x2,%x3]) \end{axiom}
Obs(2) is a 4 dimensional Frobenius Algebra
Generators of Obs(2)
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)set output abbreviate on
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)set message type off
V := OrderedVariableList [p,q]
![\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q ])](images/7148988253953892369-16.0px.png "\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q ])") | (1) |
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vars:List V := enumerate()$V
![\label{eq2}\left[ p , \: q \right]](images/7155327861761227081-16.0px.png "\label{eq2}\left[ p , \: q \right]") | (2) |
--Representation
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M := FreeMonoid V
![\label{eq3}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ]))](images/741980311099421584-16.0px.png "\label{eq3}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ]))") | (3) |
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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 ])))}](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) |
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gamma(i:Symbol,j:Symbol):Symbol == concat([string 'γ, string i, string j])::Symbol
Function declaration gamma : (SYMBOL,SYMBOL) -> SYMBOL has been added to workspace. --subscript('γ, [concat(string i, string j)::Symbol]) mass(i:Symbol):Symbol == concat("m", string i)::Symbol -- subscript('m, [i])
Function declaration mass : SYMBOL -> SYMBOL has been added to workspace. B := OrderedVariableList(concat [ _ [mass i for i in vars],_ concat [[gamma(vars i , vars j) for i in (j+1)..#vars] for j in 1..#vars] ])
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Compiling function mass with type SYMBOL -> SYMBOL
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Compiling function gamma with type (SYMBOL,SYMBOL) -> SYMBOL
![\label{eq5}\hbox{\axiomType{OVAR}\ } ([ mp , mq , �� qp ])](images/5645713701900114020-16.0px.png "\label{eq5}\hbox{\axiomType{OVAR}\ } ([ mp , mq , �� qp ])") | (5) |
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K := FRAC SMP(Integer,B)
![\label{eq6}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{SMP}\ } (\hbox{\axiomType{INT}\ } , \hbox{\axiomType{OVAR}\ } ([ mp , mq , �� qp ])))](images/7883609180541004743-16.0px.png "\label{eq6}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{SMP}\ } (\hbox{\axiomType{INT}\ } , \hbox{\axiomType{OVAR}\ } ([ mp , mq , �� qp ])))") | (6) |
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--K := Expression Integer MK := FreeModule(K,M)
![\label{eq7}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{SMP}\ } (\hbox{\axiomType{INT}\ } , \hbox{\axiomType{OVAR}\ } ([ mp , mq , �� qp ]))) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ])))](images/6015157264418409425-16.0px.png "\label{eq7}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{SMP}\ } (\hbox{\axiomType{INT}\ } , \hbox{\axiomType{OVAR}\ } ([ mp , mq , �� qp ]))) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ])))") | (7) |
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m(x:V):K == mass(x::Symbol)
Function declaration m : OVAR([p,q]) -> FRAC(SMP(INT, OVAR([mp, mq, γqp ]))) has been added to workspace. m(vars 1)
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Compiling function m with type OVAR([p,q]) -> FRAC(SMP(INT, OVAR([mp, mq, γqp])))
 | (8) |
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γ(x:V,y:V):K == if x<y then return variable(gamma(x::Symbol, y::Symbol))$B if x>y then return variable(gamma(y::Symbol, x::Symbol))$B return 1
Function declaration γ : (OVAR([p,q]), OVAR([p, q])) -> FRAC(SMP(INT, OVAR([mp, mq, γqp]))) has been added to workspace. Compiled code for gamma has been cleared. γ(vars 2, vars 1)
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Compiling function gamma with type (SYMBOL,SYMBOL) -> SYMBOL
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Compiling function γ with type (OVAR([p,q]), OVAR([p, q])) -> FRAC(SMP (INT, OVAR([mp, mq, γqp])))
 | (9) |
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--Basis basis:List M := concat(vars,concat [[i::M*j::M for j in vars | i~=j] for i in vars])
![\label{eq10}\left[ p , \: q , \:{p \ q}, \:{q \ p}\right]](images/5264386155454806585-16.0px.png "\label{eq10}\left[ p , \: q , \:{p \ q}, \:{q \ p}\right]") | (10) |
Idempotent: ii --> mᵢ γᵢᵢ i
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idem(p:MK):MK ==
-- p = c*q
q := leadingSupport p
c := leadingCoefficient p
for i in vars::List M repeat
f := divide(q,
Function declaration idem : FM(FRAC(SMP(INT,
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Compiling function idem with type FM(FRAC(SMP(INT,OVAR([mp, mq, γqp])) ), FMONOID(OVAR([p, q]))) -> FM(FRAC(SMP(INT, OVAR([mp, mq, γqp]))), FMONOID(OVAR([p, q])))
 | (11) |
Reductions: ijk --> mᵢmⱼ γᵢⱼγⱼₖ/γᵢₖ ik
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reduct(p:MK):MK ==
q := leadingSupport p
c := leadingCoefficient p
for i in vars repeat
for j in vars::List M | j ~= i repeat
for k in vars::List M | k ~= j repeat
f:=divide(q,
Function declaration reduct : FM(FRAC(SMP(INT,
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Compiling function reduct with type FM(FRAC(SMP(INT,OVAR([mp, mq, γqp] ))), FMONOID(OVAR([p, q]))) -> FM(FRAC(SMP(INT, OVAR([mp, mq, γqp]))), FMONOID(OVAR([p, q])))
 | (12) |
An endomorphism on the K-Module is defined by the fixed point of applied rules
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Y(p:MK):MK ==
repeat
r := p; p := idem reduct r
if r=p then return p
Function declaration Y : FM(FRAC(SMP(INT,
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Compiling function Y with type FM(FRAC(SMP(INT,OVAR([mp, mq, γqp]))), FMONOID(OVAR([p, q]))) -> FM(FRAC(SMP(INT, OVAR([mp, mq, γqp]))), FMONOID(OVAR([p, q])))
 | (13) |
Matrix
Algebra is the free algebra product modulo the fixed point
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MT := [[Y(i*j) for j in basis] for i in basis]
![\label{eq14}\begin{array}{@{}l}
\displaystyle
\left[{\left[{mp \ p}, \:{p \ q}, \:{mp \ p \ q}, \:{{{�� qp}^{2}}\ mq \ mp \ p}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{q \ p}, \:{mq \ q}, \:{{{�� qp}^{2}}\ mq \ mp \ q}, \:{mq \ q \ p}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{{{�� qp}^{2}}\ mq \ mp \ p}, \:{mq \ p \ q}, \:{{{�� qp}^{2}}\ mq \ mp \ p \ q}, \:{{{�� qp}^{2}}\ {{mq}^{2}}\ mp \ p}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{mp \ q \ p}, \:{{{�� qp}^{2}}\ mq \ mp \ q}, \:{{{�� qp}^{2}}\ mq \ {{mp}^{2}}\ q}, \:{{{�� qp}^{2}}\ mq \ mp \ q \ p}\right]}\right]](images/4639319588936485201-16.0px.png "\label{eq14}\begin{array}{@{}l}
\displaystyle
\left[{\left[{mp \ p}, \:{p \ q}, \:{mp \ p \ q}, \:{{{�� qp}^{2}}\ mq \ mp \ p}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{q \ p}, \:{mq \ q}, \:{{{�� qp}^{2}}\ mq \ mp \ q}, \:{mq \ q \ p}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{{{�� qp}^{2}}\ mq \ mp \ p}, \:{mq \ p \ q}, \:{{{�� qp}^{2}}\ mq \ mp \ p \ q}, \:{{{�� qp}^{2}}\ {{mq}^{2}}\ mp \ p}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{mp \ q \ p}, \:{{{�� qp}^{2}}\ mq \ mp \ q}, \:{{{�� qp}^{2}}\ mq \ {{mp}^{2}}\ q}, \:{{{�� qp}^{2}}\ mq \ mp \ q \ p}\right]}\right]") | (14) |
Structure Constants
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mat3(y:M):List List K == map(z+->map(x+->coefficient(x,y), z), MT)
Function declaration mat3 : FMONOID(OVAR([p,q])) -> LIST(LIST(FRAC( SMP(INT, OVAR([mp, mq, γqp]))))) has been added to workspace. ss:=map(mat3, basis)
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Compiling function mat3 with type FMONOID(OVAR([p,q])) -> LIST(LIST( FRAC(SMP(INT, OVAR([mp, mq, γqp])))))
![\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\left[{\left[ mp , \: 0, \: 0, \:{{{�� qp}^{2}}\ mq \ mp}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[{{{�� qp}^{2}}\ mq \ mp}, \: 0, \: 0, \:{{{�� qp}^{2}}\ {{mq}^{2}}\ mp}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}\right]](images/2432514773336941117-16.0px.png "\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\left[{\left[ mp , \: 0, \: 0, \:{{{�� qp}^{2}}\ mq \ mp}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[{{{�� qp}^{2}}\ mq \ mp}, \: 0, \: 0, \:{{{�� qp}^{2}}\ {{mq}^{2}}\ mp}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}\right]") | (15) |
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. A:=AlgebraGivenByStructuralConstants(K, #(basis)::PI, map(cats, basis), ss::Vector(Matrix K))
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Compiling function cats with type FMONOID(OVAR([p,q])) -> SYMBOL
![\label{eq16}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{SMP}\ } (\hbox{\axiomType{INT}\ } , \hbox{\axiomType{OVAR}\ } ([ mp , mq , �� qp ]))) , 4, [ p , q , pq , qp ] , [ [ [ mp , 0, 0, �� qp^2 <em> mq </em> mp ] , [ 0, 0, 0, 0 ] , [ �� qp^2 <em> mq </em> mp , 0, 0, �� qp^2 <em> mq^2 </em> mp ] , [ 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0 ] , [ 0, mq , �� qp^2 <em> mq </em> mp , 0 ] , [ 0, 0, 0, 0 ] , [ 0, �� qp^2 <em> mq </em> mp , �� qp^2 <em> mq </em> mp^2, 0 ] ] , [ [ 0, 1, mp , 0 ] , [ 0, 0, 0, 0 ] , [ 0, mq , �� qp^2 <em> mq </em> mp , 0 ] , [ 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0 ] , [ 1, 0, 0, mq ] , [ 0, 0, 0, 0 ] , [ mp , 0, 0, �� qp^2 <em> mq </em> mp ] ] ])](images/2471784023522778000-16.0px.png "\label{eq16}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{SMP}\ } (\hbox{\axiomType{INT}\ } , \hbox{\axiomType{OVAR}\ } ([ mp , mq , �� qp ]))) , 4, [ p , q , pq , qp ] , [ [ [ mp , 0, 0, �� qp^2 <em> mq </em> mp ] , [ 0, 0, 0, 0 ] , [ �� qp^2 <em> mq </em> mp , 0, 0, �� qp^2 <em> mq^2 </em> mp ] , [ 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0 ] , [ 0, mq , �� qp^2 <em> mq </em> mp , 0 ] , [ 0, 0, 0, 0 ] , [ 0, �� qp^2 <em> mq </em> mp , �� qp^2 <em> mq </em> mp^2, 0 ] ] , [ [ 0, 1, mp , 0 ] , [ 0, 0, 0, 0 ] , [ 0, mq , �� qp^2 <em> mq </em> mp , 0 ] , [ 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0 ] , [ 1, 0, 0, mq ] , [ 0, 0, 0, 0 ] , [ mp , 0, 0, �� qp^2 <em> mq </em> mp ] ] ])") | (16) |
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alternative?()$A
algebra satisfies 2*associator(a,b, b) = 0 = 2*associator(a, a, b) = 0
 | (17) |
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antiAssociative?()$A
algebra is not anti-associative
 | (18) |
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antiCommutative?()$A
algebra is not anti-commutative
 | (19) |
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associative?()$A
algebra is associative
 | (20) |
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commutative?()$A
algebra is not commutative
 | (21) |
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flexible?()$A
algebra is flexible
 | (22) |
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jacobiIdentity?()$A
Jacobi identity does not hold
 | (23) |
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jordanAdmissible?()$A
algebra is not Jordan admissible
 | (24) |
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jordanAlgebra?()$A
algebra is not commutative this is not a Jordan algebra
 | (25) |
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leftAlternative?()$A
algebra is left alternative
 | (26) |
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lieAdmissible?()$A
algebra is Lie admissible
 | (27) |
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lieAlgebra?()$A
algebra is not anti-commutative this is not a Lie algebra
 | (28) |
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--powerAssociative?()$A rightAlternative?()$A
algebra is right alternative
 | (29) |
Check Multiplication
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AB := entries basis()$A
![\label{eq30}\left[ p , \: q , \: pq , \: qp \right]](images/6685245530046166891-16.0px.png "\label{eq30}\left[ p , \: q , \: pq , \: qp \right]") | (30) |
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A2MK(z:A):MK==reduce(+,map((x:K, y:M):MK+->(x::K)*y, coordinates(z), basis))
Function declaration A2MK : ALGSC(FRAC(SMP(INT,OVAR([mp, mq, γqp]))), 4 , [p, q, pq, qp], [[[mp, 0, 0, γqp^2*mq*mp], [0, 0, 0, 0], [γqp^2*mq*mp, 0, 0, γqp^2*mq^2*mp], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, mq, γqp^2*mq*mp, 0], [0, 0, 0, 0], [0, γqp^2*mq*mp, γqp^2*mq*mp^2, 0]], [[0, 1, mp, 0], [0, 0, 0, 0], [0, mq, γqp^2*mq*mp, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [1, 0, 0, mq], [0, 0, 0, 0], [mp, 0, 0 , γqp^2*mq*mp]]]) -> FM(FRAC(SMP(INT, OVAR([mp, mq, γqp]))), FMONOID( OVAR([p, q]))) has been added to workspace. test(MT=map(x+->map(A2MK, x), [[i*j for j in AB] for i in AB]))
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Compiling function A2MK with type ALGSC(FRAC(SMP(INT,OVAR([mp, mq, γqp ]))), 4, [p, q, pq, qp], [[[mp, 0, 0, γqp^2*mq*mp], [0, 0, 0, 0], [γqp^2*mq*mp, 0, 0, γqp^2*mq^2*mp], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, mq, γqp^2*mq*mp, 0], [0, 0, 0, 0], [0, γqp^2*mq*mp, γqp^2*mq*mp^2, 0]], [[0, 1, mp, 0], [0, 0, 0, 0], [0, mq, γqp^2*mq*mp, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [1, 0, 0, mq], [0, 0, 0, 0], [mp, 0, 0, γqp^2*mq*mp]]]) -> FM(FRAC(SMP(INT, OVAR([mp, mq, γqp]))), FMONOID(OVAR([p, q])))
 | (31) |
Trace
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[rightTrace(i)$A for i in AB]
![\label{eq32}\left[{2 \ mp}, \:{2 \ mq}, \:{2 \ {{�� qp}^{2}}\ mq \ mp}, \:{2 \ {{�� qp}^{2}}\ mq \ mp}\right]](images/1659734843842288897-16.0px.png "\label{eq32}\left[{2 \ mp}, \:{2 \ mq}, \:{2 \ {{�� qp}^{2}}\ mq \ mp}, \:{2 \ {{�� qp}^{2}}\ mq \ mp}\right]") | (32) |
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[leftTrace(i)$A for i in AB]
![\label{eq33}\left[{2 \ mp}, \:{2 \ mq}, \:{2 \ {{�� qp}^{2}}\ mq \ mp}, \:{2 \ {{�� qp}^{2}}\ mq \ mp}\right]](images/582005987791018962-16.0px.png "\label{eq33}\left[{2 \ mp}, \:{2 \ mq}, \:{2 \ {{�� qp}^{2}}\ mq \ mp}, \:{2 \ {{�� qp}^{2}}\ mq \ mp}\right]") | (33) |
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trace(i)==rightTrace(i) / #vars
[trace(i) for i in AB]
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Compiling function trace with type ALGSC(FRAC(SMP(INT,OVAR([mp, mq, γqp]))), 4, [p, q, pq, qp], [[[mp, 0, 0, γqp^2*mq*mp], [0, 0, 0, 0], [γqp^2*mq* mp, 0, 0, γqp^2*mq^2*mp], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, mq, γqp^2*mq*mp, 0], [0, 0, 0, 0], [0, γqp^2*mq*mp, γqp^2*mq*mp^2, 0]], [[0, 1, mp, 0], [0, 0, 0, 0], [0, mq, γqp^2*mq*mp, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [1, 0, 0, mq], [0, 0, 0, 0], [ mp, 0, 0, γqp^2*mq*mp]]]) -> FRAC(SMP(INT, OVAR([mp, mq, γqp])))
![\label{eq34}\left[ mp , \: mq , \:{{{�� qp}^{2}}\ mq \ mp}, \:{{{�� qp}^{2}}\ mq \ mp}\right]](images/7626908039019524703-16.0px.png "\label{eq34}\left[ mp , \: mq , \:{{{�� qp}^{2}}\ mq \ mp}, \:{{{�� qp}^{2}}\ mq \ mp}\right]") | (34) |
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p:=AB(1); q:=AB(2);
test(p*p=trace(p)*p)
 | (35) |
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test(q*q=trace(q)*q)
 | (36) |
Center
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C:=basisOfCenter()$AlgebraPackage(K,A); # C
 | (37) |
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c:=C(1)
 | (38) |
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[c*i-i*c for i in AB]
![\label{eq39}\left[ 0, \: 0, \: 0, \: 0 \right]](images/2104285722250989468-16.0px.png "\label{eq39}\left[ 0, \: 0, \: 0, \: 0 \right]") | (39) |
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c*c
![\label{eq40}\begin{array}{@{}l}
\displaystyle
{{\left({{�� qp}^{2}}- 1 \right)}\ mq \ mp \ qp}+{{\left({{�� qp}^{2}}- 1 \right)}\ mq \ mp \ pq}+
\
\
\displaystyle
{{\left(-{{�� qp}^{2}}+ 1 \right)}\ mq \ {{mp}^{2}}\ q}+{{\left(-{{�� qp}^{2}}+ 1 \right)}\ {{mq}^{2}}\ mp \ p}](images/6358897017129889043-16.0px.png "\label{eq40}\begin{array}{@{}l}
\displaystyle
{{\left({{�� qp}^{2}}- 1 \right)}\ mq \ mp \ qp}+{{\left({{�� qp}^{2}}- 1 \right)}\ mq \ mp \ pq}+
\
\
\displaystyle
{{\left(-{{�� qp}^{2}}+ 1 \right)}\ mq \ {{mp}^{2}}\ q}+{{\left(-{{�� qp}^{2}}+ 1 \right)}\ {{mq}^{2}}\ mp \ p}") | (40) |
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test(c*c=c)
 | (41) |
Unit
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n := #vars/trace(c) * c
![\label{eq42}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq \ mp}}\ qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq \ mp}}\ pq}-
\
\
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq}}\ q}-{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mp}}\ p}](images/5714296297431051020-16.0px.png "\label{eq42}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq \ mp}}\ qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq \ mp}}\ pq}-
\
\
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq}}\ q}-{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mp}}\ p}") | (42) |
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test(n = unit()$A)
 | (43) |
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trace(n)
 | (44) |
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test(n*n=n)
 | (45) |
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f:=gcd map(x+->denom x,coordinates(n))
 | (46) |
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--Silberstein symmetric matrix ff:= matrix [[(i=j => 1$K; γ(i,j)) for j in vars] for i in vars]
![\label{eq47}\left[
\begin{array}{cc}
1 & �� qp
\
�� qp & 1](images/7384522912919787029-16.0px.png "\label{eq47}\left[
\begin{array}{cc}
1 & �� qp
\
�� qp & 1") | (47) |
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test(f = - determinant(ff))
 | (48) |
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(f*n)::OutputForm / f::OutputForm
 | (49) |
Orthogonal Observers
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dual(p) == trace(p)*n - p
--dual(p) == n - (1/trace(p))*p p' := dual p
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Compiling function dual with type ALGSC(FRAC(SMP(INT,OVAR([mp, mq, γqp ]))), 4, [p, q, pq, qp], [[[mp, 0, 0, γqp^2*mq*mp], [0, 0, 0, 0], [γqp^2*mq*mp, 0, 0, γqp^2*mq^2*mp], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, mq, γqp^2*mq*mp, 0], [0, 0, 0, 0], [0, γqp^2*mq*mp, γqp^2*mq*mp^2, 0]], [[0, 1, mp, 0], [0, 0, 0, 0], [0, mq, γqp^2*mq*mp, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [1, 0, 0, mq], [0, 0, 0, 0], [mp, 0, 0, γqp^2*mq*mp]]]) -> ALGSC(FRAC(SMP(INT, OVAR([mp, mq, γqp]))), 4, [ p, q, pq, qp], [[[mp, 0, 0, γqp^2*mq*mp], [0, 0, 0, 0], [γqp^2*mq*mp, 0, 0, γqp^ 2*mq^2*mp], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, mq, γqp^2*mq*mp, 0], [0, 0, 0, 0], [ 0, γqp^2*mq*mp, γqp^2*mq*mp^2, 0]], [[0, 1, mp, 0], [0, 0, 0, 0], [0, mq, γqp^2 *mq*mp, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [1, 0, 0, mq], [0, 0, 0, 0], [mp, 0, 0, γqp^ 2*mq*mp]]])
![\label{eq50}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq}}\ qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq}}\ pq}-{{mp \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq}}\ q}-
\
\
\displaystyle
{{{{�� qp}^{2}}\over{{{�� qp}^{2}}- 1}}\ p}](images/857598973879758655-16.0px.png "\label{eq50}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq}}\ qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq}}\ pq}-{{mp \over{{\left({{�� qp}^{2}}- 1 \right)}\ mq}}\ q}-
\
\
\displaystyle
{{{{�� qp}^{2}}\over{{{�� qp}^{2}}- 1}}\ p}") | (50) |
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trace p'
 | (51) |
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p'' := dual p'
 | (52) |
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trace p''
 | (53) |
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test(p' * p' = trace(p')*p')
 | (54) |
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p * p'
 | (55) |
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p' * p
 | (56) |
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q' := dual q
![\label{eq57}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mp}}\ qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mp}}\ pq}-{{{{�� qp}^{2}}\over{{{�� qp}^{2}}- 1}}\ q}-
\
\
\displaystyle
{{mq \over{{\left({{�� qp}^{2}}- 1 \right)}\ mp}}\ p}](images/33808089837502582-16.0px.png "\label{eq57}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mp}}\ qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\ mp}}\ pq}-{{{{�� qp}^{2}}\over{{{�� qp}^{2}}- 1}}\ q}-
\
\
\displaystyle
{{mq \over{{\left({{�� qp}^{2}}- 1 \right)}\ mp}}\ p}") | (57) |
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trace(q')
 | (58) |
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test(q' * q' = trace(q')*q')
 | (59) |
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q * q'
 | (60) |
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q' * q
 | (61) |
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p' * q'
 | (62) |
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q' * p'
 | (63) |
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p' * q
 | (64) |
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q * p'
 | (65) |
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p * q'
 | (66) |
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q' * p
 | (67) |
Orthogonal Observers are Derivations if there are only two observers
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test(p'*(p*q) = (p'*p)*q + p*(p'*q))
 | (68) |
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test(q'*(p*q) = (q'*p)*q + p*(q'*q))
 | (69) |
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test((p*q)*p' = (p*p')*q + p*(q*p'))
 | (70) |
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test((p*q)*q' = (p*q')*q + p*(q*q'))
 | (71) |
Momentum
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P:=reduce(+,concat [[1/γ(basis i, basis j)*AB(i)*AB(j) for j in 1..size()$V] for i in 1..size()$V])
 | (72) |
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trace(P)
 | (73) |
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u:=1/trace(P)*P
![\label{eq74}\begin{array}{@{}l}
\displaystyle
{{1 \over{{�� qp \ {{mp}^{2}}}+{2 \ {{�� qp}^{2}}\ mq \ mp}+{�� qp \ {{mq}^{2}}}}}\ qp}+
\
\
\displaystyle
{{1 \over{{�� qp \ {{mp}^{2}}}+{2 \ {{�� qp}^{2}}\ mq \ mp}+{�� qp \ {{mq}^{2}}}}}\ pq}+
\
\
\displaystyle
{{mq \over{{{mp}^{2}}+{2 \ �� qp \ mq \ mp}+{{mq}^{2}}}}\ q}+{{mp \over{{{mp}^{2}}+{2 \ �� qp \ mq \ mp}+{{mq}^{2}}}}\ p}](images/6011663980520903866-16.0px.png "\label{eq74}\begin{array}{@{}l}
\displaystyle
{{1 \over{{�� qp \ {{mp}^{2}}}+{2 \ {{�� qp}^{2}}\ mq \ mp}+{�� qp \ {{mq}^{2}}}}}\ qp}+
\
\
\displaystyle
{{1 \over{{�� qp \ {{mp}^{2}}}+{2 \ {{�� qp}^{2}}\ mq \ mp}+{�� qp \ {{mq}^{2}}}}}\ pq}+
\
\
\displaystyle
{{mq \over{{{mp}^{2}}+{2 \ �� qp \ mq \ mp}+{{mq}^{2}}}}\ q}+{{mp \over{{{mp}^{2}}+{2 \ �� qp \ mq \ mp}+{{mq}^{2}}}}\ p}") | (74) |
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u*u-u
 | (75) |
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trace(u)
 | (76) |
All idempotents
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ideq:=conditionsForIdempotents()$GCNAALG(K,#(basis)::PI, map(cats, basis), ss::Vector(Matrix K))
![\label{eq77}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
{{\left({{{�� qp}^{2}}\ {{mq}^{2}}\ mp \ \%x 3}+{{{�� qp}^{2}}\ mq \ mp \ \%x 1}\right)}\ \%x 4}+
\
\
\displaystyle
{{{�� qp}^{2}}\ mq \ mp \ \%x 1 \ \%x 3}+{mp \ {{\%x 1}^{2}}}- \%x 1](images/4707610204119568689-16.0px.png "\label{eq77}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
{{\left({{{�� qp}^{2}}\ {{mq}^{2}}\ mp \ \%x 3}+{{{�� qp}^{2}}\ mq \ mp \ \%x 1}\right)}\ \%x 4}+
\
\
\displaystyle
{{{�� qp}^{2}}\ mq \ mp \ \%x 1 \ \%x 3}+{mp \ {{\%x 1}^{2}}}- \%x 1") | (77) |
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gbs:=groebnerFactorize ideq;
#gbs
 | (78) |
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)set output tex off
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)set output algebra on
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s9:=solve(gbs.9)
(93) [ 1 1 1 [%x4 = ---------------,%x3 = ---------------, %x2 = - ------------, 2 2 2 (γqp - 1)mq mp (γqp - 1)mq mp (γqp - 1)mq 1 %x1 = - ------------] 2 (γqp - 1)mp ] i9:=represents(reverse map(rhs, s9.1))$A
(94) 1 1 1 1 --------------- qp + --------------- pq - ------------ q - ------------ p 2 2 2 2 (γqp - 1)mq mp (γqp - 1)mq mp (γqp - 1)mq (γqp - 1)mp test(i9=n)
(95) true
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s8:=solve(gbs.8)
(96) [ 1 1 1 [%x4 = ---------------,%x3 = ------------------, %x2 = - ------------, 2 4 2 2 (γqp - 1)mq mp (γqp - γqp )mq mp (γqp - 1)mq 1 %x1 = - ------------] 2 (γqp - 1)mp ] i8:=represents(reverse map(rhs, s8.1))$A
(97) 1 1 1 1 --------------- qp + ------------------ pq - ------------ q - ------------ p 2 4 2 2 2 (γqp - 1)mq mp (γqp - γqp )mq mp (γqp - 1)mq (γqp - 1)mp test(i8=n-1/trace(p*q)*p*q)
(98) true
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s7:=solve(gbs.7)
(99) [ 1 1 1 [%x4 = ------------------,%x3 = ---------------, %x2 = - ------------, 4 2 2 2 (γqp - γqp )mq mp (γqp - 1)mq mp (γqp - 1)mq 1 %x1 = - ------------] 2 (γqp - 1)mp ] i7:=represents(reverse map(rhs, s7.1))$A
(100) 1 1 1 1 ------------------ qp + --------------- pq - ------------ q - ------------ p 4 2 2 2 2 (γqp - γqp )mq mp (γqp - 1)mq mp (γqp - 1)mq (γqp - 1)mp test(i7=n-1/trace(q*p)*q*p)
(101) true
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s6:=solve(gbs.6)
1 (102) [[%x4 = ---------,%x3 = 0, %x2 = 0, %x1 = 0]] 2 γqp mq mp i6:=represents(reverse map(rhs, s6.1))$A
1 (103) --------- qp 2 γqp mq mp test(i6=1/trace(q*p)*q*p)
(104) true
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gbs.5
(105) [1] s5:=solve(gbs.5)
(106) []
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s4:=solve(gbs.4)
(107) [[%x4 = 0,%x3 = 0, %x2 = 0, %x1 = 0]] i4:=represents(reverse map(rhs, s4.1))$A
(108) 0
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s3:=solve(gbs.3)
1 (109) [[%x4 = 0,%x3 = ---------, %x2 = 0, %x1 = 0]] 2 γqp mq mp i3:=represents(reverse map(rhs, s3.1))$A
1 (110) --------- pq 2 γqp mq mp test(i3=1/trace(p*q)*p*q)
(111) true
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gbs.2
(112) 2 1 [%x4 + %x3 + ------ %x1 - ---------,2 2 γqp mq γqp mq mp 2 2 1 1 2 mp %x3 + (------ %x1 - ---------)%x3 + ------- %x1 , %x2 - -- %x1] 2 2 2 2 mq γqp mq γqp mq mp γqp mq -- apparently we need to look for solutions in a larger ring ex2:=map(x+->interpret(x::InputForm)$InputFormFunctions1(FRAC POLY INT), concat(gbs.2, [%x3-%x4]))
(113) 2 (%x4 + %x3)mp mq γqp + 2%x1 mp - 1 [-----------------------------------,2 mp mq γqp 2 2 2 2 %x3 mp mq γqp + (2%x1 %x3 mp - %x3)mq + %x1 mp %x2 mq - %x1 mp -----------------------------------------------, ---------------, 2 2 mq mp mq γqp - %x4 + %x3] s2:=solve(ex2, [%x1, %x2, %x3, %x4])
(114) [ 1 1 [%x1 = - -------------,%x2 = - -------------, 2mp γqp - 2mp 2mq γqp - 2mq 1 1 %x3 = ------------------------, %x4 = ------------------------] 2 2 2mp mq γqp - 2mp mq γqp 2mp mq γqp - 2mp mq γqp ,
1 1 1 [%x1 = -------------,%x2 = -------------, %x3 = ------------------------, 2mp γqp + 2mp 2mq γqp + 2mq 2 2mp mq γqp + 2mp mq γqp 1 %x4 = ------------------------] 2 2mp mq γqp + 2mp mq γqp ] #s2
(115) 2 -- need this to convert solution back to K (mp,mq, γqp):K
i2:=represents(map(x+->interpret(rhs(x)::InputForm)$InputFormFunctions1(K),s2.1))$A
(117) 1 1 1 ------------------- qp + ------------------- pq - ------------ q 2 2 (2γqp - 2)mq (2γqp - 2γqp)mq mp (2γqp - 2γqp)mq mp + 1 - ------------ p (2γqp - 2)mp i2':=represents(map(x+->interpret(rhs(x)::InputForm)$InputFormFunctions1(K),s2.2))$A
(118) 1 1 1 ------------------- qp + ------------------- pq + ------------ q 2 2 (2γqp + 2)mq (2γqp + 2γqp)mq mp (2γqp + 2γqp)mq mp + 1 ------------ p (2γqp + 2)mp test(n-i2=i2')
(119) true expr2:=map(x+->interpret(x::InputForm)$InputFormFunctions1(EXPR INT)=0,concat(gbs.2, []))
(120) 2 (%x4 + %x3)mp mq γqp + 2%x1 mp - 1 [----------------------------------- = 0,2 mp mq γqp 2 2 2 2 %x3 mp mq γqp + (2%x1 %x3 mp - %x3)mq + %x1 mp %x2 mq - %x1 mp ----------------------------------------------- = 0, --------------- = 0] 2 2 mq mp mq γqp solve(expr2, [%x1, %x2, %x3])
(121) [ [ %x1 = +------------------------------------------------------+ | 2 4 4 6 2 4 4 3 3 4 \|- 16%x4 mp mq γqp + (16%x4 mp mq + 16%x4 mp mq )γqp + 2 2 2 - 4%x4 mp mq γqp / 2 2 4mp mq γqp ,
%x2 = +------------------------------------------------------+ | 2 4 4 6 2 4 4 3 3 4 \|- 16%x4 mp mq γqp + (16%x4 mp mq + 16%x4 mp mq )γqp + 2 2 2 - 4%x4 mp mq γqp / 2 2 4mp mq γqp ,
%x3 = +------------------------------------------------------+ | 2 4 4 6 2 4 4 3 3 4 - \|- 16%x4 mp mq γqp + (16%x4 mp mq + 16%x4 mp mq )γqp + 2 2 4 2 2 2 - 2%x4 mp mq γqp + (4%x4 mp mq + 2mp mq)γqp / 2 2 4 2mp mq γqp ] ,
[ %x1 = +------------------------------------------------------+ | 2 4 4 6 2 4 4 3 3 4 - \|- 16%x4 mp mq γqp + (16%x4 mp mq + 16%x4 mp mq )γqp + 2 2 2 - 4%x4 mp mq γqp / 2 2 4mp mq γqp ,
%x2 = +------------------------------------------------------+ | 2 4 4 6 2 4 4 3 3 4 - \|- 16%x4 mp mq γqp + (16%x4 mp mq + 16%x4 mp mq )γqp + 2 2 2 - 4%x4 mp mq γqp / 2 2 4mp mq γqp ,
%x3 = +------------------------------------------------------+ | 2 4 4 6 2 4 4 3 3 4 \|- 16%x4 mp mq γqp + (16%x4 mp mq + 16%x4 mp mq )γqp + 2 2 4 2 2 2 - 2%x4 mp mq γqp + (4%x4 mp mq + 2mp mq)γqp / 2 2 4 2mp mq γqp ] ] gbs.1
(122) 1 1 1 [%x4 + %x3 + ------ %x2 + ------ %x1 - ---------,2 2 2 γqp mp γqp mq γqp mq mp 2 1 1 1 1 %x3 + (------ %x2 + ------ %x1 - ---------)%x3 + --------- %x1 %x2] 2 2 2 2 γqp mp γqp mq γqp mq mp γqp mq mp --s1:=solve(concat(gbs.1, [%x1-1, %x4-1])) s1:=solve(concat(gbs.1, [%x1-m('p)/trace(P), %x2-m('q)/trace(P)]))
(123) [ 1 [%x4 = ------------------------------,2 2 2 γqp mp + 2γqp mq mp + γqp mq 1 mq %x3 = ------------------------------, %x2 = ----------------------, 2 2 2 2 2 γqp mp + 2γqp mq mp + γqp mq mp + 2γqp mq mp + mq mp %x1 = ----------------------] 2 2 mp + 2γqp mq mp + mq ] i1:=represents(reverse map(rhs, s1.1))$A
(124) 1 1 ------------------------------ qp + ------------------------------ pq 2 2 2 2 2 2 γqp mp + 2γqp mq mp + γqp mq γqp mp + 2γqp mq mp + γqp mq + mq mp ---------------------- q + ---------------------- p 2 2 2 2 mp + 2γqp mq mp + mq mp + 2γqp mq mp + mq test(i1=u)
(125) true eval(gbs.1,[x=e for x in [%x1, %x2, %x3, %x4] for e in entries coordinates(u)])
(126) [0,0] eval(gbs.1, [x=e for x in [%x1, %x2, %x3, %x4] for e in entries coordinates(n-u)])
(127) [0,0] expr1a:=map(x+->interpret(x::InputForm)$InputFormFunctions1(EXPR INT)=0, concat(gbs.1, []))
(128) 2 (%x4 + %x3)mp mq γqp + %x2 mq + %x1 mp - 1 [------------------------------------------- = 0,2 mp mq γqp 2 2 %x3 mp mq γqp + %x2 %x3 mq + %x1 %x3 mp - %x3 + %x1 %x2 -------------------------------------------------------- = 0] 2 mp mq γqp solve(expr1a, [%x1, %x2])
(129) [ [ %x1 = ROOT 2 2 2 2 4 (%x4 + 2%x3 %x4 + %x3 )mp mq γqp + 2 2 2 (- 4%x3 %x4 mp mq + (- 2%x4 - 2%x3)mp mq)γqp + 1 + 2 (- %x4 - %x3)mp mq γqp + 1 / 2mp ,
%x2 = - ROOT 2 2 2 2 4 (%x4 + 2%x3 %x4 + %x3 )mp mq γqp + 2 2 2 (- 4%x3 %x4 mp mq + (- 2%x4 - 2%x3)mp mq)γqp + 1 + 2 (- %x4 - %x3)mp mq γqp + 1 / 2mq ] ,
[ %x1 = - ROOT 2 2 2 2 4 (%x4 + 2%x3 %x4 + %x3 )mp mq γqp + 2 2 2 (- 4%x3 %x4 mp mq + (- 2%x4 - 2%x3)mp mq)γqp + 1 + 2 (- %x4 - %x3)mp mq γqp + 1 / 2mp ,
%x2 = ROOT 2 2 2 2 4 (%x4 + 2%x3 %x4 + %x3 )mp mq γqp + 2 2 2 (- 4%x3 %x4 mp mq + (- 2%x4 - 2%x3)mp mq)γqp + 1 + 2 (- %x4 - %x3)mp mq γqp + 1 / 2mq ] ] expr1b:=map(x+->interpret(x::InputForm)$InputFormFunctions1(EXPR INT)=0,concat(gbs.1, [%x3-%x4]))
(130) 2 (%x4 + %x3)mp mq γqp + %x2 mq + %x1 mp - 1 [------------------------------------------- = 0,2 mp mq γqp 2 2 %x3 mp mq γqp + %x2 %x3 mq + %x1 %x3 mp - %x3 + %x1 %x2 -------------------------------------------------------- = 0, 2 mp mq γqp - %x4 + %x3 = 0] solve(expr1b, [%x1, %x2, %x3])
(131) [ [ %x1 = +------------------------------------------------------+ | 2 2 2 4 2 2 2 2 - \|4%x4 mp mq γqp + (- 4%x4 mp mq - 4%x4 mp mq)γqp + 1 + 2 - 2%x4 mp mq γqp + 1 / 2mp ,
%x2 = +------------------------------------------------------+ | 2 2 2 4 2 2 2 2 \|4%x4 mp mq γqp + (- 4%x4 mp mq - 4%x4 mp mq)γqp + 1 + 2 - 2%x4 mp mq γqp + 1 / 2mq ,%x3 = %x4] ,
[ %x1 = +------------------------------------------------------+ | 2 2 2 4 2 2 2 2 \|4%x4 mp mq γqp + (- 4%x4 mp mq - 4%x4 mp mq)γqp + 1 + 2 - 2%x4 mp mq γqp + 1 / 2mp ,
%x2 = +------------------------------------------------------+ | 2 2 2 4 2 2 2 2 - \|4%x4 mp mq γqp + (- 4%x4 mp mq - 4%x4 mp mq)γqp + 1 + 2 - 2%x4 mp mq γqp + 1 / 2mq ,%x3 = %x4] ]