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last edited 13 years ago by Bill Page |
Edit detail for SandBoxPauliAlgebra revision 6 of 7
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Editor: Bill Page
Time: 2011/06/03 19:20:56 GMT-7 |
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changed: - ( ΩX I ΩX ) / - ( I W I ) - -\end{axiom} - -\begin{axiom} \end{axiom} \begin{axiom}
The Pauli Algebra Cl(3) Is Frobenius In Many WaysLinear operators over a 8-dimensional vector space representing Pauli algebra
Ref:
- http://arxiv.org/abs/1103.5113
$S_3$-permuted Frobenius Algebras
Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)
- http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories
Joachim Kock
- http://en.wikipedia.org/wiki/Frobenius_algebra
- http://en.wikipedia.org/wiki/Pauli_matrices
- http://en.wikipedia.org/wiki/Clifford_algebra
We need the Axiom LinearOperator library. \begin{axiom} )library CARTEN ARITY CMONAL CPROP CLOP CALEY \end{axiom}
Use the following macros for convenient notation \begin{axiom} -- summation macro Σ(x,i,n)==reduce(+,[x for i in n]) -- list macro Ξ(f,i,n)==[f for i in n] -- subscript and superscripts macro sb == subscript macro sp == superscript \end{axiom}
𝐋 is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients. \begin{axiom} dim:=8 macro ℒ == List macro ℂ == CaleyDickson macro ℚ == Expression Integer 𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k,'ij,'ik,'jk,'ijk], ℚ) 𝐞:ℒ 𝐋 := basisOut() 𝐝:ℒ 𝐋 := basisIn() I:𝐋:=[1] -- identity for composition X:𝐋:=[2,1] -- twist V:𝐋:=ev(1) -- evaluation Λ:𝐋:=co(1) -- co-evaluation equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq); \end{axiom}
Now generate structure constants for Pauli Algebra
The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.
The Pauli Algebra as Cl(3)
Basis: Each B.i is a Clifford number \begin{axiom} q0:=sp('i,[2]) q1:=sp('j,[2]) q2:=sp('k,[2]) QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]]) B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[3]),monomial(1,[1,2]),monomial(1,[1,3]),monomial(1,[2,3]),monomial(1,[1,2,3])] M:Matrix QQ := matrix Ξ(Ξ(B.iB.j, i,1..dim), j,1..dim) S(y) == map(x +-> coefficient(recip(y)x,[]),M) ѕ :=map(S,B)::ℒ ℒ ℒ ℚ -- structure constants form a tensor operator Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim) matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim) XY := X/Y; \end{axiom}
Units \begin{axiom} e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8; \end{axiom}
Multiplication of arbitrary quaternions $a$ and $b$ \begin{axiom} a:=Σ(sb('a,[i])*𝐞.i, i,1..dim) b:=Σ(sb('b,[i])*𝐞.i, i,1..dim) (a*b)/Y \end{axiom}
Multiplication is Associative \begin{axiom} test( ( I Y ) / ( Y ) = ( Y I ) / _ ( Y ) ) \end{axiom}
A scalar product is denoted by the (2,0)-tensor $U = \{ u_{ij} \}$ \begin{axiom} U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim) \end{axiom}
Definition 1
We say that the scalar product is associative if the tensor equation holds:
Y = Y
U U
In other words, if the (3,0)-tensor: $$ \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(4.82,0.92) \psbezier[linewidth=0.04]?(2.2,0.9)(2.2,0.1)(2.6,0.1)(2.6,0.9) \psline[linewidth=0.04cm]?(2.4,0.3)(2.4,-0.1) \psbezier[linewidth=0.04]?(2.4,-0.1)(2.4,-0.9)(3.0,-0.9)(3.0,-0.1) \psline[linewidth=0.04cm]?(3.0,-0.1)(3.0,0.9) \psbezier[linewidth=0.04]?(4.8,0.9)(4.8,0.1)(4.4,0.1)(4.4,0.9) \psline[linewidth=0.04cm]?(4.6,0.3)(4.6,-0.1) \psbezier[linewidth=0.04]?(4.6,-0.1)(4.6,-0.9)(4.0,-0.9)(4.0,-0.1) \psline[linewidth=0.04cm]?(4.0,-0.1)(4.0,0.9) \usefont{T1}{ptm}{m}{n} \rput(3.4948437,0.205){-} \psline[linewidth=0.04cm]?(0.6,-0.7)(0.6,0.9) \psbezier[linewidth=0.04]?(0.0,-0.1)(0.0,-0.9)(1.2,-0.9)(1.2,-0.1) \psline[linewidth=0.04cm]?(0.0,-0.1)(0.0,0.9) \psline[linewidth=0.04cm]?(1.2,-0.1)(1.2,0.9) \usefont{T1}{ptm}{m}{n} \rput(1.6948438,0.205){=} \end{pspicture} } $$
\begin{equation} \label{eq1} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \} \end{equation} (three-point function) is zero.
Using the LinearOperator? domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.
\begin{axiom}
ω:𝐋 := ( Y I ) / U - ( I Y ) / U;
\end{axiom}
Definition 2
An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.
The Cartan-Killing Trace \begin{axiom}
Ú:= ( Y Λ ) / ( Y I ) / V Ù:= ( Λ Y ) / ( I Y ) / V
test(Ù=Ú)
\end{axiom} forms a non-degenerate associative scalar product for Y \begin{axiom} Ũ := Ù test ( Y I ) / Ũ = ( I Y ) / Ũ determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim) \end{axiom}
General Solution
Frobenius Form (co-unit) \begin{axiom} d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4+εij*𝐝.5+εik*𝐝.6+εjk*𝐝.7+εijk*𝐝.8 Ų:= Y/d \end{axiom}
In general the pairing is not symmetric! \begin{axiom} u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim) --eigenvectors(u1::Matrix FRAC POLY INT) \end{axiom}
The scalar product must be non-degenerate: \begin{axiom} --Ů:=determinant u1 --factor(numer Ů)/factor(denom Ů) 1 \end{axiom}
Cartan-Killing is a special case \begin{axiom} ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]?).1 \end{axiom}
Frobenius scalar product of "vector" quaternions $a$ and $b$ \begin{axiom} a:=sb('a,[1]?)*i+sb('a,[2]?)*j+sb('a,[3]?)*k b:=sb('b,[1]?)*i+sb('b,[2]?)*j+sb('b,[3]?)*k (a,a)/Ų (b,b)/Ų (a,b)/Ų \end{axiom}
Definition 3
Co-scalar product
Solve the [Snake Relation]? as a system of linear equations. \begin{axiom} mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim); Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim); ΩX:=Ω/X; --matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim) \end{axiom}
The common demoninator is $1/\sqrt{\mathring{U}}$ \begin{axiom} --squareFreePart factor denom Ů / squareFreePart factor numer Ů matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim) \end{axiom}
Check "dimension" and the snake relations. \begin{axiom}
O:𝐋:= Ω / Ų
test ( I ΩX ) / ( Ų I ) = I
test ( ΩX I ) / ( I Ų ) = I
\end{axiom}
Cartan-Killing co-scalar \begin{axiom} eval(Ω,ck) \end{axiom}
Definition 4
Co-algebra
Compute the "three-point" function and use it to define co-multiplication. \begin{axiom}
W:= (Y I) / Ų;
\end{axiom}
\begin{axiom}
λ:= _ ( I ΩX ) / _ ( Y I );
test ( ΩX I ) / ( I Y ) = λ
\end{axiom}
Cartan-Killing co-multiplication \begin{axiom} eval(λ,ck) \end{axiom}
Co-associativity \begin{axiom} test( ( λ ) / _ ( I λ ) = _ ( λ ) / _ ( λ I ) ) \end{axiom}
\begin{axiom}
test e / λ = ΩX
\end{axiom}
Some or all expressions may not have rendered properly, because Axiom returned the following error:
Error: export AXIOM=/usr/local/lib/fricas/target/x86_64-unknown-linux; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-unknown-linux/lib; LANG=en_US.UTF-8 $AXIOM/bin/AXIOMsys < /var/zope2/var/LatexWiki/110887736120721107-25px.axm KilledChecking for foreign routines AXIOM="/usr/local/lib/fricas/target/x86_64-unknown-linux" spad-lib="/usr/local/lib/fricas/target/x86_64-unknown-linux/lib/libspad.so" foreign routines found openServer result -2 FriCAS (AXIOM fork) Computer Algebra System Version: FriCAS 2010-12-08 Timestamp: Tuesday April 5, 2011 at 13:07:45 ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave FriCAS and return to shell. -----------------------------------------------------------------------------
(1) -> (1) -> (1) -> (1) -> (1) -> )library CARTEN ARITY CMONAL CPROP CLOP CALEY
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/zope2/var/LatexWiki/CARTEN.NRLIB/CARTEN Arity is now explicitly exposed in frame initial Arity will be automatically loaded when needed from /var/zope2/var/LatexWiki/ARITY.NRLIB/ARITY ClosedMonoidal is now explicitly exposed in frame initial ClosedMonoidal will be automatically loaded when needed from /var/zope2/var/LatexWiki/CMONAL.NRLIB/CMONAL ClosedProp is now explicitly exposed in frame initial ClosedProp will be automatically loaded when needed from /var/zope2/var/LatexWiki/CPROP.NRLIB/CPROP ClosedLinearOperator is now explicitly exposed in frame initial ClosedLinearOperator will be automatically loaded when needed from /var/zope2/var/LatexWiki/CLOP.NRLIB/CLOP CaleyDickson is now explicitly exposed in frame initial CaleyDickson will be automatically loaded when needed from /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY (1) -> -- summation macro Σ(x,i,n)==reduce(+,[x for i in n])
- Type: Void
- list macro Ξ(f,i,n)==[f for i in n]
- Type: Void
- subscript and superscripts macro sb == subscript
Type: Void macro sp == superscript
Type: Void (5) -> dim:=8
$$ 8 \leqno(5) $$
Type: PositiveInteger macro ℒ == List
Type: Void macro ℂ == CaleyDickson
Type: Void macro ℚ == Expression Integer
Type: Void 𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k,'ij,'ik,'jk,'ijk], ℚ)
$$ ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) \leqno(9) $$
Type: Type 𝐞:ℒ 𝐋 := basisOut()
$$ \left[ {| \sb {{ \ 1}}}, \: {| \sb {{ \ i}}}, \: {| \sb {{ \ j}}}, \: {| \sb {{ \ k}}}, \: {| \sb {{ \ ij}}}, \: {| \sb {{ \ ik}}}, \: {| \sb {{ \ jk}}}, \: {| \sb {{ \ ijk}}} \right] \leqno(10) $$
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))) 𝐝:ℒ 𝐋 := basisIn()
$$ \left[ {| \sp {{ \ 1}}}, \: {| \sp {{ \ i}}}, \: {| \sp {{ \ j}}}, \: {| \sp {{ \ k}}}, \: {| \sp {{ \ ij}}}, \: {| \sp {{ \ ik}}}, \: {| \sp {{ \ jk}}}, \: {| \sp {{ \ ijk}}} \right] \leqno(11) $$
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))) I:𝐋:=[1] -- identity for composition
$$ {| \sb {{ \ 1}} \sp {{ \ 1}}}+{| \sb {{ \ i}} \sp {{ \ i}}}+{| \sb {{ \ j}} \sp {{ \ j}}}+{| \sb {{ \ k}} \sp {{ \ k}}}+{| \sb {{ \ ij}} \sp {{ \ ij}}}+{| \sb {{ \ ik}} \sp {{ \ ik}}}+{| \sb {{ \ jk}} \sp {{ \ jk}}}+{| \sb {{ \ ijk}} \sp {{ \ ijk}}} \leqno(12) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) X:𝐋:=[2,1] -- twist
$$ {| \sb {{ \ 1 \ 1}} \sp {{ \ 1 \ 1}}}+{| \sb {{ \ i \ 1}} \sp {{ \ 1 \ i}}}+{| \sb {{ \ j \ 1}} \sp {{ \ 1 \ j}}}+{| \sb {{ \ k \ 1}} \sp {{ \ 1 \ k}}}+{| \sb {{ \ ij \ 1}} \sp {{ \ 1 \ ij}}}+{| \sb {{ \ ik \ 1}} \sp {{ \ 1 \ ik}}}+{| \sb {{ \ jk \ 1}} \sp {{ \ 1 \ jk}}}+{| \sb {{ \ ijk \ 1}} \sp {{ \ 1 \ ijk}}}+{| \sb {{ \ 1 \ i}} \sp {{ \ i \ 1}}}+{| \sb {{ \ i \ i}} \sp {{ \ i \ i}}}+{| \sb {{ \ j \ i}} \sp {{ \ i \ j}}}+{| \sb {{ \ k \ i}} \sp {{ \ i \ k}}}+{| \sb {{ \ ij \ i}} \sp {{ \ i \ ij}}}+{| \sb {{ \ ik \ i}} \sp {{ \ i \ ik}}}+{| \sb {{ \ jk \ i}} \sp {{ \ i \ jk}}}+{| \sb {{ \ ijk \ i}} \sp {{ \ i \ ijk}}}+{| \sb {{ \ 1 \ j}} \sp {{ \ j \ 1}}}+{| \sb {{ \ i \ j}} \sp {{ \ j \ i}}}+{| \sb {{ \ j \ j}} \sp {{ \ j \ j}}}+{| \sb {{ \ k \ j}} \sp {{ \ j \ k}}}+{| \sb {{ \ ij \ j}} \sp {{ \ j \ ij}}}+{| \sb {{ \ ik \ j}} \sp {{ \ j \ ik}}}+{| \sb {{ \ jk \ j}} \sp {{ \ j \ jk}}}+{| \sb {{ \ ijk \ j}} \sp {{ \ j \ ijk}}}+{| \sb {{ \ 1 \ k}} \sp {{ \ k \ 1}}}+{| \sb {{ \ i \ k}} \sp {{ \ k \ i}}}+{| \sb {{ \ j \ k}} \sp {{ \ k \ j}}}+{| \sb {{ \ k \ k}} \sp {{ \ k \ k}}}+{| \sb {{ \ ij \ k}} \sp {{ \ k \ ij}}}+{| \sb {{ \ ik \ k}} \sp {{ \ k \ ik}}}+{| \sb {{ \ jk \ k}} \sp {{ \ k \ jk}}}+{| \sb {{ \ ijk \ k}} \sp {{ \ k \ ijk}}}+{| \sb {{ \ 1 \ ij}} \sp {{ \ ij \ 1}}}+{| \sb {{ \ i \ ij}} \sp {{ \ ij \ i}}}+{| \sb {{ \ j \ ij}} \sp {{ \ ij \ j}}}+{| \sb {{ \ k \ ij}} \sp {{ \ ij \ k}}}+{| \sb {{ \ ij \ ij}} \sp {{ \ ij \ ij}}}+{| \sb {{ \ ik \ ij}} \sp {{ \ ij \ ik}}}+{| \sb {{ \ jk \ ij}} \sp {{ \ ij \ jk}}}+{| \sb {{ \ ijk \ ij}} \sp {{ \ ij \ ijk}}}+{| \sb {{ \ 1 \ ik}} \sp {{ \ ik \ 1}}}+{| \sb {{ \ i \ ik}} \sp {{ \ ik \ i}}}+{| \sb {{ \ j \ ik}} \sp {{ \ ik \ j}}}+{| \sb {{ \ k \ ik}} \sp {{ \ ik \ k}}}+{| \sb {{ \ ij \ ik}} \sp {{ \ ik \ ij}}}+{| \sb {{ \ ik \ ik}} \sp {{ \ ik \ ik}}}+{| \sb {{ \ jk \ ik}} \sp {{ \ ik \ jk}}}+{| \sb {{ \ ijk \ ik}} \sp {{ \ ik \ ijk}}}+{| \sb {{ \ 1 \ jk}} \sp {{ \ jk \ 1}}}+{| \sb {{ \ i \ jk}} \sp {{ \ jk \ i}}}+{| \sb {{ \ j \ jk}} \sp {{ \ jk \ j}}}+{| \sb {{ \ k \ jk}} \sp {{ \ jk \ k}}}+{| \sb {{ \ ij \ jk}} \sp {{ \ jk \ ij}}}+{| \sb {{ \ ik \ jk}} \sp {{ \ jk \ ik}}}+{| \sb {{ \ jk \ jk}} \sp {{ \ jk \ jk}}}+{| \sb {{ \ ijk \ jk}} \sp {{ \ jk \ ijk}}}+{| \sb {{ \ 1 \ ijk}} \sp {{ \ ijk \ 1}}}+{| \sb {{ \ i \ ijk}} \sp {{ \ ijk \ i}}}+{| \sb {{ \ j \ ijk}} \sp {{ \ ijk \ j}}}+{| \sb {{ \ k \ ijk}} \sp {{ \ ijk \ k}}}+{| \sb {{ \ ij \ ijk}} \sp {{ \ ijk \ ij}}}+{| \sb {{ \ ik \ ijk}} \sp {{ \ ijk \ ik}}}+{| \sb {{ \ jk \ ijk}} \sp {{ \ ijk \ jk}}}+{| \sb {{ \ ijk \ ijk}} \sp {{ \ ijk \ ijk}}} \leqno(13) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) V:𝐋:=ev(1) -- evaluation
$$ {| \sp {{ \ 1 \ 1}}}+{| \sp {{ \ i \ i}}}+{| \sp {{ \ j \ j}}}+{| \sp {{ \ k \ k}}}+{| \sp {{ \ ij \ ij}}}+{| \sp {{ \ ik \ ik}}}+{| \sp {{ \ jk \ jk}}}+{| \sp {{ \ ijk \ ijk}}} \leqno(14) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) Λ:𝐋:=co(1) -- co-evaluation
$$ {| \sb {{ \ 1 \ 1}}}+{| \sb {{ \ i \ i}}}+{| \sb {{ \ j \ j}}}+{| \sb {{ \ k \ k}}}+{| \sb {{ \ ij \ ij}}}+{| \sb {{ \ ik \ ik}}}+{| \sb {{ \ jk \ jk}}}+{| \sb {{ \ ijk \ ijk}}} \leqno(15) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
Type: Void (17) -> q0:=sp('i,[2])
$$ i \sp {2} \leqno(17) $$
Type: Symbol q1:=sp('j,[2])
$$ j \sp {2} \leqno(18) $$
Type: Symbol q2:=sp('k,[2])
$$ k \sp {2} \leqno(19) $$
Type: Symbol QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]])
$$ CliffordAlgebra(3,Expression(Integer),[[*001i(2),0,0],[0,*001j(2),0],[0,0,*001k(2)]]) \leqno(20) $$
Type: Type B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[3]),monomial(1,[1,2]),monomial(1,[1,3]),monomial(1,[2,3]),monomial(1,[1,2,3])]
$$ \left[ 1, \: {e \sb {1}}, \: {e \sb {2}}, \: {e \sb {3}}, \: {{e \sb {1}} \ {e \sb {2}}}, \: {{e \sb {1}} \ {e \sb {3}}}, \: {{e \sb {2}} \ {e \sb {3}}}, \: {{e \sb {1}} \ {e \sb {2}} \ {e \sb {3}}} \right] \leqno(21) $$
Type: List(CliffordAlgebra(3,Expression(Integer),[[*001i(2),0,0],[0,*001j(2),0],[0,0,*001k(2)]])) M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)
$$ \left[ \begin{array}{cccccccc} 1 & {e \sb {1}} & {e \sb {2}} & {e \sb {3}} & {{e \sb {1}} \ {e \sb {2}}} & {{e \sb {1}} \ {e \sb {3}}} & {{e \sb {2}} \ {e \sb {3}}} & {{e \sb {1}} \ {e \sb {2}} \ {e \sb {3}}} \ {e \sb {1}} & {i \sp {2}} & -{{e \sb {1}} \ {e \sb {2}}} & -{{e \sb {1}} \ {e \sb {3}}} & -{{i \sp {2}} \ {e \sb {2}}} & -{{i \sp {2}} \ {e \sb {3}}} & {{e \sb {1}} \ {e \sb {2}} \ {e \sb {3}}} & {{i \sp {2}} \ {e \sb {2}} \ {e \sb {3}}} \ {e \sb {2}} & {{e \sb {1}} \ {e \sb {2}}} & {j \sp {2}} & -{{e \sb {2}} \ {e \sb {3}}} & {{j \sp {2}} \ {e \sb {1}}} & -{{e \sb {1}} \ {e \sb {2}} \ {e \sb {3}}} & -{{j \sp {2}} \ {e \sb {3}}} & -{{j \sp {2}} \ {e \sb {1}} \ {e \sb {3}}} \ {e \sb {3}} & {{e \sb {1}} \ {e \sb {3}}} & {{e \sb {2}} \ {e \sb {3}}} & {k \sp {2}} & {{e \sb {1}} \ {e \sb {2}} \ {e \sb {3}}} & {{k \sp {2}} \ {e \sb {1}}} & {{k \sp {2}} \ {e \sb {2}}} & {{k \sp {2}} \ {e \sb {1}} \ {e \sb {2}}} \ {{e \sb {1}} \ {e \sb {2}}} & {{i \sp {2}} \ {e \sb {2}}} & -{{j \sp {2}} \ {e \sb {1}}} & {{e \sb {1}} \ {e \sb {2}} \ {e \sb {3}}} & -{{i \sp {2}} \ {j \sp {2}}} & {{i \sp {2}} \ {e \sb {2}} \ {e \sb {3}}} & -{{j \sp {2}} \ {e \sb {1}} \ {e \sb {3}}} & -{{i \sp {2}} \ {j \sp {2}} \ {e \sb {3}}} \ {{e \sb {1}} \ {e \sb {3}}} & {{i \sp {2}} \ {e \sb {3}}} & -{{e \sb {1}} \ {e \sb {2}} \ {e \sb {3}}} & -{{k \sp {2}} \ {e \sb {1}}} & -{{i \sp {2}} \ {e \sb {2}} \ {e \sb {3}}} & -{{i \sp {2}} \ {k \sp {2}}} & {{k \sp {2}} \ {e \sb {1}} \ {e \sb {2}}} & {{i \sp {2}} \ {k \sp {2}} \ {e \sb {2}}} \ {{e \sb {2}} \ {e \sb {3}}} & {{e \sb {1}} \ {e \sb {2}} \ {e \sb {3}}} & {{j \sp {2}} \ {e \sb {3}}} & -{{k \sp {2}} \ {e \sb {2}}} & {{j \sp {2}} \ {e \sb {1}} \ {e \sb {3}}} & -{{k \sp {2}} \ {e \sb {1}} \ {e \sb {2}}} & -{{j \sp {2}} \ {k \sp {2}}} & -{{j \sp {2}} \ {k \sp {2}} \ {e \sb {1}}} \ {{e \sb {1}} \ {e \sb {2}} \ {e \sb {3}}} & {{i \sp {2}} \ {e \sb {2}} \ {e \sb {3}}} & -{{j \sp {2}} \ {e \sb {1}} \ {e \sb {3}}} & {{k \sp {2}} \ {e \sb {1}} \ {e \sb {2}}} & -{{i \sp {2}} \ {j \sp {2}} \ {e \sb {3}}} & {{i \sp {2}} \ {k \sp {2}} \ {e \sb {2}}} & -{{j \sp {2}} \ {k \sp {2}} \ {e \sb {1}}} & -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}}} \end{array} \right] \leqno(22) $$
Type: Matrix(CliffordAlgebra(3,Expression(Integer),[[*001i(2),0,0],[0,*001j(2),0],[0,0,001k(2)]])) S(y) == map(x +-> coefficient(recip(y)x,[]),M)
Type: Void ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
Compiling function S with type CliffordAlgebra(3,Expression(Integer) ,[[*001i(2),0,0],[0,*001j(2),0],[0,0,*001k(2)]]) -> Matrix( Expression(Integer))
$$ \left[ {\left[ {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: {i \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: {j \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: {k \sp {2}}, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \ {j \sp {2}}}, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \ {k \sp {2}}}, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{j \sp {2}} \ {k \sp {2}}}, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}}} \right]} \right]}, \: {\left[ {\left[ 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: {j \sp {2}}, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: {k \sp {2}}, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: -{j \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: -{k \sp {2}}, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{j \sp {2}} \ {k \sp {2}}} \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{j \sp {2}} \ {k \sp {2}}}, \: 0 \right]} \right]}, \: {\left[ {\left[ 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: -{i \sp {2}}, \: 0, \: 0, \: 0 \right]}, \: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {k \sp {2}}, \: 0 \right]}, \: {\left[ 0, \: {i \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {{i \sp {2}} \ {k \sp {2}}} \right]}, \: {\left[ 0, \: 0, \: 0, \: -{k \sp {2}}, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: {{i \sp {2}} \ {k \sp {2}}}, \: 0, \: 0 \right]} \right]}, \: {\left[ {\left[ 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -{i \sp {2}}, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{j \sp {2}}, \: 0 \right]}, \: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \ {j \sp {2}}} \right]}, \: {\left[ 0, \: {i \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: {j \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \ {j \sp {2}}}, \: 0, \: 0, \: 0 \right]} \right]}, \: {\left[ {\left[ 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: -1, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {k \sp {2}} \right]}, \: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {k \sp {2}}, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -{k \sp {2}}, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: {k \sp {2}}, \: 0, \: 0, \: 0, \: 0 \right]} \right]}, \: {\left[ {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: -1, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{j \sp {2}} \right]}, \: {\left[ 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{j \sp {2}}, \: 0 \right]}, \: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: {j \sp {2}}, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: -{j \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0 \right]} \right]}, \: {\left[ {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {i \sp {2}} \right]}, \: {\left[ 0, \: 0, \: 0, \: -1, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: {i \sp {2}}, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: -{i \sp {2}}, \: 0, \: 0, \: 0 \right]}, \: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: {i \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]} \right]}, \: {\left[ {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 1 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -1, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 0, \: -1, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]} \right]} \right] \leqno(24) $$
- Type: List(List(List(Expression(Integer))))
- structure constants form a tensor operator Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim)
$$ {| \sb {{ \ 1}} \sp {{ \ 1 \ 1}}}+{| \sb {{ \ i}} \sp {{ \ 1 \ i}}}+{| \sb {{ \ j}} \sp {{ \ 1 \ j}}}+{| \sb {{ \ k}} \sp {{ \ 1 \ k}}}+{| \sb {{ \ ij}} \sp {{ \ 1 \ ij}}}+{| \sb {{ \ ik}} \sp {{ \ 1 \ ik}}}+{| \sb {{ \ jk}} \sp {{ \ 1 \ jk}}}+{| \sb {{ \ ijk}} \sp {{ \ 1 \ ijk}}}+{| \sb {{ \ i}} \sp {{ \ i \ 1}}}+{{i \sp {2}} \ {| \sb {{ \ 1}} \sp {{ \ i \ i}}}}+{| \sb {{ \ ij}} \sp {{ \ i \ j}}}+{| \sb {{ \ ik}} \sp {{ \ i \ k}}}+{{i \sp {2}} \ {| \sb {{ \ j}} \sp {{ \ i \ ij}}}}+{{i \sp {2}} \ {| \sb {{ \ k}} \sp {{ \ i \ ik}}}}+{| \sb {{ \ ijk}} \sp {{ \ i \ jk}}}+{{i \sp {2}} \ {| \sb {{ \ jk}} \sp {{ \ i \ ijk}}}}+{| \sb {{ \ j}} \sp {{ \ j \ 1}}} -{| \sb {{ \ ij}} \sp {{ \ j \ i}}}+{{j \sp {2}} \ {| \sb {{ \ 1}} \sp {{ \ j \ j}}}}+{| \sb {{ \ jk}} \sp {{ \ j \ k}}} -{{j \sp {2}} \ {| \sb {{ \ i}} \sp {{ \ j \ ij}}}} -{| \sb {{ \ ijk}} \sp {{ \ j \ ik}}}+{{j \sp {2}} \ {| \sb {{ \ k}} \sp {{ \ j \ jk}}}} -{{j \sp {2}} \ {| \sb {{ \ ik}} \sp {{ \ j \ ijk}}}}+{| \sb {{ \ k}} \sp {{ \ k \ 1}}} -{| \sb {{ \ ik}} \sp {{ \ k \ i}}} -{| \sb {{ \ jk}} \sp {{ \ k \ j}}}+{{k \sp {2}} \ {| \sb {{ \ 1}} \sp {{ \ k \ k}}}}+{| \sb {{ \ ijk}} \sp {{ \ k \ ij}}} -{{k \sp {2}} \ {| \sb {{ \ i}} \sp {{ \ k \ ik}}}} -{{k \sp {2}} \ {| \sb {{ \ j}} \sp {{ \ k \ jk}}}}+{{k \sp {2}} \ {| \sb {{ \ ij}} \sp {{ \ k \ ijk}}}}+{| \sb {{ \ ij}} \sp {{ \ ij \ 1}}} -{{i \sp {2}} \ {| \sb {{ \ j}} \sp {{ \ ij \ i}}}}+{{j \sp {2}} \ {| \sb {{ \ i}} \sp {{ \ ij \ j}}}}+{| \sb {{ \ ijk}} \sp {{ \ ij \ k}}} -{{i \sp {2}} \ {j \sp {2}} \ {| \sb {{ \ 1}} \sp {{ \ ij \ ij}}}} -{{i \sp {2}} \ {| \sb {{ \ jk}} \sp {{ \ ij \ ik}}}}+{{j \sp {2}} \ {| \sb {{ \ ik}} \sp {{ \ ij \ jk}}}} -{{i \sp {2}} \ {j \sp {2}} \ {| \sb {{ \ k}} \sp {{ \ ij \ ijk}}}}+{| \sb {{ \ ik}} \sp {{ \ ik \ 1}}} -{{i \sp {2}} \ {| \sb {{ \ k}} \sp {{ \ ik \ i}}}} -{| \sb {{ \ ijk}} \sp {{ \ ik \ j}}}+{{k \sp {2}} \ {| \sb {{ \ i}} \sp {{ \ ik \ k}}}}+{{i \sp {2}} \ {| \sb {{ \ jk}} \sp {{ \ ik \ ij}}}} -{{i \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ 1}} \sp {{ \ ik \ ik}}}} -{{k \sp {2}} \ {| \sb {{ \ ij}} \sp {{ \ ik \ jk}}}}+{{i \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ j}} \sp {{ \ ik \ ijk}}}}+{| \sb {{ \ jk}} \sp {{ \ jk \ 1}}}+{| \sb {{ \ ijk}} \sp {{ \ jk \ i}}} -{{j \sp {2}} \ {| \sb {{ \ k}} \sp {{ \ jk \ j}}}}+{{k \sp {2}} \ {| \sb {{ \ j}} \sp {{ \ jk \ k}}}} -{{j \sp {2}} \ {| \sb {{ \ ik}} \sp {{ \ jk \ ij}}}}+{{k \sp {2}} \ {| \sb {{ \ ij}} \sp {{ \ jk \ ik}}}} -{{j \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ 1}} \sp {{ \ jk \ jk}}}} -{{j \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ i}} \sp {{ \ jk \ ijk}}}}+{| \sb {{ \ ijk}} \sp {{ \ ijk \ 1}}}+{{i \sp {2}} \ {| \sb {{ \ jk}} \sp {{ \ ijk \ i}}}} -{{j \sp {2}} \ {| \sb {{ \ ik}} \sp {{ \ ijk \ j}}}}+{{k \sp {2}} \ {| \sb {{ \ ij}} \sp {{ \ ijk \ k}}}} -{{i \sp {2}} \ {j \sp {2}} \ {| \sb {{ \ k}} \sp {{ \ ijk \ ij}}}}+{{i \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ j}} \sp {{ \ ijk \ ik}}}} -{{j \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ i}} \sp {{ \ ijk \ jk}}}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ 1}} \sp {{ \ ijk \ ijk}}}} \leqno(25) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)
$$ \left[ \begin{array}{cccccccc} {| \sb {{ \ 1}}} & {| \sb {{ \ i}}} & {| \sb {{ \ j}}} & {| \sb {{ \ k}}} & {| \sb {{ \ ij}}} & {| \sb {{ \ ik}}} & {| \sb {{ \ jk}}} & {| \sb {{ \ ijk}}} \ {| \sb {{ \ i}}} & {{i \sp {2}} \ {| \sb {{ \ 1}}}} & -{| \sb {{ \ ij}}} & -{| \sb {{ \ ik}}} & -{{i \sp {2}} \ {| \sb {{ \ j}}}} & -{{i \sp {2}} \ {| \sb {{ \ k}}}} & {| \sb {{ \ ijk}}} & {{i \sp {2}} \ {| \sb {{ \ jk}}}} \ {| \sb {{ \ j}}} & {| \sb {{ \ ij}}} & {{j \sp {2}} \ {| \sb {{ \ 1}}}} & -{| \sb {{ \ jk}}} & {{j \sp {2}} \ {| \sb {{ \ i}}}} & -{| \sb {{ \ ijk}}} & -{{j \sp {2}} \ {| \sb {{ \ k}}}} & -{{j \sp {2}} \ {| \sb {{ \ ik}}}} \ {| \sb {{ \ k}}} & {| \sb {{ \ ik}}} & {| \sb {{ \ jk}}} & {{k \sp {2}} \ {| \sb {{ \ 1}}}} & {| \sb {{ \ ijk}}} & {{k \sp {2}} \ {| \sb {{ \ i}}}} & {{k \sp {2}} \ {| \sb {{ \ j}}}} & {{k \sp {2}} \ {| \sb {{ \ ij}}}} \ {| \sb {{ \ ij}}} & {{i \sp {2}} \ {| \sb {{ \ j}}}} & -{{j \sp {2}} \ {| \sb {{ \ i}}}} & {| \sb {{ \ ijk}}} & -{{i \sp {2}} \ {j \sp {2}} \ {| \sb {{ \ 1}}}} & {{i \sp {2}} \ {| \sb {{ \ jk}}}} & -{{j \sp {2}} \ {| \sb {{ \ ik}}}} & -{{i \sp {2}} \ {j \sp {2}} \ {| \sb {{ \ k}}}} \ {| \sb {{ \ ik}}} & {{i \sp {2}} \ {| \sb {{ \ k}}}} & -{| \sb {{ \ ijk}}} & -{{k \sp {2}} \ {| \sb {{ \ i}}}} & -{{i \sp {2}} \ {| \sb {{ \ jk}}}} & -{{i \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ 1}}}} & {{k \sp {2}} \ {| \sb {{ \ ij}}}} & {{i \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ j}}}} \ {| \sb {{ \ jk}}} & {| \sb {{ \ ijk}}} & {{j \sp {2}} \ {| \sb {{ \ k}}}} & -{{k \sp {2}} \ {| \sb {{ \ j}}}} & {{j \sp {2}} \ {| \sb {{ \ ik}}}} & -{{k \sp {2}} \ {| \sb {{ \ ij}}}} & -{{j \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ 1}}}} & -{{j \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ i}}}} \ {| \sb {{ \ ijk}}} & {{i \sp {2}} \ {| \sb {{ \ jk}}}} & -{{j \sp {2}} \ {| \sb {{ \ ik}}}} & {{k \sp {2}} \ {| \sb {{ \ ij}}}} & -{{i \sp {2}} \ {j \sp {2}} \ {| \sb {{ \ k}}}} & {{i \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ j}}}} & -{{j \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ i}}}} & -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {| \sb {{ \ 1}}}} \end{array} \right] \leqno(26) $$
Type: Matrix(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))) XY := X/Y;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (28) -> e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (29) -> a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)
$$ {{a \sb {1}} \ {| \sb {{ \ 1}}}}+{{a \sb {2}} \ {| \sb {{ \ i}}}}+{{a \sb {3}} \ {| \sb {{ \ j}}}}+{{a \sb {4}} \ {| \sb {{ \ k}}}}+{{a \sb {5}} \ {| \sb {{ \ ij}}}}+{{a \sb {6}} \ {| \sb {{ \ ik}}}}+{{a \sb {7}} \ {| \sb {{ \ jk}}}}+{{a \sb {8}} \ {| \sb {{ \ ijk}}}} \leqno(29) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)
$$ {{b \sb {1}} \ {| \sb {{ \ 1}}}}+{{b \sb {2}} \ {| \sb {{ \ i}}}}+{{b \sb {3}} \ {| \sb {{ \ j}}}}+{{b \sb {4}} \ {| \sb {{ \ k}}}}+{{b \sb {5}} \ {| \sb {{ \ ij}}}}+{{b \sb {6}} \ {| \sb {{ \ ik}}}}+{{b \sb {7}} \ {| \sb {{ \ jk}}}}+{{b \sb {8}} \ {| \sb {{ \ ijk}}}} \leqno(30) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (a*b)/Y
$$ {{\left( -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {a \sb {8}} \ {b \sb {8}}} -{{j \sp {2}} \ {k \sp {2}} \ {a \sb {7}} \ {b \sb {7}}} -{{i \sp {2}} \ {k \sp {2}} \ {a \sb {6}} \ {b \sb {6}}} -{{i \sp {2}} \ {j \sp {2}} \ {a \sb {5}} \ {b \sb {5}}}+{{k \sp {2}} \ {a \sb {4}} \ {b \sb {4}}}+{{j \sp {2}} \ {a \sb {3}} \ {b \sb {3}}}+{{i \sp {2}} \ {a \sb {2}} \ {b \sb {2}}}+{{a \sb {1}} \ {b \sb {1}}} \right)} \ {| \sb {{ \ 1}}}}+{{\left( -{{j \sp {2}} \ {k \sp {2}} \ {a \sb {7}} \ {b \sb {8}}} -{{j \sp {2}} \ {k \sp {2}} \ {a \sb {8}} \ {b \sb {7}}} -{{k \sp {2}} \ {a \sb {4}} \ {b \sb {6}}} -{{j \sp {2}} \ {a \sb {3}} \ {b \sb {5}}}+{{k \sp {2}} \ {a \sb {6}} \ {b \sb {4}}}+{{j \sp {2}} \ {a \sb {5}} \ {b \sb {3}}}+{{a \sb {1}} \ {b \sb {2}}}+{{a \sb {2}} \ {b \sb {1}}} \right)} \ {| \sb {{ \ i}}}}+{{\left( {{i \sp {2}} \ {k \sp {2}} \ {a \sb {6}} \ {b \sb {8}}} -{{k \sp {2}} \ {a \sb {4}} \ {b \sb {7}}}+{{i \sp {2}} \ {k \sp {2}} \ {a \sb {8}} \ {b \sb {6}}}+{{i \sp {2}} \ {a \sb {2}} \ {b \sb {5}}}+{{k \sp {2}} \ {a \sb {7}} \ {b \sb {4}}}+{{a \sb {1}} \ {b \sb {3}}} -{{i \sp {2}} \ {a \sb {5}} \ {b \sb {2}}}+{{a \sb {3}} \ {b \sb {1}}} \right)} \ {| \sb {{ \ j}}}}+{{\left( -{{i \sp {2}} \ {j \sp {2}} \ {a \sb {5}} \ {b \sb {8}}}+{{j \sp {2}} \ {a \sb {3}} \ {b \sb {7}}}+{{i \sp {2}} \ {a \sb {2}} \ {b \sb {6}}} -{{i \sp {2}} \ {j \sp {2}} \ {a \sb {8}} \ {b \sb {5}}}+{{a \sb {1}} \ {b \sb {4}}} -{{j \sp {2}} \ {a \sb {7}} \ {b \sb {3}}} -{{i \sp {2}} \ {a \sb {6}} \ {b \sb {2}}}+{{a \sb {4}} \ {b \sb {1}}} \right)} \ {| \sb {{ \ k}}}}+{{\left( {{k \sp {2}} \ {a \sb {4}} \ {b \sb {8}}} -{{k \sp {2}} \ {a \sb {6}} \ {b \sb {7}}}+{{k \sp {2}} \ {a \sb {7}} \ {b \sb {6}}}+{{a \sb {1}} \ {b \sb {5}}}+{{k \sp {2}} \ {a \sb {8}} \ {b \sb {4}}}+{{a \sb {2}} \ {b \sb {3}}} -{{a \sb {3}} \ {b \sb {2}}}+{{a \sb {5}} \ {b \sb {1}}} \right)} \ {| \sb {{ \ ij}}}}+{{\left( -{{j \sp {2}} \ {a \sb {3}} \ {b \sb {8}}}+{{j \sp {2}} \ {a \sb {5}} \ {b \sb {7}}}+{{a \sb {1}} \ {b \sb {6}}} -{{j \sp {2}} \ {a \sb {7}} \ {b \sb {5}}}+{{a \sb {2}} \ {b \sb {4}}} -{{j \sp {2}} \ {a \sb {8}} \ {b \sb {3}}} -{{a \sb {4}} \ {b \sb {2}}}+{{a \sb {6}} \ {b \sb {1}}} \right)} \ {| \sb {{ \ ik}}}}+{{\left( {{i \sp {2}} \ {a \sb {2}} \ {b \sb {8}}}+{{a \sb {1}} \ {b \sb {7}}} -{{i \sp {2}} \ {a \sb {5}} \ {b \sb {6}}}+{{i \sp {2}} \ {a \sb {6}} \ {b \sb {5}}}+{{a \sb {3}} \ {b \sb {4}}} -{{a \sb {4}} \ {b \sb {3}}}+{{i \sp {2}} \ {a \sb {8}} \ {b \sb {2}}}+{{a \sb {7}} \ {b \sb {1}}} \right)} \ {| \sb {{ \ jk}}}}+{{\left( {{a \sb {1}} \ {b \sb {8}}}+{{a \sb {2}} \ {b \sb {7}}} -{{a \sb {3}} \ {b \sb {6}}}+{{a \sb {4}} \ {b \sb {5}}}+{{a \sb {5}} \ {b \sb {4}}} -{{a \sb {6}} \ {b \sb {3}}}+{{a \sb {7}} \ {b \sb {2}}}+{{a \sb {8}} \ {b \sb {1}}} \right)} \ {| \sb {{ \ ijk}}}} \leqno(31) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (32) -> test( ( I Y ) / ( Y ) = ( Y I ) / _ ( Y ) )
$$ true \leqno(32) $$
Type: Boolean (33) -> U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
$$ {{u \sp {{1, \: 1}}} \ {| \sp {{ \ 1 \ 1}}}}+{{u \sp {{1, \: 2}}} \ {| \sp {{ \ 1 \ i}}}}+{{u \sp {{1, \: 3}}} \ {| \sp {{ \ 1 \ j}}}}+{{u \sp {{1, \: 4}}} \ {| \sp {{ \ 1 \ k}}}}+{{u \sp {{1, \: 5}}} \ {| \sp {{ \ 1 \ ij}}}}+{{u \sp {{1, \: 6}}} \ {| \sp {{ \ 1 \ ik}}}}+{{u \sp {{1, \: 7}}} \ {| \sp {{ \ 1 \ jk}}}}+{{u \sp {{1, \: 8}}} \ {| \sp {{ \ 1 \ ijk}}}}+{{u \sp {{2, \: 1}}} \ {| \sp {{ \ i \ 1}}}}+{{u \sp {{2, \: 2}}} \ {| \sp {{ \ i \ i}}}}+{{u \sp {{2, \: 3}}} \ {| \sp {{ \ i \ j}}}}+{{u \sp {{2, \: 4}}} \ {| \sp {{ \ i \ k}}}}+{{u \sp {{2, \: 5}}} \ {| \sp {{ \ i \ ij}}}}+{{u \sp {{2, \: 6}}} \ {| \sp {{ \ i \ ik}}}}+{{u \sp {{2, \: 7}}} \ {| \sp {{ \ i \ jk}}}}+{{u \sp {{2, \: 8}}} \ {| \sp {{ \ i \ ijk}}}}+{{u \sp {{3, \: 1}}} \ {| \sp {{ \ j \ 1}}}}+{{u \sp {{3, \: 2}}} \ {| \sp {{ \ j \ i}}}}+{{u \sp {{3, \: 3}}} \ {| \sp {{ \ j \ j}}}}+{{u \sp {{3, \: 4}}} \ {| \sp {{ \ j \ k}}}}+{{u \sp {{3, \: 5}}} \ {| \sp {{ \ j \ ij}}}}+{{u \sp {{3, \: 6}}} \ {| \sp {{ \ j \ ik}}}}+{{u \sp {{3, \: 7}}} \ {| \sp {{ \ j \ jk}}}}+{{u \sp {{3, \: 8}}} \ {| \sp {{ \ j \ ijk}}}}+{{u \sp {{4, \: 1}}} \ {| \sp {{ \ k \ 1}}}}+{{u \sp {{4, \: 2}}} \ {| \sp {{ \ k \ i}}}}+{{u \sp {{4, \: 3}}} \ {| \sp {{ \ k \ j}}}}+{{u \sp {{4, \: 4}}} \ {| \sp {{ \ k \ k}}}}+{{u \sp {{4, \: 5}}} \ {| \sp {{ \ k \ ij}}}}+{{u \sp {{4, \: 6}}} \ {| \sp {{ \ k \ ik}}}}+{{u \sp {{4, \: 7}}} \ {| \sp {{ \ k \ jk}}}}+{{u \sp {{4, \: 8}}} \ {| \sp {{ \ k \ ijk}}}}+{{u \sp {{5, \: 1}}} \ {| \sp {{ \ ij \ 1}}}}+{{u \sp {{5, \: 2}}} \ {| \sp {{ \ ij \ i}}}}+{{u \sp {{5, \: 3}}} \ {| \sp {{ \ ij \ j}}}}+{{u \sp {{5, \: 4}}} \ {| \sp {{ \ ij \ k}}}}+{{u \sp {{5, \: 5}}} \ {| \sp {{ \ ij \ ij}}}}+{{u \sp {{5, \: 6}}} \ {| \sp {{ \ ij \ ik}}}}+{{u \sp {{5, \: 7}}} \ {| \sp {{ \ ij \ jk}}}}+{{u \sp {{5, \: 8}}} \ {| \sp {{ \ ij \ ijk}}}}+{{u \sp {{6, \: 1}}} \ {| \sp {{ \ ik \ 1}}}}+{{u \sp {{6, \: 2}}} \ {| \sp {{ \ ik \ i}}}}+{{u \sp {{6, \: 3}}} \ {| \sp {{ \ ik \ j}}}}+{{u \sp {{6, \: 4}}} \ {| \sp {{ \ ik \ k}}}}+{{u \sp {{6, \: 5}}} \ {| \sp {{ \ ik \ ij}}}}+{{u \sp {{6, \: 6}}} \ {| \sp {{ \ ik \ ik}}}}+{{u \sp {{6, \: 7}}} \ {| \sp {{ \ ik \ jk}}}}+{{u \sp {{6, \: 8}}} \ {| \sp {{ \ ik \ ijk}}}}+{{u \sp {{7, \: 1}}} \ {| \sp {{ \ jk \ 1}}}}+{{u \sp {{7, \: 2}}} \ {| \sp {{ \ jk \ i}}}}+{{u \sp {{7, \: 3}}} \ {| \sp {{ \ jk \ j}}}}+{{u \sp {{7, \: 4}}} \ {| \sp {{ \ jk \ k}}}}+{{u \sp {{7, \: 5}}} \ {| \sp {{ \ jk \ ij}}}}+{{u \sp {{7, \: 6}}} \ {| \sp {{ \ jk \ ik}}}}+{{u \sp {{7, \: 7}}} \ {| \sp {{ \ jk \ jk}}}}+{{u \sp {{7, \: 8}}} \ {| \sp {{ \ jk \ ijk}}}}+{{u \sp {{8, \: 1}}} \ {| \sp {{ \ ijk \ 1}}}}+{{u \sp {{8, \: 2}}} \ {| \sp {{ \ ijk \ i}}}}+{{u \sp {{8, \: 3}}} \ {| \sp {{ \ ijk \ j}}}}+{{u \sp {{8, \: 4}}} \ {| \sp {{ \ ijk \ k}}}}+{{u \sp {{8, \: 5}}} \ {| \sp {{ \ ijk \ ij}}}}+{{u \sp {{8, \: 6}}} \ {| \sp {{ \ ijk \ ik}}}}+{{u \sp {{8, \: 7}}} \ {| \sp {{ \ ijk \ jk}}}}+{{u \sp {{8, \: 8}}} \ {| \sp {{ \ ijk \ ijk}}}} \leqno(33) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (34) -> ω:𝐋 := ( Y I ) / U - ( I Y ) / U;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (35) -> Ú:= ( Y Λ ) / ( Y I ) / V
$$ {8 \ {| \sp {{ \ 1 \ 1}}}}+{8 \ {i \sp {2}} \ {| \sp {{ \ i \ i}}}}+{8 \ {j \sp {2}} \ {| \sp {{ \ j \ j}}}}+{8 \ {k \sp {2}} \ {| \sp {{ \ k \ k}}}} -{8 \ {i \sp {2}} \ {j \sp {2}} \ {| \sp {{ \ ij \ ij}}}} -{8 \ {i \sp {2}} \ {k \sp {2}} \ {| \sp {{ \ ik \ ik}}}} -{8 \ {j \sp {2}} \ {k \sp {2}} \ {| \sp {{ \ jk \ jk}}}} -{8 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {| \sp {{ \ ijk \ ijk}}}} \leqno(35) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) Ù:= ( Λ Y ) / ( I Y ) / V
$$ {8 \ {| \sp {{ \ 1 \ 1}}}}+{8 \ {i \sp {2}} \ {| \sp {{ \ i \ i}}}}+{8 \ {j \sp {2}} \ {| \sp {{ \ j \ j}}}}+{8 \ {k \sp {2}} \ {| \sp {{ \ k \ k}}}} -{8 \ {i \sp {2}} \ {j \sp {2}} \ {| \sp {{ \ ij \ ij}}}} -{8 \ {i \sp {2}} \ {k \sp {2}} \ {| \sp {{ \ ik \ ik}}}} -{8 \ {j \sp {2}} \ {k \sp {2}} \ {| \sp {{ \ jk \ jk}}}} -{8 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {| \sp {{ \ ijk \ ijk}}}} \leqno(36) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
test(Ù=Ú)
$$ true \leqno(37) $$
Type: Boolean (38) -> Ũ := Ù
$$ {8 \ {| \sp {{ \ 1 \ 1}}}}+{8 \ {i \sp {2}} \ {| \sp {{ \ i \ i}}}}+{8 \ {j \sp {2}} \ {| \sp {{ \ j \ j}}}}+{8 \ {k \sp {2}} \ {| \sp {{ \ k \ k}}}} -{8 \ {i \sp {2}} \ {j \sp {2}} \ {| \sp {{ \ ij \ ij}}}} -{8 \ {i \sp {2}} \ {k \sp {2}} \ {| \sp {{ \ ik \ ik}}}} -{8 \ {j \sp {2}} \ {k \sp {2}} \ {| \sp {{ \ jk \ jk}}}} -{8 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {| \sp {{ \ ijk \ ijk}}}} \leqno(38) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) test ( Y I ) / Ũ = ( I Y ) / Ũ
$$ true \leqno(39) $$
Type: Boolean determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)
$$ {16777216} \ {{i \sp {2}} \sp 4} \ {{j \sp {2}} \sp 4} \ {{k \sp {2}} \sp 4} \leqno(40) $$
Type: Expression(Integer) (41) -> d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4+εij*𝐝.5+εik*𝐝.6+εjk*𝐝.7+εijk*𝐝.8
$$ {ε1 \ {| \sp {{ \ 1}}}}+{εi \ {| \sp {{ \ i}}}}+{εj \ {| \sp {{ \ j}}}}+{εk \ {| \sp {{ \ k}}}}+{εij \ {| \sp {{ \ ij}}}}+{εik \ {| \sp {{ \ ik}}}}+{εjk \ {| \sp {{ \ jk}}}}+{εijk \ {| \sp {{ \ ijk}}}} \leqno(41) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) Ų:= Y/d
$$ {ε1 \ {| \sp {{ \ 1 \ 1}}}}+{εi \ {| \sp {{ \ 1 \ i}}}}+{εj \ {| \sp {{ \ 1 \ j}}}}+{εk \ {| \sp {{ \ 1 \ k}}}}+{εij \ {| \sp {{ \ 1 \ ij}}}}+{εik \ {| \sp {{ \ 1 \ ik}}}}+{εjk \ {| \sp {{ \ 1 \ jk}}}}+{εijk \ {| \sp {{ \ 1 \ ijk}}}}+{εi \ {| \sp {{ \ i \ 1}}}}+{{i \sp {2}} \ ε1 \ {| \sp {{ \ i \ i}}}}+{εij \ {| \sp {{ \ i \ j}}}}+{εik \ {| \sp {{ \ i \ k}}}}+{{i \sp {2}} \ εj \ {| \sp {{ \ i \ ij}}}}+{{i \sp {2}} \ εk \ {| \sp {{ \ i \ ik}}}}+{εijk \ {| \sp {{ \ i \ jk}}}}+{{i \sp {2}} \ εjk \ {| \sp {{ \ i \ ijk}}}}+{εj \ {| \sp {{ \ j \ 1}}}} -{εij \ {| \sp {{ \ j \ i}}}}+{{j \sp {2}} \ ε1 \ {| \sp {{ \ j \ j}}}}+{εjk \ {| \sp {{ \ j \ k}}}} -{{j \sp {2}} \ εi \ {| \sp {{ \ j \ ij}}}} -{εijk \ {| \sp {{ \ j \ ik}}}}+{{j \sp {2}} \ εk \ {| \sp {{ \ j \ jk}}}} -{{j \sp {2}} \ εik \ {| \sp {{ \ j \ ijk}}}}+{εk \ {| \sp {{ \ k \ 1}}}} -{εik \ {| \sp {{ \ k \ i}}}} -{εjk \ {| \sp {{ \ k \ j}}}}+{{k \sp {2}} \ ε1 \ {| \sp {{ \ k \ k}}}}+{εijk \ {| \sp {{ \ k \ ij}}}} -{{k \sp {2}} \ εi \ {| \sp {{ \ k \ ik}}}} -{{k \sp {2}} \ εj \ {| \sp {{ \ k \ jk}}}}+{{k \sp {2}} \ εij \ {| \sp {{ \ k \ ijk}}}}+{εij \ {| \sp {{ \ ij \ 1}}}} -{{i \sp {2}} \ εj \ {| \sp {{ \ ij \ i}}}}+{{j \sp {2}} \ εi \ {| \sp {{ \ ij \ j}}}}+{εijk \ {| \sp {{ \ ij \ k}}}} -{{i \sp {2}} \ {j \sp {2}} \ ε1 \ {| \sp {{ \ ij \ ij}}}} -{{i \sp {2}} \ εjk \ {| \sp {{ \ ij \ ik}}}}+{{j \sp {2}} \ εik \ {| \sp {{ \ ij \ jk}}}} -{{i \sp {2}} \ {j \sp {2}} \ εk \ {| \sp {{ \ ij \ ijk}}}}+{εik \ {| \sp {{ \ ik \ 1}}}} -{{i \sp {2}} \ εk \ {| \sp {{ \ ik \ i}}}} -{εijk \ {| \sp {{ \ ik \ j}}}}+{{k \sp {2}} \ εi \ {| \sp {{ \ ik \ k}}}}+{{i \sp {2}} \ εjk \ {| \sp {{ \ ik \ ij}}}} -{{i \sp {2}} \ {k \sp {2}} \ ε1 \ {| \sp {{ \ ik \ ik}}}} -{{k \sp {2}} \ εij \ {| \sp {{ \ ik \ jk}}}}+{{i \sp {2}} \ {k \sp {2}} \ εj \ {| \sp {{ \ ik \ ijk}}}}+{εjk \ {| \sp {{ \ jk \ 1}}}}+{εijk \ {| \sp {{ \ jk \ i}}}} -{{j \sp {2}} \ εk \ {| \sp {{ \ jk \ j}}}}+{{k \sp {2}} \ εj \ {| \sp {{ \ jk \ k}}}} -{{j \sp {2}} \ εik \ {| \sp {{ \ jk \ ij}}}}+{{k \sp {2}} \ εij \ {| \sp {{ \ jk \ ik}}}} -{{j \sp {2}} \ {k \sp {2}} \ ε1 \ {| \sp {{ \ jk \ jk}}}} -{{j \sp {2}} \ {k \sp {2}} \ εi \ {| \sp {{ \ jk \ ijk}}}}+{εijk \ {| \sp {{ \ ijk \ 1}}}}+{{i \sp {2}} \ εjk \ {| \sp {{ \ ijk \ i}}}} -{{j \sp {2}} \ εik \ {| \sp {{ \ ijk \ j}}}}+{{k \sp {2}} \ εij \ {| \sp {{ \ ijk \ k}}}} -{{i \sp {2}} \ {j \sp {2}} \ εk \ {| \sp {{ \ ijk \ ij}}}}+{{i \sp {2}} \ {k \sp {2}} \ εj \ {| \sp {{ \ ijk \ ik}}}} -{{j \sp {2}} \ {k \sp {2}} \ εi \ {| \sp {{ \ ijk \ jk}}}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {| \sp {{ \ ijk \ ijk}}}} \leqno(42) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (43) -> u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)
$$ \left[ \begin{array}{cccccccc} ε1 & εi & εj & εk & εij & εik & εjk & εijk \ εi & {{i \sp {2}} \ ε1} & -εij & -εik & -{{i \sp {2}} \ εj} & -{{i \sp {2}} \ εk} & εijk & {{i \sp {2}} \ εjk} \ εj & εij & {{j \sp {2}} \ ε1} & -εjk & {{j \sp {2}} \ εi} & -εijk & -{{j \sp {2}} \ εk} & -{{j \sp {2}} \ εik} \ εk & εik & εjk & {{k \sp {2}} \ ε1} & εijk & {{k \sp {2}} \ εi} & {{k \sp {2}} \ εj} & {{k \sp {2}} \ εij} \ εij & {{i \sp {2}} \ εj} & -{{j \sp {2}} \ εi} & εijk & -{{i \sp {2}} \ {j \sp {2}} \ ε1} & {{i \sp {2}} \ εjk} & -{{j \sp {2}} \ εik} & -{{i \sp {2}} \ {j \sp {2}} \ εk} \ εik & {{i \sp {2}} \ εk} & -εijk & -{{k \sp {2}} \ εi} & -{{i \sp {2}} \ εjk} & -{{i \sp {2}} \ {k \sp {2}} \ ε1} & {{k \sp {2}} \ εij} & {{i \sp {2}} \ {k \sp {2}} \ εj} \ εjk & εijk & {{j \sp {2}} \ εk} & -{{k \sp {2}} \ εj} & {{j \sp {2}} \ εik} & -{{k \sp {2}} \ εij} & -{{j \sp {2}} \ {k \sp {2}} \ ε1} & -{{j \sp {2}} \ {k \sp {2}} \ εi} \ εijk & {{i \sp {2}} \ εjk} & -{{j \sp {2}} \ εik} & {{k \sp {2}} \ εij} & -{{i \sp {2}} \ {j \sp {2}} \ εk} & {{i \sp {2}} \ {k \sp {2}} \ εj} & -{{j \sp {2}} \ {k \sp {2}} \ εi} & -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1} \end{array} \right] \leqno(43) $$
Type: Matrix(Expression(Integer)) (44) -> --Ů:=determinant u1 --factor(numer Ů)/factor(denom Ů) 1
$$ 1 \leqno(44) $$
Type: PositiveInteger (45) -> ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]).1
Compiling function equate with type Equation(ClosedLinearOperator( OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))) -> List(Equation(Expression(Integer)))
$$ \left[ {ε1=8}, \: {εi=0}, \: {εj=0}, \: {εk=0}, \: {εij=0}, \: {εik=0}, \: {εjk=0}, \: {εijk=0} \right] \leqno(45) $$
Type: List(Equation(Expression(Integer))) (46) -> a:=sb('a,[1])*i+sb('a,[2])*j+sb('a,[3])*k
$$ {{a \sb {1}} \ {| \sb {{ \ i}}}}+{{a \sb {2}} \ {| \sb {{ \ j}}}}+{{a \sb {3}} \ {| \sb {{ \ k}}}} \leqno(46) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) b:=sb('b,[1])*i+sb('b,[2])*j+sb('b,[3])*k
$$ {{b \sb {1}} \ {| \sb {{ \ i}}}}+{{b \sb {2}} \ {| \sb {{ \ j}}}}+{{b \sb {3}} \ {| \sb {{ \ k}}}} \leqno(47) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (a,a)/Ų
$$ {\left( {{k \sp {2}} \ {{a \sb {3}} \sp 2}}+{{j \sp {2}} \ {{a \sb {2}} \sp 2}}+{{i \sp {2}} \ {{a \sb {1}} \sp 2}} \right)} \ ε1 \leqno(48) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (b,b)/Ų
$$ {\left( {{k \sp {2}} \ {{b \sb {3}} \sp 2}}+{{j \sp {2}} \ {{b \sb {2}} \sp 2}}+{{i \sp {2}} \ {{b \sb {1}} \sp 2}} \right)} \ ε1 \leqno(49) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (a,b)/Ų
$$ {{\left( {{a \sb {2}} \ {b \sb {3}}} -{{a \sb {3}} \ {b \sb {2}}} \right)} \ εjk}+{{\left( {{a \sb {1}} \ {b \sb {3}}} -{{a \sb {3}} \ {b \sb {1}}} \right)} \ εik}+{{\left( {{a \sb {1}} \ {b \sb {2}}} -{{a \sb {2}} \ {b \sb {1}}} \right)} \ εij}+{{\left( {{k \sp {2}} \ {a \sb {3}} \ {b \sb {3}}}+{{j \sp {2}} \ {a \sb {2}} \ {b \sb {2}}}+{{i \sp {2}} \ {a \sb {1}} \ {b \sb {1}}} \right)} \ ε1} \leqno(50) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (51) -> mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);
Type: Union(Matrix(Expression(Integer)),...) Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim);
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) ΩX:=Ω/X;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (54) -> --squareFreePart factor denom Ů / squareFreePart factor numer Ů matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)
$$ \left[ \begin{array}{cccccccc} {-{{{i \sp {2}} \sp 2} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ ε1 \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εijk \ εk}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εijk \ εjk} -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ ε1 \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εijk \ εik \ εj}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ ε1 \ {εik \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εijk \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ ε1 \ {εij \sp 2}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ ε1 \ {εi \sp 2}}+{{{i \sp {2}} \sp 2} \ {{j \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 3}}} & {{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ εi \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εjk \ εk}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ {εjk \sp 2}}+{{\left( -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk} \right)} \ εjk}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ εi \ {εj \sp 2}} -{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ εi \ {εik \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ εi \ {εijk \sp 2}} -{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ εi \ {εij \sp 2}}+{{{j \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ {εi \sp 3}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2} \ εi}} & {{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ εj \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εik \ εk} -{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ εj \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εjk}+{{{i \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ {εj \sp 3}}+{{\left( {{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εik \sp 2}}+{{i \sp {2}} \ {k \sp {2}} \ {εijk \sp 2}} -{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ {εij \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 2}} -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2}} \right)} \ εj}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εik}} & {{{{i \sp {2}} \sp 2} \ {{j \sp {2}} \sp 2} \ {εk \sp 3}}+{{\left( -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {εjk \sp 2}}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {εj \sp 2}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {εik \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {εijk \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εij \sp 2}}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 2}} -{{{i \sp {2}} \sp 2} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εk}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ {εk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk} \right)} \ εk}+{{i \sp {2}} \ {k \sp {2}} \ εij \ {εjk \sp 2}} -{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ εij \ {εj \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ εij \ {εik \sp 2}} -{{k \sp {2}} \ εij \ {εijk \sp 2}}+{{{k \sp {2}} \sp 2} \ {εij \sp 3}}+{{\left( -{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2}} \right)} \ εij}} & {-{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ εik \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εj \ εk}+{{i \sp {2}} \ {j \sp {2}} \ εik \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εj \ εjk}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εj}+{{{j \sp {2}} \sp 2} \ {εik \sp 3}}+{{\left( -{{j \sp {2}} \ {εijk \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εij \sp 2}} -{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εik}} & {-{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ εjk \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εk}+{{{i \sp {2}} \sp 2} \ {εjk \sp 3}}+{{\left( -{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ {εj \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {εik \sp 2}} -{{i \sp {2}} \ {εijk \sp 2}}+{{i \sp {2}} \ {k \sp {2}} \ {εij \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ εijk \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εk} -{{i \sp {2}} \ εijk \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ εijk \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εik \ εj} -{{j \sp {2}} \ εijk \ {εik \sp 2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εijk}} \ {{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ εi \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εjk \ εk}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ {εjk \sp 2}}+{{\left( -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk} \right)} \ εjk}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ εi \ {εj \sp 2}} -{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ εi \ {εik \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ εi \ {εijk \sp 2}} -{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ εi \ {εij \sp 2}}+{{{j \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ {εi \sp 3}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2} \ εi}} & {-{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ ε1 \ {εk \sp 2}} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εij \ εijk \ εk}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εjk \sp 2}} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εijk \ εjk} -{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ ε1 \ {εj \sp 2}}+{2 \ {j \sp {2}} \ {k \sp {2}} \ εijk \ εik \ εj}+{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ ε1 \ {εik \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ ε1 \ {εijk \sp 2}}+{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ ε1 \ {εij \sp 2}} -{{{j \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ ε1 \ {εi \sp 2}}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 3}}} & {-{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ {εk \sp 2}}+{{\left( -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εjk}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ εj}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk} \right)} \ εk} -{{i \sp {2}} \ {k \sp {2}} \ εij \ {εjk \sp 2}}+{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ εij \ {εj \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ εij \ {εik \sp 2}}+{{k \sp {2}} \ εij \ {εijk \sp 2}} -{{{k \sp {2}} \sp 2} \ {εij \sp 3}}+{{\left( {{j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2}} \right)} \ εij}} & {{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ εik \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εj \ εk} -{{i \sp {2}} \ {j \sp {2}} \ εik \ {εjk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εj \ εjk} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ {εj \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εj} -{{{j \sp {2}} \sp 2} \ {εik \sp 3}}+{{\left( {{j \sp {2}} \ {εijk \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εij \sp 2}}+{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εik}} & {-{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εj \ {εk \sp 2}}+{2 \ {j \sp {2}} \ {k \sp {2}} \ εij \ εik \ εk}+{{i \sp {2}} \ {k \sp {2}} \ εj \ {εjk \sp 2}}+{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εjk} -{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ {εj \sp 3}}+{{\left( -{{j \sp {2}} \ {k \sp {2}} \ {εik \sp 2}} -{{k \sp {2}} \ {εijk \sp 2}}+{{{k \sp {2}} \sp 2} \ {εij \sp 2}} -{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2}} \right)} \ εj} -{2 \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εik}} & {-{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {εk \sp 3}}+{{\left( {{i \sp {2}} \ {j \sp {2}} \ {εjk \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εj \sp 2}}+{{{j \sp {2}} \sp 2} \ {εik \sp 2}} -{{j \sp {2}} \ {εijk \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εij \sp 2}} -{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εk} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εjk}+{2 \ {j \sp {2}} \ {k \sp {2}} \ εij \ εik \ εj}+{2 \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ εijk \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εk} -{{i \sp {2}} \ εijk \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ εijk \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εik \ εj} -{{j \sp {2}} \ εijk \ {εik \sp 2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εijk}} & {-{{i \sp {2}} \ {j \sp {2}} \ εjk \ {εk \sp 2}}+{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εk}+{{i \sp {2}} \ {εjk \sp 3}}+{{\left( -{{i \sp {2}} \ {k \sp {2}} \ {εj \sp 2}}+{{j \sp {2}} \ {εik \sp 2}} -{εijk \sp 2}+{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εjk} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εj} -{2 \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εijk}} \ {{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ εj \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εik \ εk} -{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ εj \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εjk}+{{{i \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ {εj \sp 3}}+{{\left( {{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εik \sp 2}}+{{i \sp {2}} \ {k \sp {2}} \ {εijk \sp 2}} -{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ {εij \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 2}} -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2}} \right)} \ εj}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εik}} & {{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ {εk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk} \right)} \ εk}+{{i \sp {2}} \ {k \sp {2}} \ εij \ {εjk \sp 2}} -{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ εij \ {εj \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ εij \ {εik \sp 2}} -{{k \sp {2}} \ εij \ {εijk \sp 2}}+{{{k \sp {2}} \sp 2} \ {εij \sp 3}}+{{\left( -{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2}} \right)} \ εij}} & {-{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {k \sp {2}} \ εij \ εijk \ εk}+{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ ε1 \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {k \sp {2}} \ εi \ εijk \ εjk} -{{{i \sp {2}} \sp 2} \ {{k \sp {2}} \sp 2} \ ε1 \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εijk \ εik \ εj}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εik \sp 2}}+{{i \sp {2}} \ {k \sp {2}} \ ε1 \ {εijk \sp 2}}+{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ ε1 \ {εij \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ ε1 \ {εi \sp 2}}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 3}}} & {{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ εjk \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εk} -{{{i \sp {2}} \sp 2} \ {εjk \sp 3}}+{{\left( {{{i \sp {2}} \sp 2} \ {k \sp {2}} \ {εj \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {εik \sp 2}}+{{i \sp {2}} \ {εijk \sp 2}} -{{i \sp {2}} \ {k \sp {2}} \ {εij \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εjk}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εj}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εij \ εjk \ εk}+{{i \sp {2}} \ {k \sp {2}} \ εi \ {εjk \sp 2}}+{{\left( -{2 \ {i \sp {2}} \ {k \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {k \sp {2}} \ ε1 \ εijk} \right)} \ εjk}+{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ εi \ {εj \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ εi \ {εik \sp 2}}+{{k \sp {2}} \ εi \ {εijk \sp 2}} -{{{k \sp {2}} \sp 2} \ εi \ {εij \sp 2}}+{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 3}} -{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2} \ εi}} & {-{{i \sp {2}} \ {j \sp {2}} \ εijk \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εk}+{{i \sp {2}} \ εijk \ {εjk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εjk} -{{i \sp {2}} \ {k \sp {2}} \ εijk \ {εj \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εik \ εj}+{{j \sp {2}} \ εijk \ {εik \sp 2}} -{εijk \sp 3}+{{\left( {{k \sp {2}} \ {εij \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εijk}} & {-{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {εk \sp 3}}+{{\left( {{{i \sp {2}} \sp 2} \ {εjk \sp 2}} -{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ {εj \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {εik \sp 2}} -{{i \sp {2}} \ {εijk \sp 2}} -{{i \sp {2}} \ {k \sp {2}} \ {εij \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εk} -{2 \ {i \sp {2}} \ {k \sp {2}} \ εi \ εij \ εjk}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εij \ εik \ εj}+{2 \ {i \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ εik \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εij \ εj \ εk} -{{i \sp {2}} \ εik \ {εjk \sp 2}}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εi \ εj \ εjk} -{{i \sp {2}} \ {k \sp {2}} \ εik \ {εj \sp 2}} -{2 \ {i \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εj} -{{j \sp {2}} \ {εik \sp 3}}+{{\left( {εijk \sp 2} -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εik}} \ {{{{i \sp {2}} \sp 2} \ {{j \sp {2}} \sp 2} \ {εk \sp 3}}+{{\left( -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {εjk \sp 2}}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {εj \sp 2}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {εik \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {εijk \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εij \sp 2}}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 2}} -{{{i \sp {2}} \sp 2} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εk}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εijk}} & {-{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ εik \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εj \ εk}+{{i \sp {2}} \ {j \sp {2}} \ εik \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εj \ εjk}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εj}+{{{j \sp {2}} \sp 2} \ {εik \sp 3}}+{{\left( -{{j \sp {2}} \ {εijk \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εij \sp 2}} -{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εik}} & {-{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ εjk \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εk}+{{{i \sp {2}} \sp 2} \ {εjk \sp 3}}+{{\left( -{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ {εj \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {εik \sp 2}} -{{i \sp {2}} \ {εijk \sp 2}}+{{i \sp {2}} \ {k \sp {2}} \ {εij \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εijk}} & {-{{{i \sp {2}} \sp 2} \ {{j \sp {2}} \sp 2} \ ε1 \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ εij \ εijk \ εk}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ ε1 \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ εi \ εijk \ εjk} -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ εijk \ εik \ εj}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ ε1 \ {εik \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ ε1 \ {εijk \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εij \sp 2}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ ε1 \ {εi \sp 2}}+{{{i \sp {2}} \sp 2} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 3}}} & {{{i \sp {2}} \ {j \sp {2}} \ εijk \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εk} -{{i \sp {2}} \ εijk \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ εijk \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εik \ εj} -{{j \sp {2}} \ εijk \ {εik \sp 2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εijk}} & {{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ εi \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ εij \ εjk \ εk}+{{i \sp {2}} \ {j \sp {2}} \ εi \ {εjk \sp 2}}+{{\left( -{2 \ {i \sp {2}} \ {j \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ ε1 \ εijk} \right)} \ εjk}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ {εj \sp 2}} -{{{j \sp {2}} \sp 2} \ εi \ {εik \sp 2}}+{{j \sp {2}} \ εi \ {εijk \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ εi \ {εij \sp 2}}+{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 3}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2} \ εi}} & {{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ εj \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ εij \ εik \ εk} -{{{i \sp {2}} \sp 2} \ εj \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ εi \ εik \ εjk}+{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ {εj \sp 3}}+{{\left( {{i \sp {2}} \ {j \sp {2}} \ {εik \sp 2}}+{{i \sp {2}} \ {εijk \sp 2}} -{{i \sp {2}} \ {k \sp {2}} \ {εij \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εj}+{2 \ {i \sp {2}} \ {j \sp {2}} \ ε1 \ εijk \ εik}} & {{{i \sp {2}} \ {j \sp {2}} \ εij \ {εk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ {j \sp {2}} \ εi \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ ε1 \ εijk} \right)} \ εk}+{{i \sp {2}} \ εij \ {εjk \sp 2}} -{{i \sp {2}} \ {k \sp {2}} \ εij \ {εj \sp 2}}+{{j \sp {2}} \ εij \ {εik \sp 2}} -{εij \ {εijk \sp 2}}+{{k \sp {2}} \ {εij \sp 3}}+{{\left( -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εij}} \ {{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ {εk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk} \right)} \ εk}+{{i \sp {2}} \ {k \sp {2}} \ εij \ {εjk \sp 2}} -{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ εij \ {εj \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ εij \ {εik \sp 2}} -{{k \sp {2}} \ εij \ {εijk \sp 2}}+{{{k \sp {2}} \sp 2} \ {εij \sp 3}}+{{\left( -{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2}} \right)} \ εij}} & {{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εj \ {εk \sp 2}} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εij \ εik \ εk} -{{i \sp {2}} \ {k \sp {2}} \ εj \ {εjk \sp 2}} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εjk}+{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ {εj \sp 3}}+{{\left( {{j \sp {2}} \ {k \sp {2}} \ {εik \sp 2}}+{{k \sp {2}} \ {εijk \sp 2}} -{{{k \sp {2}} \sp 2} \ {εij \sp 2}}+{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2}} \right)} \ εj}+{2 \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εik}} & {-{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {k \sp {2}} \ εij \ εjk \ εk} -{{i \sp {2}} \ {k \sp {2}} \ εi \ {εjk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ {k \sp {2}} \ εik \ εj}+{2 \ {i \sp {2}} \ {k \sp {2}} \ ε1 \ εijk} \right)} \ εjk} -{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ εi \ {εj \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ εi \ {εik \sp 2}} -{{k \sp {2}} \ εi \ {εijk \sp 2}}+{{{k \sp {2}} \sp 2} \ εi \ {εij \sp 2}} -{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ {εi \sp 3}}+{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 2} \ εi}} & {{{i \sp {2}} \ {j \sp {2}} \ εijk \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εk} -{{i \sp {2}} \ εijk \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ εijk \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εik \ εj} -{{j \sp {2}} \ εijk \ {εik \sp 2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εk \sp 2}}+{2 \ {k \sp {2}} \ εij \ εijk \ εk} -{{i \sp {2}} \ {k \sp {2}} \ ε1 \ {εjk \sp 2}}+{2 \ {k \sp {2}} \ εi \ εijk \ εjk}+{{i \sp {2}} \ {{k \sp {2}} \sp 2} \ ε1 \ {εj \sp 2}} -{2 \ {k \sp {2}} \ εijk \ εik \ εj} -{{j \sp {2}} \ {k \sp {2}} \ ε1 \ {εik \sp 2}} -{{k \sp {2}} \ ε1 \ {εijk \sp 2}} -{{{k \sp {2}} \sp 2} \ ε1 \ {εij \sp 2}}+{{j \sp {2}} \ {{k \sp {2}} \sp 2} \ ε1 \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {{k \sp {2}} \sp 2} \ {ε1 \sp 3}}} & {-{{i \sp {2}} \ {j \sp {2}} \ εjk \ {εk \sp 2}}+{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εk}+{{i \sp {2}} \ {εjk \sp 3}}+{{\left( -{{i \sp {2}} \ {k \sp {2}} \ {εj \sp 2}}+{{j \sp {2}} \ {εik \sp 2}} -{εijk \sp 2}+{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εjk} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εj} -{2 \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ εik \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εij \ εj \ εk} -{{i \sp {2}} \ εik \ {εjk \sp 2}}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εi \ εj \ εjk} -{{i \sp {2}} \ {k \sp {2}} \ εik \ {εj \sp 2}} -{2 \ {i \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εj} -{{j \sp {2}} \ {εik \sp 3}}+{{\left( {εijk \sp 2} -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εik}} & {-{{i \sp {2}} \ {j \sp {2}} \ {εk \sp 3}}+{{\left( {{i \sp {2}} \ {εjk \sp 2}} -{{i \sp {2}} \ {k \sp {2}} \ {εj \sp 2}}+{{j \sp {2}} \ {εik \sp 2}} -{εijk \sp 2} -{{k \sp {2}} \ {εij \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εk} -{2 \ {k \sp {2}} \ εi \ εij \ εjk}+{2 \ {k \sp {2}} \ εij \ εik \ εj}+{2 \ {k \sp {2}} \ ε1 \ εij \ εijk}} \ {-{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ εik \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εij \ εj \ εk}+{{i \sp {2}} \ {j \sp {2}} \ εik \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εj \ εjk}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εik \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εj}+{{{j \sp {2}} \sp 2} \ {εik \sp 3}}+{{\left( -{{j \sp {2}} \ {εijk \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εij \sp 2}} -{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εik}} & {{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {εk \sp 3}}+{{\left( -{{i \sp {2}} \ {j \sp {2}} \ {εjk \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εj \sp 2}} -{{{j \sp {2}} \sp 2} \ {εik \sp 2}}+{{j \sp {2}} \ {εijk \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εij \sp 2}}+{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εk}+{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εjk} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εij \ εik \ εj} -{2 \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εijk}} & {-{{i \sp {2}} \ {j \sp {2}} \ εijk \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εk}+{{i \sp {2}} \ εijk \ {εjk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εjk} -{{i \sp {2}} \ {k \sp {2}} \ εijk \ {εj \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εik \ εj}+{{j \sp {2}} \ εijk \ {εik \sp 2}} -{εijk \sp 3}+{{\left( {{k \sp {2}} \ {εij \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εijk}} & {-{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ εi \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ εij \ εjk \ εk} -{{i \sp {2}} \ {j \sp {2}} \ εi \ {εjk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ {j \sp {2}} \ εik \ εj}+{2 \ {i \sp {2}} \ {j \sp {2}} \ ε1 \ εijk} \right)} \ εjk} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ {εj \sp 2}}+{{{j \sp {2}} \sp 2} \ εi \ {εik \sp 2}} -{{j \sp {2}} \ εi \ {εijk \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ εi \ {εij \sp 2}} -{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ {εi \sp 3}}+{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 2} \ εi}} & {{{i \sp {2}} \ {j \sp {2}} \ εjk \ {εk \sp 2}} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εk} -{{i \sp {2}} \ {εjk \sp 3}}+{{\left( {{i \sp {2}} \ {k \sp {2}} \ {εj \sp 2}} -{{j \sp {2}} \ {εik \sp 2}}+{εijk \sp 2} -{{k \sp {2}} \ {εij \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εjk}+{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εj}+{2 \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εijk}} & {{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ ε1 \ {εk \sp 2}}+{2 \ {j \sp {2}} \ εij \ εijk \ εk} -{{i \sp {2}} \ {j \sp {2}} \ ε1 \ {εjk \sp 2}}+{2 \ {j \sp {2}} \ εi \ εijk \ εjk}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εj \sp 2}} -{2 \ {j \sp {2}} \ εijk \ εik \ εj} -{{{j \sp {2}} \sp 2} \ ε1 \ {εik \sp 2}} -{{j \sp {2}} \ ε1 \ {εijk \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ ε1 \ {εij \sp 2}}+{{{j \sp {2}} \sp 2} \ {k \sp {2}} \ ε1 \ {εi \sp 2}} -{{i \sp {2}} \ {{j \sp {2}} \sp 2} \ {k \sp {2}} \ {ε1 \sp 3}}} & {{{i \sp {2}} \ {j \sp {2}} \ εij \ {εk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ {j \sp {2}} \ εi \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ ε1 \ εijk} \right)} \ εk}+{{i \sp {2}} \ εij \ {εjk \sp 2}} -{{i \sp {2}} \ {k \sp {2}} \ εij \ {εj \sp 2}}+{{j \sp {2}} \ εij \ {εik \sp 2}} -{εij \ {εijk \sp 2}}+{{k \sp {2}} \ {εij \sp 3}}+{{\left( -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εij}} & {{{i \sp {2}} \ {j \sp {2}} \ εj \ {εk \sp 2}} -{2 \ {j \sp {2}} \ εij \ εik \ εk} -{{i \sp {2}} \ εj \ {εjk \sp 2}} -{2 \ {j \sp {2}} \ εi \ εik \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ {εj \sp 3}}+{{\left( {{j \sp {2}} \ {εik \sp 2}}+{εijk \sp 2} -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εj}+{2 \ {j \sp {2}} \ ε1 \ εijk \ εik}} \ {-{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ εjk \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εk}+{{{i \sp {2}} \sp 2} \ {εjk \sp 3}}+{{\left( -{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ {εj \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {εik \sp 2}} -{{i \sp {2}} \ {εijk \sp 2}}+{{i \sp {2}} \ {k \sp {2}} \ {εij \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ εijk \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εk} -{{i \sp {2}} \ εijk \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ εijk \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εik \ εj} -{{j \sp {2}} \ εijk \ {εik \sp 2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εijk}} & {{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {εk \sp 3}}+{{\left( -{{{i \sp {2}} \sp 2} \ {εjk \sp 2}}+{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ {εj \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {εik \sp 2}}+{{i \sp {2}} \ {εijk \sp 2}}+{{i \sp {2}} \ {k \sp {2}} \ {εij \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εk}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εi \ εij \ εjk} -{2 \ {i \sp {2}} \ {k \sp {2}} \ εij \ εik \ εj} -{2 \ {i \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εijk}} & {-{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ εj \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ εij \ εik \ εk}+{{{i \sp {2}} \sp 2} \ εj \ {εjk \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ εi \ εik \ εjk} -{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ {εj \sp 3}}+{{\left( -{{i \sp {2}} \ {j \sp {2}} \ {εik \sp 2}} -{{i \sp {2}} \ {εijk \sp 2}}+{{i \sp {2}} \ {k \sp {2}} \ {εij \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ ε1 \ εijk \ εik}} & {-{{i \sp {2}} \ {j \sp {2}} \ εik \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {k \sp {2}} \ εij \ εj \ εk}+{{i \sp {2}} \ εik \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {k \sp {2}} \ εi \ εj \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ εik \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εj}+{{j \sp {2}} \ {εik \sp 3}}+{{\left( -{εijk \sp 2}+{{k \sp {2}} \ {εij \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εik}} & {-{{i \sp {2}} \ {j \sp {2}} \ εij \ {εk \sp 2}}+{{\left( -{2 \ {i \sp {2}} \ {j \sp {2}} \ εi \ εjk}+{2 \ {i \sp {2}} \ {j \sp {2}} \ εik \ εj}+{2 \ {i \sp {2}} \ {j \sp {2}} \ ε1 \ εijk} \right)} \ εk} -{{i \sp {2}} \ εij \ {εjk \sp 2}}+{{i \sp {2}} \ {k \sp {2}} \ εij \ {εj \sp 2}} -{{j \sp {2}} \ εij \ {εik \sp 2}}+{εij \ {εijk \sp 2}} -{{k \sp {2}} \ {εij \sp 3}}+{{\left( {{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εij}} & {{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ ε1 \ {εk \sp 2}}+{2 \ {i \sp {2}} \ εij \ εijk \ εk} -{{{i \sp {2}} \sp 2} \ ε1 \ {εjk \sp 2}}+{2 \ {i \sp {2}} \ εi \ εijk \ εjk}+{{{i \sp {2}} \sp 2} \ {k \sp {2}} \ ε1 \ {εj \sp 2}} -{2 \ {i \sp {2}} \ εijk \ εik \ εj} -{{i \sp {2}} \ {j \sp {2}} \ ε1 \ {εik \sp 2}} -{{i \sp {2}} \ ε1 \ {εijk \sp 2}} -{{i \sp {2}} \ {k \sp {2}} \ ε1 \ {εij \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ {εi \sp 2}} -{{{i \sp {2}} \sp 2} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 3}}} & {-{{i \sp {2}} \ {j \sp {2}} \ εi \ {εk \sp 2}} -{2 \ {i \sp {2}} \ εij \ εjk \ εk} -{{i \sp {2}} \ εi \ {εjk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ εik \ εj}+{2 \ {i \sp {2}} \ ε1 \ εijk} \right)} \ εjk} -{{i \sp {2}} \ {k \sp {2}} \ εi \ {εj \sp 2}}+{{j \sp {2}} \ εi \ {εik \sp 2}} -{εi \ {εijk \sp 2}}+{{k \sp {2}} \ εi \ {εij \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 3}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2} \ εi}} \ {{{i \sp {2}} \ {j \sp {2}} \ εijk \ {εk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εij \ εk} -{{i \sp {2}} \ εijk \ {εjk \sp 2}} -{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ εijk \ {εj \sp 2}}+{2 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εik \ εj} -{{j \sp {2}} \ εijk \ {εik \sp 2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εijk}} & {-{{i \sp {2}} \ {j \sp {2}} \ εjk \ {εk \sp 2}}+{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εij \ εk}+{{i \sp {2}} \ {εjk \sp 3}}+{{\left( -{{i \sp {2}} \ {k \sp {2}} \ {εj \sp 2}}+{{j \sp {2}} \ {εik \sp 2}} -{εijk \sp 2}+{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εjk} -{2 \ {j \sp {2}} \ {k \sp {2}} \ εi \ εik \ εj} -{2 \ {j \sp {2}} \ {k \sp {2}} \ ε1 \ εi \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ εik \ {εk \sp 2}}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εij \ εj \ εk} -{{i \sp {2}} \ εik \ {εjk \sp 2}}+{2 \ {i \sp {2}} \ {k \sp {2}} \ εi \ εj \ εjk} -{{i \sp {2}} \ {k \sp {2}} \ εik \ {εj \sp 2}} -{2 \ {i \sp {2}} \ {k \sp {2}} \ ε1 \ εijk \ εj} -{{j \sp {2}} \ {εik \sp 3}}+{{\left( {εijk \sp 2} -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εik}} & {{{i \sp {2}} \ {j \sp {2}} \ εij \ {εk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ {j \sp {2}} \ εi \ εjk} -{2 \ {i \sp {2}} \ {j \sp {2}} \ εik \ εj} -{2 \ {i \sp {2}} \ {j \sp {2}} \ ε1 \ εijk} \right)} \ εk}+{{i \sp {2}} \ εij \ {εjk \sp 2}} -{{i \sp {2}} \ {k \sp {2}} \ εij \ {εj \sp 2}}+{{j \sp {2}} \ εij \ {εik \sp 2}} -{εij \ {εijk \sp 2}}+{{k \sp {2}} \ {εij \sp 3}}+{{\left( -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εij}} & {-{{i \sp {2}} \ {j \sp {2}} \ {εk \sp 3}}+{{\left( {{i \sp {2}} \ {εjk \sp 2}} -{{i \sp {2}} \ {k \sp {2}} \ {εj \sp 2}}+{{j \sp {2}} \ {εik \sp 2}} -{εijk \sp 2} -{{k \sp {2}} \ {εij \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εk} -{2 \ {k \sp {2}} \ εi \ εij \ εjk}+{2 \ {k \sp {2}} \ εij \ εik \ εj}+{2 \ {k \sp {2}} \ ε1 \ εij \ εijk}} & {{{i \sp {2}} \ {j \sp {2}} \ εj \ {εk \sp 2}} -{2 \ {j \sp {2}} \ εij \ εik \ εk} -{{i \sp {2}} \ εj \ {εjk \sp 2}} -{2 \ {j \sp {2}} \ εi \ εik \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ {εj \sp 3}}+{{\left( {{j \sp {2}} \ {εik \sp 2}}+{εijk \sp 2} -{{k \sp {2}} \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2}} \right)} \ εj}+{2 \ {j \sp {2}} \ ε1 \ εijk \ εik}} & {-{{i \sp {2}} \ {j \sp {2}} \ εi \ {εk \sp 2}} -{2 \ {i \sp {2}} \ εij \ εjk \ εk} -{{i \sp {2}} \ εi \ {εjk \sp 2}}+{{\left( {2 \ {i \sp {2}} \ εik \ εj}+{2 \ {i \sp {2}} \ ε1 \ εijk} \right)} \ εjk} -{{i \sp {2}} \ {k \sp {2}} \ εi \ {εj \sp 2}}+{{j \sp {2}} \ εi \ {εik \sp 2}} -{εi \ {εijk \sp 2}}+{{k \sp {2}} \ εi \ {εij \sp 2}} -{{j \sp {2}} \ {k \sp {2}} \ {εi \sp 3}}+{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 2} \ εi}} & {{{i \sp {2}} \ {j \sp {2}} \ ε1 \ {εk \sp 2}}+{2 \ εij \ εijk \ εk} -{{i \sp {2}} \ ε1 \ {εjk \sp 2}}+{2 \ εi \ εijk \ εjk}+{{i \sp {2}} \ {k \sp {2}} \ ε1 \ {εj \sp 2}} -{2 \ εijk \ εik \ εj} -{{j \sp {2}} \ ε1 \ {εik \sp 2}} -{ε1 \ {εijk \sp 2}} -{{k \sp {2}} \ ε1 \ {εij \sp 2}}+{{j \sp {2}} \ {k \sp {2}} \ ε1 \ {εi \sp 2}} -{{i \sp {2}} \ {j \sp {2}} \ {k \sp {2}} \ {ε1 \sp 3}}} \end{array} \right] \leqno(54) $$
Type: Matrix(SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))) (55) -> O:𝐋:= Ω / Ų
$$ 8 \leqno(55) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
test ( I ΩX ) / ( Ų I ) = I
$$ true \leqno(56) $$
Type: Boolean
test ( ΩX I ) / ( I Ų ) = I
$$ true \leqno(57) $$
Type: Boolean (58) -> eval(Ω,ck)
$$ {{1 \over 8} \ {| \sb {{ \ 1 \ 1}}}}+{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ i \ i}}}}+{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ j \ j}}}}+{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ k \ k}}}} -{{1 \over {8 \ {i \sp {2}} \ {j \sp {2}}}} \ {| \sb {{ \ ij \ ij}}}} -{{1 \over {8 \ {i \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ ik \ ik}}}} -{{1 \over {8 \ {j \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ jk \ jk}}}} -{{1 \over {8 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ ijk \ ijk}}}} \leqno(58) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (59) -> W:= (Y I) / Ų;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (60) -> λ:= _ ( I ΩX ) / _ ( Y I );
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
test ( ΩX I ) / ( I Y ) = λ
$$ true \leqno(61) $$
Type: Boolean (62) -> eval(λ,ck)
$$ {{1 \over 8} \ {| \sb {{ \ 1 \ 1}} \sp {{ \ 1}}}}+{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ i \ i}} \sp {{ \ 1}}}}+{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ j \ j}} \sp {{ \ 1}}}}+{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ k \ k}} \sp {{ \ 1}}}} -{{1 \over {8 \ {i \sp {2}} \ {j \sp {2}}}} \ {| \sb {{ \ ij \ ij}} \sp {{ \ 1}}}} -{{1 \over {8 \ {i \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ ik \ ik}} \sp {{ \ 1}}}} -{{1 \over {8 \ {j \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ jk \ jk}} \sp {{ \ 1}}}} -{{1 \over {8 \ {i \sp {2}} \ {j \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ ijk \ ijk}} \sp {{ \ 1}}}}+{{1 \over 8} \ {| \sb {{ \ 1 \ i}} \sp {{ \ i}}}}+{{1 \over 8} \ {| \sb {{ \ i \ 1}} \sp {{ \ i}}}} -{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ j \ ij}} \sp {{ \ i}}}} -{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ k \ ik}} \sp {{ \ i}}}}+{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ ij \ j}} \sp {{ \ i}}}}+{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ ik \ k}} \sp {{ \ i}}}} -{{1 \over {8 \ {j \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ jk \ ijk}} \sp {{ \ i}}}} -{{1 \over {8 \ {j \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ ijk \ jk}} \sp {{ \ i}}}}+{{1 \over 8} \ {| \sb {{ \ 1 \ j}} \sp {{ \ j}}}}+{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ i \ ij}} \sp {{ \ j}}}}+{{1 \over 8} \ {| \sb {{ \ j \ 1}} \sp {{ \ j}}}} -{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ k \ jk}} \sp {{ \ j}}}} -{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ ij \ i}} \sp {{ \ j}}}}+{{1 \over {8 \ {i \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ ik \ ijk}} \sp {{ \ j}}}}+{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ jk \ k}} \sp {{ \ j}}}}+{{1 \over {8 \ {i \sp {2}} \ {k \sp {2}}}} \ {| \sb {{ \ ijk \ ik}} \sp {{ \ j}}}}+{{1 \over 8} \ {| \sb {{ \ 1 \ k}} \sp {{ \ k}}}}+{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ i \ ik}} \sp {{ \ k}}}}+{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ j \ jk}} \sp {{ \ k}}}}+{{1 \over 8} \ {| \sb {{ \ k \ 1}} \sp {{ \ k}}}} -{{1 \over {8 \ {i \sp {2}} \ {j \sp {2}}}} \ {| \sb {{ \ ij \ ijk}} \sp {{ \ k}}}} -{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ ik \ i}} \sp {{ \ k}}}} -{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ jk \ j}} \sp {{ \ k}}}} -{{1 \over {8 \ {i \sp {2}} \ {j \sp {2}}}} \ {| \sb {{ \ ijk \ ij}} \sp {{ \ k}}}}+{{1 \over 8} \ {| \sb {{ \ 1 \ ij}} \sp {{ \ ij}}}}+{{1 \over 8} \ {| \sb {{ \ i \ j}} \sp {{ \ ij}}}} -{{1 \over 8} \ {| \sb {{ \ j \ i}} \sp {{ \ ij}}}}+{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ k \ ijk}} \sp {{ \ ij}}}}+{{1 \over 8} \ {| \sb {{ \ ij \ 1}} \sp {{ \ ij}}}} -{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ ik \ jk}} \sp {{ \ ij}}}}+{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ jk \ ik}} \sp {{ \ ij}}}}+{{1 \over {8 \ {k \sp {2}}}} \ {| \sb {{ \ ijk \ k}} \sp {{ \ ij}}}}+{{1 \over 8} \ {| \sb {{ \ 1 \ ik}} \sp {{ \ ik}}}}+{{1 \over 8} \ {| \sb {{ \ i \ k}} \sp {{ \ ik}}}} -{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ j \ ijk}} \sp {{ \ ik}}}} -{{1 \over 8} \ {| \sb {{ \ k \ i}} \sp {{ \ ik}}}}+{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ ij \ jk}} \sp {{ \ ik}}}}+{{1 \over 8} \ {| \sb {{ \ ik \ 1}} \sp {{ \ ik}}}} -{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ jk \ ij}} \sp {{ \ ik}}}} -{{1 \over {8 \ {j \sp {2}}}} \ {| \sb {{ \ ijk \ j}} \sp {{ \ ik}}}}+{{1 \over 8} \ {| \sb {{ \ 1 \ jk}} \sp {{ \ jk}}}}+{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ i \ ijk}} \sp {{ \ jk}}}}+{{1 \over 8} \ {| \sb {{ \ j \ k}} \sp {{ \ jk}}}} -{{1 \over 8} \ {| \sb {{ \ k \ j}} \sp {{ \ jk}}}} -{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ ij \ ik}} \sp {{ \ jk}}}}+{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ ik \ ij}} \sp {{ \ jk}}}}+{{1 \over 8} \ {| \sb {{ \ jk \ 1}} \sp {{ \ jk}}}}+{{1 \over {8 \ {i \sp {2}}}} \ {| \sb {{ \ ijk \ i}} \sp {{ \ jk}}}}+{{1 \over 8} \ {| \sb {{ \ 1 \ ijk}} \sp {{ \ ijk}}}}+{{1 \over 8} \ {| \sb {{ \ i \ jk}} \sp {{ \ ijk}}}} -{{1 \over 8} \ {| \sb {{ \ j \ ik}} \sp {{ \ ijk}}}}+{{1 \over 8} \ {| \sb {{ \ k \ ij}} \sp {{ \ ijk}}}}+{{1 \over 8} \ {| \sb {{ \ ij \ k}} \sp {{ \ ijk}}}} -{{1 \over 8} \ {| \sb {{ \ ik \ j}} \sp {{ \ ijk}}}}+{{1 \over 8} \ {| \sb {{ \ jk \ i}} \sp {{ \ ijk}}}}+{{1 \over 8} \ {| \sb {{ \ ijk \ 1}} \sp {{ \ ijk}}}} \leqno(62) $$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) (63) -> test( ( λ ) / _ ( I λ ) = _ ( λ ) / _ ( λ I ) )
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Missing $ inserted. <inserted text> $ l.155 ...atrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)
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l.159 ...j*𝐝.k, i,1..dim), j,1..dim), k,1..dim)
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[2] [3] Missing $ inserted. <inserted text> $ l.169 a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)
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