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last edited 11 years ago by Bill Page |
Edit detail for Octonion Algebra is Frobenius in Just One Way revision 8 of 8
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Editor: Bill Page
Time: 2013/04/01 17:53:40 GMT+0 |
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| Note: update | ||
changed: -)library CARTEN MONAL PROP LOP CALEY -\end{axiom} )library CARTEN ARITY CMONAL CPROP CLOP CALEY \end{axiom} removed: --- list -macro Ξ(f,i,n)==[f for i in n] changed: -𝐋 is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients. ℒ is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients. removed: -macro ℒ == List changed: -𝐋 := LinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7], ℚ) -𝐞:ℒ 𝐋 := basisOut() -𝐝:ℒ 𝐋 := basisIn() -I:𝐋:=[1]; -- identity for composition -X:𝐋:=[2,1]; -- twist -\end{axiom} ℒ := ClosedLinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7], ℚ) ⅇ:List ℒ := basisOut() ⅆ:List ℒ := basisIn() I:ℒ:=[1]; -- identity for composition X:ℒ:=[2,1]; -- twist V:ℒ:=ev(1) -- evaluation Λ:ℒ:=co(1) -- co-evaluation \end{axiom} changed: -B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ) B:List QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::List List ℚ) changed: -M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim) M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim] changed: -ѕ :=map(S,B)::ℒ ℒ ℒ ℚ; ѕ :=map(S,B)::List List List ℚ; changed: -Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim) Y := Σ(Σ(Σ(ѕ(i)(k)(j)*ⅇ.i*ⅆ.j*ⅆ.k, i,1..dim), j,1..dim), k,1..dim) changed: -matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim) -\end{axiom} matrix [[(ⅇ.i*ⅇ.j)/Y for i in 1..dim] for j in 1..dim] \end{axiom} changed: -U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim) -\end{axiom} U:=Σ(Σ(script('u,[[],[i,j]])*ⅆ.i*ⅆ.j, i,1..dim), j,1..dim) \end{axiom} changed: -ω:𝐋 :=(Y*I)/U - (I*Y)/U; -\end{axiom} ω:ℒ :=(Y*I)/U - (I*Y)/U; \end{axiom} changed: -We may consider the problem where multiplication Y is given, 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 [[retract((ⅇ.i * ⅇ.j)/Ũ) for j in 1..dim] for i in 1..dim] \end{axiom} General Solution We may consider the problem where multiplication Y is given, changed: -J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol); -u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol]; J := jacobian(ravel ω,concat map(variables,ravel U)::List Symbol); u := transpose matrix [concat map(variables,ravel U)::List Symbol]; changed: -Ų:𝐋 := eval(U,ℰ) -matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų, i,1..dim), j,1..dim) -\end{axiom} Ų:ℒ := eval(U,ℰ) matrix [[(ⅇ.i ⅇ.j)/Ų for i in 1..dim] for j in 1..dim] \end{axiom} changed: -Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,1..dim) Ů:=determinant [[retract((ⅇ.i * ⅇ.j)/Ų) for j in 1..dim] for i in 1..dim] changed: -Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/Ų, i,1..dim), j,1..dim) Um:=matrix [[(ⅇ.i*ⅇ.j)/Ų for i in 1..dim] for j in 1..dim] changed: -Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim) -matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim) -\end{axiom} Ω:=Σ(Σ(mU(i,j)*(ⅇ.i*ⅇ.j), i,1..dim), j,1..dim) matrix [[Ω/(ⅆ.i*ⅆ.j) for i in 1..dim] for j in 1..dim] \end{axiom} changed: -d:𝐋:= d:ℒ:= changed: -Χ := H := changed: -Χr := (λ,I)/(I,Y) -test(Χr = Χ ) - -Χl := (I,λ)/(Y,I); -test( Χl = Χ ) -test( Χr = Χl ) - -\end{axiom} Hr := (λ,I)/(I,Y) test(Hr = H ) Hl := (I,λ)/(Y,I); test( Hl = H ) test( Hr = Hl ) \end{axiom} changed: -i:=𝐞.1 i:=ⅇ.1 changed: -Handle -\begin{axiom} - -H:𝐋 := Handle and handle element \begin{axiom} Φ:ℒ := changed: -\end{axiom} Φ1:= i / Φ \end{axiom} changed: -ι:𝐋:= ι:ℒ:= changed: -Ų0:𝐋 :=eval(Ų,ex1) -Ω0:𝐋 :=eval(Ω,ex1)$𝐋 -λ0:𝐋 :=eval(λ,ex1)$𝐋 -H0:𝐋 :=eval(H,ex1)$𝐋 -\end{axiom} Ų0:ℒ :=eval(Ų,ex1) Ω0:ℒ :=eval(Ω,ex1)$ℒ λ0:ℒ :=eval(λ,ex1)$ℒ H0:ℒ :=eval(H,ex1)$ℒ \end{axiom}
Octonion Algebra Is Frobenius In Just One Way
Linear operators over a 8-dimensional vector space representing octonnion algebra
Ref:
- http://arxiv.org/abs/1103.5113
-permuted Frobenius AlgebrasZbigniew 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/Octonion
We need the Axiom LinearOperator library.
fricas
)library CARTEN ARITY CMONAL CPROP CLOP CALEY
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN Arity is now explicitly exposed in frame initial Arity will be automatically loaded when needed from /var/aw/var/LatexWiki/ARITY.NRLIB/ARITY ClosedMonoidal is now explicitly exposed in frame initial ClosedMonoidal will be automatically loaded when needed from /var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL ClosedProp is now explicitly exposed in frame initial ClosedProp will be automatically loaded when needed from /var/aw/var/LatexWiki/CPROP.NRLIB/CPROP ClosedLinearOperator is now explicitly exposed in frame initial ClosedLinearOperator will be automatically loaded when needed from /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP CaleyDickson is now explicitly exposed in frame initial CaleyDickson will be automatically loaded when needed from /var/aw/var/LatexWiki/CALEY.NRLIB/CALEY
Use the following macros for convenient notation
fricas
-- summation macro Σ(x,i, n)==reduce(+, [x for i in n])
Type: Void
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-- subscript macro sb == subscript
Type: Void
ℒ is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.
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dim:=8
 | (1) |
Type: PositiveInteger?
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macro ℂ == CaleyDickson
Type: Void
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macro ℚ == Expression Integer
Type: Void
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ℒ := ClosedLinearOperator(OVAR ['0,'1, '2, '3, '4, '5, '6, '7], ℚ)
![\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 0, 1, 2, 3, 4, 5, 6, 7 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))](images/2951597409152818973-16.0px.png "\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 0, 1, 2, 3, 4, 5, 6, 7 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))") | (2) |
Type: Type
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ⅇ:List ℒ := basisOut()
![\label{eq3}\left[{|_{\ 0}}, \:{|_{\ 1}}, \:{|_{\ 2}}, \:{|_{\ 3}}, \:{|_{\ 4}}, \:{|_{\ 5}}, \:{|_{\ 6}}, \:{|_{\ 7}}\right]](images/8958326163756196923-16.0px.png "\label{eq3}\left[{|_{\ 0}}, \:{|_{\ 1}}, \:{|_{\ 2}}, \:{|_{\ 3}}, \:{|_{\ 4}}, \:{|_{\ 5}}, \:{|_{\ 6}}, \:{|_{\ 7}}\right]") | (3) |
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ⅆ:List ℒ := basisIn()
![\label{eq4}\left[{|^{\ 0}}, \:{|^{\ 1}}, \:{|^{\ 2}}, \:{|^{\ 3}}, \:{|^{\ 4}}, \:{|^{\ 5}}, \:{|^{\ 6}}, \:{|^{\ 7}}\right]](images/4543563693985852424-16.0px.png "\label{eq4}\left[{|^{\ 0}}, \:{|^{\ 1}}, \:{|^{\ 2}}, \:{|^{\ 3}}, \:{|^{\ 4}}, \:{|^{\ 5}}, \:{|^{\ 6}}, \:{|^{\ 7}}\right]") | (4) |
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I:ℒ:=[1]; -- identity for composition
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X:ℒ:=[2,1]; -- twist
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V:ℒ:=ev(1) -- evaluation
 | (5) |
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Λ:ℒ:=co(1) -- co-evaluation
 | (6) |
Now generate structure constants for Octonion 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.
Split-complex, co-quaternions and split-octonions can be specified by Caley-Dickson parameters
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--q0:=sb('q,
 | (7) |
Type: PositiveInteger?
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--q1:=sb('q,
 | (8) |
Type: PositiveInteger?
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q2:=sb('q,
 | (9) |
Type: Symbol
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--q2:=1 -- split-octonion QQ := ℂ(ℂ(ℂ(ℚ,'i, q0), 'j, q1), 'k, q2);
Type: Type
Basis: Each B.i is a octonion number
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B:List QQ := map(x +-> hyper x,1$SQMATRIX(dim, ℚ)::List List ℚ)
![\label{eq10}\left[ 1, \: i , \: j , \:{ij}, \: k , \:{ik}, \:{jk}, \:{{ij}k}\right]](images/3483064029485616143-16.0px.png "\label{eq10}\left[ 1, \: i , \: j , \:{ij}, \: k , \:{ik}, \:{jk}, \:{{ij}k}\right]") | (10) |
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-- Multiplication table: M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]
![\label{eq11}\left[
\begin{array}{cccccccc}
1 & i & j &{ij}& k &{ik}&{jk}&{{ij}k}
\
i & - 1 & -{ij}& j &{- ik}& k &{{ij}k}& -{jk}
\
j &{ij}& - 1 & - i & -{jk}&{-{ij}k}& k &{ik}
\
{ij}& - j & i & - 1 &{-{ij}k}&{jk}&{- ik}& k
\
k &{ik}&{jk}&{{ij}k}& -{q_{2}}&{-{q_{2}}i}&{-{q_{2}}j}&{{-{\overline{q_{2}}}i}j}
\
{ik}& - k &{{ij}k}& -{jk}&{{q_{2}}i}& -{q_{2}}&{{{q_{2}}i}j}&{-{\overline{q_{2}}}j}
\
{jk}&{-{ij}k}& - k &{ik}&{{q_{2}}j}&{{-{q_{2}}i}j}& -{q_{2}}&{{\overline{q_{2}}}i}
\
{{ij}k}&{jk}&{- ik}& - k &{{{\overline{q_{2}}}i}j}&{{\overline{q_{2}}}j}&{-{\overline{q_{2}}}i}& -{q_{2}}](images/5311415551034158175-16.0px.png "\label{eq11}\left[
\begin{array}{cccccccc}
1 & i & j &{ij}& k &{ik}&{jk}&{{ij}k}
\
i & - 1 & -{ij}& j &{- ik}& k &{{ij}k}& -{jk}
\
j &{ij}& - 1 & - i & -{jk}&{-{ij}k}& k &{ik}
\
{ij}& - j & i & - 1 &{-{ij}k}&{jk}&{- ik}& k
\
k &{ik}&{jk}&{{ij}k}& -{q_{2}}&{-{q_{2}}i}&{-{q_{2}}j}&{{-{\overline{q_{2}}}i}j}
\
{ik}& - k &{{ij}k}& -{jk}&{{q_{2}}i}& -{q_{2}}&{{{q_{2}}i}j}&{-{\overline{q_{2}}}j}
\
{jk}&{-{ij}k}& - k &{ik}&{{q_{2}}j}&{{-{q_{2}}i}j}& -{q_{2}}&{{\overline{q_{2}}}i}
\
{{ij}k}&{jk}&{- ik}& - k &{{{\overline{q_{2}}}i}j}&{{\overline{q_{2}}}j}&{-{\overline{q_{2}}}i}& -{q_{2}}") | (11) |
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-- Function to divide the matrix entries by a basis element S(y) == map(x +-> real real real(x/y),M)
Type: Void
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-- The result is a nested list ѕ :=map(S,B)::List List List ℚ;
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Compiling function S with type CaleyDickson(CaleyDickson(
CaleyDickson(Expression(Integer),Type: List(List(List(Expression(Integer))))
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-- 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)
![\label{eq12}\begin{array}{@{}l}
\displaystyle
{|_{\ 0}^{\ 0 \ 0}}+{|_{\ 1}^{\ 0 \ 1}}+{|_{\ 2}^{\ 0 \ 2}}+{|_{\ 3}^{\ 0 \ 3}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 0 \ 4}}}+{|_{\ 5}^{\ 0 \ 5}}+
\
\
\displaystyle
{|_{\ 6}^{\ 0 \ 6}}+{|_{\ 7}^{\ 0 \ 7}}+{|_{\ 1}^{\ 1 \ 0}}-{|_{\ 0}^{\ 1 \ 1}}+{|_{\ 3}^{\ 1 \ 2}}-{|_{\ 2}^{\ 1 \ 3}}+
\
\
\displaystyle
{|_{\ 5}^{\ 1 \ 4}}-{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 1 \ 5}}}-{|_{\ 7}^{\ 1 \ 6}}+{|_{\ 6}^{\ 1 \ 7}}+{|_{\ 2}^{\ 2 \ 0}}-{|_{\ 3}^{\ 2 \ 1}}-
\
\
\displaystyle
{|_{\ 0}^{\ 2 \ 2}}+{|_{\ 1}^{\ 2 \ 3}}+{|_{\ 6}^{\ 2 \ 4}}+{|_{\ 7}^{\ 2 \ 5}}-{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 2 \ 6}}}-{|_{\ 5}^{\ 2 \ 7}}+
\
\
\displaystyle
{|_{\ 3}^{\ 3 \ 0}}+{|_{\ 2}^{\ 3 \ 1}}-{|_{\ 1}^{\ 3 \ 2}}-{|_{\ 0}^{\ 3 \ 3}}+{|_{\ 7}^{\ 3 \ 4}}-{|_{\ 6}^{\ 3 \ 5}}+
\
\
\displaystyle
{|_{\ 5}^{\ 3 \ 6}}-{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 3 \ 7}}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 4 \ 0}}}-{|_{\ 5}^{\ 4 \ 1}}-{|_{\ 6}^{\ 4 \ 2}}-{|_{\ 7}^{\ 4 \ 3}}-
\
\
\displaystyle
{{q_{2}}\ {|_{\ 0}^{\ 4 \ 4}}}+{{q_{2}}\ {|_{\ 1}^{\ 4 \ 5}}}+{{q_{2}}\ {|_{\ 2}^{\ 4 \ 6}}}+{{q_{2}}\ {|_{\ 3}^{\ 4 \ 7}}}+{|_{\ 5}^{\ 5 \ 0}}+
\
\
\displaystyle
{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 5 \ 1}}}-{|_{\ 7}^{\ 5 \ 2}}+{|_{\ 6}^{\ 5 \ 3}}-{{q_{2}}\ {|_{\ 1}^{\ 5 \ 4}}}-{{q_{2}}\ {|_{\ 0}^{\ 5 \ 5}}}-
\
\
\displaystyle
{{\overline{q_{2}}}\ {|_{\ 3}^{\ 5 \ 6}}}+{{\overline{q_{2}}}\ {|_{\ 2}^{\ 5 \ 7}}}+{|_{\ 6}^{\ 6 \ 0}}+{|_{\ 7}^{\ 6 \ 1}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 6 \ 2}}}-
\
\
\displaystyle
{|_{\ 5}^{\ 6 \ 3}}-{{q_{2}}\ {|_{\ 2}^{\ 6 \ 4}}}+{{\overline{q_{2}}}\ {|_{\ 3}^{\ 6 \ 5}}}-{{q_{2}}\ {|_{\ 0}^{\ 6 \ 6}}}-{{\overline{q_{2}}}\ {|_{\ 1}^{\ 6 \ 7}}}+
\
\
\displaystyle
{|_{\ 7}^{\ 7 \ 0}}-{|_{\ 6}^{\ 7 \ 1}}+{|_{\ 5}^{\ 7 \ 2}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 7 \ 3}}}-{{q_{2}}\ {|_{\ 3}^{\ 7 \ 4}}}-
\
\
\displaystyle
{{\overline{q_{2}}}\ {|_{\ 2}^{\ 7 \ 5}}}+{{\overline{q_{2}}}\ {|_{\ 1}^{\ 7 \ 6}}}-{{q_{2}}\ {|_{\ 0}^{\ 7 \ 7}}}](images/8439159333734992001-16.0px.png "\label{eq12}\begin{array}{@{}l}
\displaystyle
{|_{\ 0}^{\ 0 \ 0}}+{|_{\ 1}^{\ 0 \ 1}}+{|_{\ 2}^{\ 0 \ 2}}+{|_{\ 3}^{\ 0 \ 3}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 0 \ 4}}}+{|_{\ 5}^{\ 0 \ 5}}+
\
\
\displaystyle
{|_{\ 6}^{\ 0 \ 6}}+{|_{\ 7}^{\ 0 \ 7}}+{|_{\ 1}^{\ 1 \ 0}}-{|_{\ 0}^{\ 1 \ 1}}+{|_{\ 3}^{\ 1 \ 2}}-{|_{\ 2}^{\ 1 \ 3}}+
\
\
\displaystyle
{|_{\ 5}^{\ 1 \ 4}}-{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 1 \ 5}}}-{|_{\ 7}^{\ 1 \ 6}}+{|_{\ 6}^{\ 1 \ 7}}+{|_{\ 2}^{\ 2 \ 0}}-{|_{\ 3}^{\ 2 \ 1}}-
\
\
\displaystyle
{|_{\ 0}^{\ 2 \ 2}}+{|_{\ 1}^{\ 2 \ 3}}+{|_{\ 6}^{\ 2 \ 4}}+{|_{\ 7}^{\ 2 \ 5}}-{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 2 \ 6}}}-{|_{\ 5}^{\ 2 \ 7}}+
\
\
\displaystyle
{|_{\ 3}^{\ 3 \ 0}}+{|_{\ 2}^{\ 3 \ 1}}-{|_{\ 1}^{\ 3 \ 2}}-{|_{\ 0}^{\ 3 \ 3}}+{|_{\ 7}^{\ 3 \ 4}}-{|_{\ 6}^{\ 3 \ 5}}+
\
\
\displaystyle
{|_{\ 5}^{\ 3 \ 6}}-{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 3 \ 7}}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 4 \ 0}}}-{|_{\ 5}^{\ 4 \ 1}}-{|_{\ 6}^{\ 4 \ 2}}-{|_{\ 7}^{\ 4 \ 3}}-
\
\
\displaystyle
{{q_{2}}\ {|_{\ 0}^{\ 4 \ 4}}}+{{q_{2}}\ {|_{\ 1}^{\ 4 \ 5}}}+{{q_{2}}\ {|_{\ 2}^{\ 4 \ 6}}}+{{q_{2}}\ {|_{\ 3}^{\ 4 \ 7}}}+{|_{\ 5}^{\ 5 \ 0}}+
\
\
\displaystyle
{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 5 \ 1}}}-{|_{\ 7}^{\ 5 \ 2}}+{|_{\ 6}^{\ 5 \ 3}}-{{q_{2}}\ {|_{\ 1}^{\ 5 \ 4}}}-{{q_{2}}\ {|_{\ 0}^{\ 5 \ 5}}}-
\
\
\displaystyle
{{\overline{q_{2}}}\ {|_{\ 3}^{\ 5 \ 6}}}+{{\overline{q_{2}}}\ {|_{\ 2}^{\ 5 \ 7}}}+{|_{\ 6}^{\ 6 \ 0}}+{|_{\ 7}^{\ 6 \ 1}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 6 \ 2}}}-
\
\
\displaystyle
{|_{\ 5}^{\ 6 \ 3}}-{{q_{2}}\ {|_{\ 2}^{\ 6 \ 4}}}+{{\overline{q_{2}}}\ {|_{\ 3}^{\ 6 \ 5}}}-{{q_{2}}\ {|_{\ 0}^{\ 6 \ 6}}}-{{\overline{q_{2}}}\ {|_{\ 1}^{\ 6 \ 7}}}+
\
\
\displaystyle
{|_{\ 7}^{\ 7 \ 0}}-{|_{\ 6}^{\ 7 \ 1}}+{|_{\ 5}^{\ 7 \ 2}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 7 \ 3}}}-{{q_{2}}\ {|_{\ 3}^{\ 7 \ 4}}}-
\
\
\displaystyle
{{\overline{q_{2}}}\ {|_{\ 2}^{\ 7 \ 5}}}+{{\overline{q_{2}}}\ {|_{\ 1}^{\ 7 \ 6}}}-{{q_{2}}\ {|_{\ 0}^{\ 7 \ 7}}}") | (12) |
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arity Y
 | (13) |
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matrix [[(ⅇ.i*ⅇ.j)/Y for i in 1..dim] for j in 1..dim]
![\label{eq14}\left[
\begin{array}{cccccccc}
{|_{\ 0}}&{|_{\ 1}}&{|_{\ 2}}&{|_{\ 3}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 5}}&{|_{\ 6}}&{|_{\ 7}}
\
{|_{\ 1}}& -{|_{\ 0}}& -{|_{\ 3}}&{|_{\ 2}}& -{|_{\ 5}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 7}}& -{|_{\ 6}}
\
{|_{\ 2}}&{|_{\ 3}}& -{|_{\ 0}}& -{|_{\ 1}}& -{|_{\ 6}}& -{|_{\ 7}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 5}}
\
{|_{\ 3}}& -{|_{\ 2}}&{|_{\ 1}}& -{|_{\ 0}}& -{|_{\ 7}}&{|_{\ 6}}& -{|_{\ 5}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}
\
{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 5}}&{|_{\ 6}}&{|_{\ 7}}& -{{q_{2}}\ {|_{\ 0}}}& -{{q_{2}}\ {|_{\ 1}}}& -{{q_{2}}\ {|_{\ 2}}}& -{{q_{2}}\ {|_{\ 3}}}
\
{|_{\ 5}}& -{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 7}}& -{|_{\ 6}}&{{q_{2}}\ {|_{\ 1}}}& -{{q_{2}}\ {|_{\ 0}}}&{{\overline{q_{2}}}\ {|_{\ 3}}}& -{{\overline{q_{2}}}\ {|_{\ 2}}}
\
{|_{\ 6}}& -{|_{\ 7}}& -{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 5}}&{{q_{2}}\ {|_{\ 2}}}& -{{\overline{q_{2}}}\ {|_{\ 3}}}& -{{q_{2}}\ {|_{\ 0}}}&{{\overline{q_{2}}}\ {|_{\ 1}}}
\
{|_{\ 7}}&{|_{\ 6}}& -{|_{\ 5}}& -{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{{q_{2}}\ {|_{\ 3}}}&{{\overline{q_{2}}}\ {|_{\ 2}}}& -{{\overline{q_{2}}}\ {|_{\ 1}}}& -{{q_{2}}\ {|_{\ 0}}}](images/2789247758362211731-16.0px.png "\label{eq14}\left[
\begin{array}{cccccccc}
{|_{\ 0}}&{|_{\ 1}}&{|_{\ 2}}&{|_{\ 3}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 5}}&{|_{\ 6}}&{|_{\ 7}}
\
{|_{\ 1}}& -{|_{\ 0}}& -{|_{\ 3}}&{|_{\ 2}}& -{|_{\ 5}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 7}}& -{|_{\ 6}}
\
{|_{\ 2}}&{|_{\ 3}}& -{|_{\ 0}}& -{|_{\ 1}}& -{|_{\ 6}}& -{|_{\ 7}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 5}}
\
{|_{\ 3}}& -{|_{\ 2}}&{|_{\ 1}}& -{|_{\ 0}}& -{|_{\ 7}}&{|_{\ 6}}& -{|_{\ 5}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}
\
{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 5}}&{|_{\ 6}}&{|_{\ 7}}& -{{q_{2}}\ {|_{\ 0}}}& -{{q_{2}}\ {|_{\ 1}}}& -{{q_{2}}\ {|_{\ 2}}}& -{{q_{2}}\ {|_{\ 3}}}
\
{|_{\ 5}}& -{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 7}}& -{|_{\ 6}}&{{q_{2}}\ {|_{\ 1}}}& -{{q_{2}}\ {|_{\ 0}}}&{{\overline{q_{2}}}\ {|_{\ 3}}}& -{{\overline{q_{2}}}\ {|_{\ 2}}}
\
{|_{\ 6}}& -{|_{\ 7}}& -{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{|_{\ 5}}&{{q_{2}}\ {|_{\ 2}}}& -{{\overline{q_{2}}}\ {|_{\ 3}}}& -{{q_{2}}\ {|_{\ 0}}}&{{\overline{q_{2}}}\ {|_{\ 1}}}
\
{|_{\ 7}}&{|_{\ 6}}& -{|_{\ 5}}& -{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\ 4}}}&{{q_{2}}\ {|_{\ 3}}}&{{\overline{q_{2}}}\ {|_{\ 2}}}& -{{\overline{q_{2}}}\ {|_{\ 1}}}& -{{q_{2}}\ {|_{\ 0}}}") | (14) |
A scalar product is denoted by the (2,0)-tensor
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U:=Σ(Σ(script('u,
![\label{eq15}\begin{array}{@{}l}
\displaystyle
{{u^{1, \: 1}}\ {|^{\ 0 \ 0}}}+{{u^{1, \: 2}}\ {|^{\ 0 \ 1}}}+{{u^{1, \: 3}}\ {|^{\ 0 \ 2}}}+{{u^{1, \: 4}}\ {|^{\ 0 \ 3}}}+
\
\
\displaystyle
{{u^{1, \: 5}}\ {|^{\ 0 \ 4}}}+{{u^{1, \: 6}}\ {|^{\ 0 \ 5}}}+{{u^{1, \: 7}}\ {|^{\ 0 \ 6}}}+{{u^{1, \: 8}}\ {|^{\ 0 \ 7}}}+
\
\
\displaystyle
{{u^{2, \: 1}}\ {|^{\ 1 \ 0}}}+{{u^{2, \: 2}}\ {|^{\ 1 \ 1}}}+{{u^{2, \: 3}}\ {|^{\ 1 \ 2}}}+{{u^{2, \: 4}}\ {|^{\ 1 \ 3}}}+
\
\
\displaystyle
{{u^{2, \: 5}}\ {|^{\ 1 \ 4}}}+{{u^{2, \: 6}}\ {|^{\ 1 \ 5}}}+{{u^{2, \: 7}}\ {|^{\ 1 \ 6}}}+{{u^{2, \: 8}}\ {|^{\ 1 \ 7}}}+
\
\
\displaystyle
{{u^{3, \: 1}}\ {|^{\ 2 \ 0}}}+{{u^{3, \: 2}}\ {|^{\ 2 \ 1}}}+{{u^{3, \: 3}}\ {|^{\ 2 \ 2}}}+{{u^{3, \: 4}}\ {|^{\ 2 \ 3}}}+
\
\
\displaystyle
{{u^{3, \: 5}}\ {|^{\ 2 \ 4}}}+{{u^{3, \: 6}}\ {|^{\ 2 \ 5}}}+{{u^{3, \: 7}}\ {|^{\ 2 \ 6}}}+{{u^{3, \: 8}}\ {|^{\ 2 \ 7}}}+
\
\
\displaystyle
{{u^{4, \: 1}}\ {|^{\ 3 \ 0}}}+{{u^{4, \: 2}}\ {|^{\ 3 \ 1}}}+{{u^{4, \: 3}}\ {|^{\ 3 \ 2}}}+{{u^{4, \: 4}}\ {|^{\ 3 \ 3}}}+
\
\
\displaystyle
{{u^{4, \: 5}}\ {|^{\ 3 \ 4}}}+{{u^{4, \: 6}}\ {|^{\ 3 \ 5}}}+{{u^{4, \: 7}}\ {|^{\ 3 \ 6}}}+{{u^{4, \: 8}}\ {|^{\ 3 \ 7}}}+
\
\
\displaystyle
{{u^{5, \: 1}}\ {|^{\ 4 \ 0}}}+{{u^{5, \: 2}}\ {|^{\ 4 \ 1}}}+{{u^{5, \: 3}}\ {|^{\ 4 \ 2}}}+{{u^{5, \: 4}}\ {|^{\ 4 \ 3}}}+
\
\
\displaystyle
{{u^{5, \: 5}}\ {|^{\ 4 \ 4}}}+{{u^{5, \: 6}}\ {|^{\ 4 \ 5}}}+{{u^{5, \: 7}}\ {|^{\ 4 \ 6}}}+{{u^{5, \: 8}}\ {|^{\ 4 \ 7}}}+
\
\
\displaystyle
{{u^{6, \: 1}}\ {|^{\ 5 \ 0}}}+{{u^{6, \: 2}}\ {|^{\ 5 \ 1}}}+{{u^{6, \: 3}}\ {|^{\ 5 \ 2}}}+{{u^{6, \: 4}}\ {|^{\ 5 \ 3}}}+
\
\
\displaystyle
{{u^{6, \: 5}}\ {|^{\ 5 \ 4}}}+{{u^{6, \: 6}}\ {|^{\ 5 \ 5}}}+{{u^{6, \: 7}}\ {|^{\ 5 \ 6}}}+{{u^{6, \: 8}}\ {|^{\ 5 \ 7}}}+
\
\
\displaystyle
{{u^{7, \: 1}}\ {|^{\ 6 \ 0}}}+{{u^{7, \: 2}}\ {|^{\ 6 \ 1}}}+{{u^{7, \: 3}}\ {|^{\ 6 \ 2}}}+{{u^{7, \: 4}}\ {|^{\ 6 \ 3}}}+
\
\
\displaystyle
{{u^{7, \: 5}}\ {|^{\ 6 \ 4}}}+{{u^{7, \: 6}}\ {|^{\ 6 \ 5}}}+{{u^{7, \: 7}}\ {|^{\ 6 \ 6}}}+{{u^{7, \: 8}}\ {|^{\ 6 \ 7}}}+
\
\
\displaystyle
{{u^{8, \: 1}}\ {|^{\ 7 \ 0}}}+{{u^{8, \: 2}}\ {|^{\ 7 \ 1}}}+{{u^{8, \: 3}}\ {|^{\ 7 \ 2}}}+{{u^{8, \: 4}}\ {|^{\ 7 \ 3}}}+
\
\
\displaystyle
{{u^{8, \: 5}}\ {|^{\ 7 \ 4}}}+{{u^{8, \: 6}}\ {|^{\ 7 \ 5}}}+{{u^{8, \: 7}}\ {|^{\ 7 \ 6}}}+{{u^{8, \: 8}}\ {|^{\ 7 \ 7}}}](images/8916129074682016888-16.0px.png "\label{eq15}\begin{array}{@{}l}
\displaystyle
{{u^{1, \: 1}}\ {|^{\ 0 \ 0}}}+{{u^{1, \: 2}}\ {|^{\ 0 \ 1}}}+{{u^{1, \: 3}}\ {|^{\ 0 \ 2}}}+{{u^{1, \: 4}}\ {|^{\ 0 \ 3}}}+
\
\
\displaystyle
{{u^{1, \: 5}}\ {|^{\ 0 \ 4}}}+{{u^{1, \: 6}}\ {|^{\ 0 \ 5}}}+{{u^{1, \: 7}}\ {|^{\ 0 \ 6}}}+{{u^{1, \: 8}}\ {|^{\ 0 \ 7}}}+
\
\
\displaystyle
{{u^{2, \: 1}}\ {|^{\ 1 \ 0}}}+{{u^{2, \: 2}}\ {|^{\ 1 \ 1}}}+{{u^{2, \: 3}}\ {|^{\ 1 \ 2}}}+{{u^{2, \: 4}}\ {|^{\ 1 \ 3}}}+
\
\
\displaystyle
{{u^{2, \: 5}}\ {|^{\ 1 \ 4}}}+{{u^{2, \: 6}}\ {|^{\ 1 \ 5}}}+{{u^{2, \: 7}}\ {|^{\ 1 \ 6}}}+{{u^{2, \: 8}}\ {|^{\ 1 \ 7}}}+
\
\
\displaystyle
{{u^{3, \: 1}}\ {|^{\ 2 \ 0}}}+{{u^{3, \: 2}}\ {|^{\ 2 \ 1}}}+{{u^{3, \: 3}}\ {|^{\ 2 \ 2}}}+{{u^{3, \: 4}}\ {|^{\ 2 \ 3}}}+
\
\
\displaystyle
{{u^{3, \: 5}}\ {|^{\ 2 \ 4}}}+{{u^{3, \: 6}}\ {|^{\ 2 \ 5}}}+{{u^{3, \: 7}}\ {|^{\ 2 \ 6}}}+{{u^{3, \: 8}}\ {|^{\ 2 \ 7}}}+
\
\
\displaystyle
{{u^{4, \: 1}}\ {|^{\ 3 \ 0}}}+{{u^{4, \: 2}}\ {|^{\ 3 \ 1}}}+{{u^{4, \: 3}}\ {|^{\ 3 \ 2}}}+{{u^{4, \: 4}}\ {|^{\ 3 \ 3}}}+
\
\
\displaystyle
{{u^{4, \: 5}}\ {|^{\ 3 \ 4}}}+{{u^{4, \: 6}}\ {|^{\ 3 \ 5}}}+{{u^{4, \: 7}}\ {|^{\ 3 \ 6}}}+{{u^{4, \: 8}}\ {|^{\ 3 \ 7}}}+
\
\
\displaystyle
{{u^{5, \: 1}}\ {|^{\ 4 \ 0}}}+{{u^{5, \: 2}}\ {|^{\ 4 \ 1}}}+{{u^{5, \: 3}}\ {|^{\ 4 \ 2}}}+{{u^{5, \: 4}}\ {|^{\ 4 \ 3}}}+
\
\
\displaystyle
{{u^{5, \: 5}}\ {|^{\ 4 \ 4}}}+{{u^{5, \: 6}}\ {|^{\ 4 \ 5}}}+{{u^{5, \: 7}}\ {|^{\ 4 \ 6}}}+{{u^{5, \: 8}}\ {|^{\ 4 \ 7}}}+
\
\
\displaystyle
{{u^{6, \: 1}}\ {|^{\ 5 \ 0}}}+{{u^{6, \: 2}}\ {|^{\ 5 \ 1}}}+{{u^{6, \: 3}}\ {|^{\ 5 \ 2}}}+{{u^{6, \: 4}}\ {|^{\ 5 \ 3}}}+
\
\
\displaystyle
{{u^{6, \: 5}}\ {|^{\ 5 \ 4}}}+{{u^{6, \: 6}}\ {|^{\ 5 \ 5}}}+{{u^{6, \: 7}}\ {|^{\ 5 \ 6}}}+{{u^{6, \: 8}}\ {|^{\ 5 \ 7}}}+
\
\
\displaystyle
{{u^{7, \: 1}}\ {|^{\ 6 \ 0}}}+{{u^{7, \: 2}}\ {|^{\ 6 \ 1}}}+{{u^{7, \: 3}}\ {|^{\ 6 \ 2}}}+{{u^{7, \: 4}}\ {|^{\ 6 \ 3}}}+
\
\
\displaystyle
{{u^{7, \: 5}}\ {|^{\ 6 \ 4}}}+{{u^{7, \: 6}}\ {|^{\ 6 \ 5}}}+{{u^{7, \: 7}}\ {|^{\ 6 \ 6}}}+{{u^{7, \: 8}}\ {|^{\ 6 \ 7}}}+
\
\
\displaystyle
{{u^{8, \: 1}}\ {|^{\ 7 \ 0}}}+{{u^{8, \: 2}}\ {|^{\ 7 \ 1}}}+{{u^{8, \: 3}}\ {|^{\ 7 \ 2}}}+{{u^{8, \: 4}}\ {|^{\ 7 \ 3}}}+
\
\
\displaystyle
{{u^{8, \: 5}}\ {|^{\ 7 \ 4}}}+{{u^{8, \: 6}}\ {|^{\ 7 \ 5}}}+{{u^{8, \: 7}}\ {|^{\ 7 \ 6}}}+{{u^{8, \: 8}}\ {|^{\ 7 \ 7}}}") | (15) |
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:
(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}
}](images/2375189246716000159-16.0px.png "\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}
}") |
 | (16) |
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.
fricas
ω:ℒ :=(Y*I)/U - (I*Y)/U;
Definition 2
An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.
The Cartan-Killing Trace
fricas
Ú:=
( Y Λ ) / _
( Y I ) / _
V
![\label{eq17}\begin{array}{@{}l}
\displaystyle
{{{{7 \ {\overline{q_{2}}}}+{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 0 \ 0}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 1 \ 1}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 2 \ 2}}}+
\
\
\displaystyle
{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 3 \ 3}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 4 \ 4}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 5 \ 5}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 6 \ 6}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 7 \ 7}}}](images/7668440731659621428-16.0px.png "\label{eq17}\begin{array}{@{}l}
\displaystyle
{{{{7 \ {\overline{q_{2}}}}+{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 0 \ 0}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 1 \ 1}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 2 \ 2}}}+
\
\
\displaystyle
{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 3 \ 3}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 4 \ 4}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 5 \ 5}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 6 \ 6}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 7 \ 7}}}") | (17) |
fricas
Ù:=
( Λ Y ) / _
( I Y ) / _
V
![\label{eq18}\begin{array}{@{}l}
\displaystyle
{{{{7 \ {\overline{q_{2}}}}+{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 0 \ 0}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 1 \ 1}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 2 \ 2}}}+
\
\
\displaystyle
{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 3 \ 3}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 4 \ 4}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 5 \ 5}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 6 \ 6}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 7 \ 7}}}](images/351765943747435795-16.0px.png "\label{eq18}\begin{array}{@{}l}
\displaystyle
{{{{7 \ {\overline{q_{2}}}}+{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 0 \ 0}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 1 \ 1}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 2 \ 2}}}+
\
\
\displaystyle
{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 3 \ 3}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 4 \ 4}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 5 \ 5}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 6 \ 6}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 7 \ 7}}}") | (18) |
fricas
test(Ù=Ú)
 | (19) |
Type: Boolean
forms a non-degenerate associative scalar product for Y
fricas
Ũ := Ù
![\label{eq20}\begin{array}{@{}l}
\displaystyle
{{{{7 \ {\overline{q_{2}}}}+{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 0 \ 0}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 1 \ 1}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 2 \ 2}}}+
\
\
\displaystyle
{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 3 \ 3}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 4 \ 4}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 5 \ 5}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 6 \ 6}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 7 \ 7}}}](images/9004498164049069466-16.0px.png "\label{eq20}\begin{array}{@{}l}
\displaystyle
{{{{7 \ {\overline{q_{2}}}}+{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 0 \ 0}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 1 \ 1}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 2 \ 2}}}+
\
\
\displaystyle
{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\ 3 \ 3}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 4 \ 4}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 5 \ 5}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 6 \ 6}}}+
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\ 7 \ 7}}}") | (20) |
fricas
test
( Y I ) /
Ũ =
( I Y ) /
Ũ
 | (21) |
Type: Boolean
fricas
determinant [[retract((ⅇ.i * ⅇ.j)/Ũ) for j in 1..dim] for i in 1..dim]
![\label{eq22}{\left(
\begin{array}{@{}l}
\displaystyle
-{{5764801}\ {{q_{2}}^{4}}\ {{\overline{q_{2}}}^{8}}}-{{65883
44}\ {{q_{2}}^{5}}\ {{\overline{q_{2}}}^{7}}}-
\
\
\displaystyle
{{3294172}\ {{q_{2}}^{6}}\ {{\overline{q_{2}}}^{6}}}-{{941192}\ {{q_{2}}^{7}}\ {{\overline{q_{2}}}^{5}}}-{{168070}\ {{q_{2}}^{8}}\ {{\overline{q_{2}}}^{4}}}-
\
\
\displaystyle
{{19208}\ {{q_{2}}^{9}}\ {{\overline{q_{2}}}^{3}}}-{{1372}\ {{q_{2}}^{10}}\ {{\overline{q_{2}}}^{2}}}-{{56}\ {{q_{2}}^{11}}\ {\overline{q_{2}}}}-{{q_{2}}^{12}}](images/5292919296824870720-16.0px.png "\label{eq22}{\left(
\begin{array}{@{}l}
\displaystyle
-{{5764801}\ {{q_{2}}^{4}}\ {{\overline{q_{2}}}^{8}}}-{{65883
44}\ {{q_{2}}^{5}}\ {{\overline{q_{2}}}^{7}}}-
\
\
\displaystyle
{{3294172}\ {{q_{2}}^{6}}\ {{\overline{q_{2}}}^{6}}}-{{941192}\ {{q_{2}}^{7}}\ {{\overline{q_{2}}}^{5}}}-{{168070}\ {{q_{2}}^{8}}\ {{\overline{q_{2}}}^{4}}}-
\
\
\displaystyle
{{19208}\ {{q_{2}}^{9}}\ {{\overline{q_{2}}}^{3}}}-{{1372}\ {{q_{2}}^{10}}\ {{\overline{q_{2}}}^{2}}}-{{56}\ {{q_{2}}^{11}}\ {\overline{q_{2}}}}-{{q_{2}}^{12}}") | (22) |
Type: Expression(Integer)
General Solution
We may consider the problem where multiplication Y is given,
and look for all associative scalar products 
This problem can be solved using linear algebra.
fricas
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial J := jacobian(ravel ω,concat map(variables, ravel U)::List Symbol);
Type: Matrix(Expression(Integer))
fricas
u := transpose matrix [concat map(variables,ravel U)::List Symbol];
Type: Matrix(Polynomial(Integer))
fricas
J::OutputForm * u::OutputForm = 0
 | (23) |
Type: Equation(OutputForm?)
fricas
nrows(J),ncols(J)
![\label{eq24}\left[{512}, \:{64}\right]](images/7899213068589509046-16.0px.png "\label{eq24}\left[{512}, \:{64}\right]") | (24) |
Type: Tuple(PositiveInteger?)
The matrix J transforms the coefficients of the tensor 
into coefficients of the tensor 
. We are looking for
the general linear family of tensors 
such that
J transforms into 
for any such .
If the null space of the J matrix is not empty we can use
the basis to find all non-trivial solutions for U:
fricas
Ñ:=nullSpace(J);
Type: List(Vector(Expression(Integer)))
fricas
ℰ:=map((x,y)+->x=y, concat map(variables, ravel U), entries Σ(sb('p, [i])*Ñ.i, i, 1..#Ñ) )
>> Error detected within library code: reducing over an empty list needs the 3 argument form
This defines a family of Frobenius algebras:
fricas
zero? eval(ω,ℰ)
There are 12 exposed and 6 unexposed library operations named eval having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse,or issue )display op eval to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval with argument type(s) ClosedLinearOperator(OrderedVariableList([0,1, 2, 3, 4, 5, 6, 7]), Expression(Integer)) Variable(ℰ)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need.
The pairing is necessarily diagonal!
fricas
Ų:ℒ := eval(U,ℰ)
There are 12 exposed and 6 unexposed library operations named eval having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse,or issue )display op eval to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval with argument type(s) ClosedLinearOperator(OrderedVariableList([0,1, 2, 3, 4, 5, 6, 7]), Expression(Integer)) Variable(ℰ)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need.
The scalar product must be non-degenerate:
fricas
Ů:=determinant [[retract((ⅇ.i * ⅇ.j)/Ų) for j in 1..dim] for i in 1..dim]
0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 +
>> Error detected within library code: failed
Definition 3
Co-pairing
Solve the Snake Relation as a system of linear equations.
fricas
Um:=matrix [[(ⅇ.i*ⅇ.j)/Ų for i in 1..dim] for j in 1..dim]
0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 +
![\label{eq25}\left[
\begin{array}{cccccccc}
{�� \ {|_{\ 0 \ 0}}}&{�� \ {|_{\ 1 \ 0}}}&{�� \ {|_{\ 2 \ 0}}}&{�� \ {|_{\ 3 \ 0}}}&{�� \ {|_{\ 4 \ 0}}}&{�� \ {|_{\ 5 \ 0}}}&{�� \ {|_{\ 6 \ 0}}}&{�� \ {|_{\ 7 \ 0}}}
\
{�� \ {|_{\ 0 \ 1}}}&{�� \ {|_{\ 1 \ 1}}}&{�� \ {|_{\ 2 \ 1}}}&{�� \ {|_{\ 3 \ 1}}}&{�� \ {|_{\ 4 \ 1}}}&{�� \ {|_{\ 5 \ 1}}}&{�� \ {|_{\ 6 \ 1}}}&{�� \ {|_{\ 7 \ 1}}}
\
{�� \ {|_{\ 0 \ 2}}}&{�� \ {|_{\ 1 \ 2}}}&{�� \ {|_{\ 2 \ 2}}}&{�� \ {|_{\ 3 \ 2}}}&{�� \ {|_{\ 4 \ 2}}}&{�� \ {|_{\ 5 \ 2}}}&{�� \ {|_{\ 6 \ 2}}}&{�� \ {|_{\ 7 \ 2}}}
\
{�� \ {|_{\ 0 \ 3}}}&{�� \ {|_{\ 1 \ 3}}}&{�� \ {|_{\ 2 \ 3}}}&{�� \ {|_{\ 3 \ 3}}}&{�� \ {|_{\ 4 \ 3}}}&{�� \ {|_{\ 5 \ 3}}}&{�� \ {|_{\ 6 \ 3}}}&{�� \ {|_{\ 7 \ 3}}}
\
{�� \ {|_{\ 0 \ 4}}}&{�� \ {|_{\ 1 \ 4}}}&{�� \ {|_{\ 2 \ 4}}}&{�� \ {|_{\ 3 \ 4}}}&{�� \ {|_{\ 4 \ 4}}}&{�� \ {|_{\ 5 \ 4}}}&{�� \ {|_{\ 6 \ 4}}}&{�� \ {|_{\ 7 \ 4}}}
\
{�� \ {|_{\ 0 \ 5}}}&{�� \ {|_{\ 1 \ 5}}}&{�� \ {|_{\ 2 \ 5}}}&{�� \ {|_{\ 3 \ 5}}}&{�� \ {|_{\ 4 \ 5}}}&{�� \ {|_{\ 5 \ 5}}}&{�� \ {|_{\ 6 \ 5}}}&{�� \ {|_{\ 7 \ 5}}}
\
{�� \ {|_{\ 0 \ 6}}}&{�� \ {|_{\ 1 \ 6}}}&{�� \ {|_{\ 2 \ 6}}}&{�� \ {|_{\ 3 \ 6}}}&{�� \ {|_{\ 4 \ 6}}}&{�� \ {|_{\ 5 \ 6}}}&{�� \ {|_{\ 6 \ 6}}}&{�� \ {|_{\ 7 \ 6}}}
\
{�� \ {|_{\ 0 \ 7}}}&{�� \ {|_{\ 1 \ 7}}}&{�� \ {|_{\ 2 \ 7}}}&{�� \ {|_{\ 3 \ 7}}}&{�� \ {|_{\ 4 \ 7}}}&{�� \ {|_{\ 5 \ 7}}}&{�� \ {|_{\ 6 \ 7}}}&{�� \ {|_{\ 7 \ 7}}}](images/9140469345711941437-16.0px.png "\label{eq25}\left[
\begin{array}{cccccccc}
{�� \ {|_{\ 0 \ 0}}}&{�� \ {|_{\ 1 \ 0}}}&{�� \ {|_{\ 2 \ 0}}}&{�� \ {|_{\ 3 \ 0}}}&{�� \ {|_{\ 4 \ 0}}}&{�� \ {|_{\ 5 \ 0}}}&{�� \ {|_{\ 6 \ 0}}}&{�� \ {|_{\ 7 \ 0}}}
\
{�� \ {|_{\ 0 \ 1}}}&{�� \ {|_{\ 1 \ 1}}}&{�� \ {|_{\ 2 \ 1}}}&{�� \ {|_{\ 3 \ 1}}}&{�� \ {|_{\ 4 \ 1}}}&{�� \ {|_{\ 5 \ 1}}}&{�� \ {|_{\ 6 \ 1}}}&{�� \ {|_{\ 7 \ 1}}}
\
{�� \ {|_{\ 0 \ 2}}}&{�� \ {|_{\ 1 \ 2}}}&{�� \ {|_{\ 2 \ 2}}}&{�� \ {|_{\ 3 \ 2}}}&{�� \ {|_{\ 4 \ 2}}}&{�� \ {|_{\ 5 \ 2}}}&{�� \ {|_{\ 6 \ 2}}}&{�� \ {|_{\ 7 \ 2}}}
\
{�� \ {|_{\ 0 \ 3}}}&{�� \ {|_{\ 1 \ 3}}}&{�� \ {|_{\ 2 \ 3}}}&{�� \ {|_{\ 3 \ 3}}}&{�� \ {|_{\ 4 \ 3}}}&{�� \ {|_{\ 5 \ 3}}}&{�� \ {|_{\ 6 \ 3}}}&{�� \ {|_{\ 7 \ 3}}}
\
{�� \ {|_{\ 0 \ 4}}}&{�� \ {|_{\ 1 \ 4}}}&{�� \ {|_{\ 2 \ 4}}}&{�� \ {|_{\ 3 \ 4}}}&{�� \ {|_{\ 4 \ 4}}}&{�� \ {|_{\ 5 \ 4}}}&{�� \ {|_{\ 6 \ 4}}}&{�� \ {|_{\ 7 \ 4}}}
\
{�� \ {|_{\ 0 \ 5}}}&{�� \ {|_{\ 1 \ 5}}}&{�� \ {|_{\ 2 \ 5}}}&{�� \ {|_{\ 3 \ 5}}}&{�� \ {|_{\ 4 \ 5}}}&{�� \ {|_{\ 5 \ 5}}}&{�� \ {|_{\ 6 \ 5}}}&{�� \ {|_{\ 7 \ 5}}}
\
{�� \ {|_{\ 0 \ 6}}}&{�� \ {|_{\ 1 \ 6}}}&{�� \ {|_{\ 2 \ 6}}}&{�� \ {|_{\ 3 \ 6}}}&{�� \ {|_{\ 4 \ 6}}}&{�� \ {|_{\ 5 \ 6}}}&{�� \ {|_{\ 6 \ 6}}}&{�� \ {|_{\ 7 \ 6}}}
\
{�� \ {|_{\ 0 \ 7}}}&{�� \ {|_{\ 1 \ 7}}}&{�� \ {|_{\ 2 \ 7}}}&{�� \ {|_{\ 3 \ 7}}}&{�� \ {|_{\ 4 \ 7}}}&{�� \ {|_{\ 5 \ 7}}}&{�� \ {|_{\ 6 \ 7}}}&{�� \ {|_{\ 7 \ 7}}}") | (25) |
fricas
mU:=transpose inverse map(retract,Um)
>> Error detected within library code: failed
Check "dimension" and the snake relations.
fricas
d:ℒ:=
Ω /
X /
Ų
2
0 +
- --
0 2
+
arity warning: ----
2
0 +
- --
0 2
+
2
+ 0
-- -
2 0
+
arity warning: ----
2
+ 0
-- -
2 0
+
![\label{eq26}\begin{array}{@{}l}
\displaystyle
{�� \ �� \ {|_{\ 0 \ 0}^{\ 0 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 0}^{\ 0 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 0}^{\ 0 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 0}^{\ 0 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 0}^{\ 0 \ 4}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 5 \ 0}^{\ 0 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 0}^{\ 0 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 0}^{\ 0 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 1}^{\ 1 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 1}^{\ 1 \ 1}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 2 \ 1}^{\ 1 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 1}^{\ 1 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 1}^{\ 1 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 1}^{\ 1 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 1}^{\ 1 \ 6}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 7 \ 1}^{\ 1 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 2}^{\ 2 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 2}^{\ 2 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 2}^{\ 2 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 2}^{\ 2 \ 3}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 4 \ 2}^{\ 2 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 2}^{\ 2 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 2}^{\ 2 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 2}^{\ 2 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 3}^{\ 3 \ 0}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 1 \ 3}^{\ 3 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 3}^{\ 3 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 3}^{\ 3 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 3}^{\ 3 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 3}^{\ 3 \ 5}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 6 \ 3}^{\ 3 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 3}^{\ 3 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 4}^{\ 4 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 4}^{\ 4 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 4}^{\ 4 \ 2}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 3 \ 4}^{\ 4 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 4}^{\ 4 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 4}^{\ 4 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 4}^{\ 4 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 4}^{\ 4 \ 7}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 0 \ 5}^{\ 5 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 5}^{\ 5 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 5}^{\ 5 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 5}^{\ 5 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 5}^{\ 5 \ 4}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 5 \ 5}^{\ 5 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 5}^{\ 5 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 5}^{\ 5 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 6}^{\ 6 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 6}^{\ 6 \ 1}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 2 \ 6}^{\ 6 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 6}^{\ 6 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 6}^{\ 6 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 6}^{\ 6 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 6}^{\ 6 \ 6}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 7 \ 6}^{\ 6 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 7}^{\ 7 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 7}^{\ 7 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 7}^{\ 7 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 7}^{\ 7 \ 3}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 4 \ 7}^{\ 7 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 7}^{\ 7 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 7}^{\ 7 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 7}^{\ 7 \ 7}}}](images/3040215903008030011-16.0px.png "\label{eq26}\begin{array}{@{}l}
\displaystyle
{�� \ �� \ {|_{\ 0 \ 0}^{\ 0 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 0}^{\ 0 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 0}^{\ 0 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 0}^{\ 0 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 0}^{\ 0 \ 4}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 5 \ 0}^{\ 0 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 0}^{\ 0 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 0}^{\ 0 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 1}^{\ 1 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 1}^{\ 1 \ 1}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 2 \ 1}^{\ 1 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 1}^{\ 1 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 1}^{\ 1 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 1}^{\ 1 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 1}^{\ 1 \ 6}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 7 \ 1}^{\ 1 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 2}^{\ 2 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 2}^{\ 2 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 2}^{\ 2 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 2}^{\ 2 \ 3}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 4 \ 2}^{\ 2 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 2}^{\ 2 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 2}^{\ 2 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 2}^{\ 2 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 3}^{\ 3 \ 0}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 1 \ 3}^{\ 3 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 3}^{\ 3 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 3}^{\ 3 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 3}^{\ 3 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 3}^{\ 3 \ 5}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 6 \ 3}^{\ 3 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 3}^{\ 3 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 4}^{\ 4 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 4}^{\ 4 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 4}^{\ 4 \ 2}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 3 \ 4}^{\ 4 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 4}^{\ 4 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 4}^{\ 4 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 4}^{\ 4 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 4}^{\ 4 \ 7}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 0 \ 5}^{\ 5 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 5}^{\ 5 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 5}^{\ 5 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 5}^{\ 5 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 5}^{\ 5 \ 4}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 5 \ 5}^{\ 5 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 5}^{\ 5 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 5}^{\ 5 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 6}^{\ 6 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 6}^{\ 6 \ 1}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 2 \ 6}^{\ 6 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 6}^{\ 6 \ 3}}}+{�� \ �� \ {|_{\ 4 \ 6}^{\ 6 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 6}^{\ 6 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 6}^{\ 6 \ 6}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 7 \ 6}^{\ 6 \ 7}}}+{�� \ �� \ {|_{\ 0 \ 7}^{\ 7 \ 0}}}+{�� \ �� \ {|_{\ 1 \ 7}^{\ 7 \ 1}}}+{�� \ �� \ {|_{\ 2 \ 7}^{\ 7 \ 2}}}+{�� \ �� \ {|_{\ 3 \ 7}^{\ 7 \ 3}}}+
\
\
\displaystyle
{�� \ �� \ {|_{\ 4 \ 7}^{\ 7 \ 4}}}+{�� \ �� \ {|_{\ 5 \ 7}^{\ 7 \ 5}}}+{�� \ �� \ {|_{\ 6 \ 7}^{\ 7 \ 6}}}+{�� \ �� \ {|_{\ 7 \ 7}^{\ 7 \ 7}}}") | (26) |
fricas
test
( I Ω ) /
( Ų I ) = I
 | (27) |
Type: Boolean
fricas
test
( Ω I ) /
( I Ų ) = I
There are no library operations named Ω
Use HyperDoc Browse or issue
)what op Ω
to learn if there is any operation containing " Ω " in its name.
Cannot find a definition or applicable library operation named Ω
with argument type(s)
ClosedLinearOperator(OrderedVariableList([0,
Perhaps you should use "@" to indicate the required return type,
Definition 4
Co-algebra
Compute the "three-point" function and use it to define co-multiplication.
fricas
W:=(Y,I)/Ų
3 + 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 +
![\label{eq28}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0 \ 0}^{\ 0 \ 0 \ 0}}}+{�� \ {|_{\ 0 \ 1}^{\ 0 \ 0 \ 1}}}+{�� \ {|_{\ 0 \ 2}^{\ 0 \ 0 \ 2}}}+{�� \ {|_{\ 0 \ 3}^{\ 0 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 4}^{\ 0 \ 0 \ 4}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 0 \ 5}}}+{�� \ {|_{\ 0 \ 6}^{\ 0 \ 0 \ 6}}}+{�� \ {|_{\ 0 \ 7}^{\ 0 \ 0 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 0}^{\ 0 \ 1 \ 0}}}+{�� \ {|_{\ 1 \ 1}^{\ 0 \ 1 \ 1}}}+{�� \ {|_{\ 1 \ 2}^{\ 0 \ 1 \ 2}}}+{�� \ {|_{\ 1 \ 3}^{\ 0 \ 1 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 4}^{\ 0 \ 1 \ 4}}}+{�� \ {|_{\ 1 \ 5}^{\ 0 \ 1 \ 5}}}+{�� \ {|_{\ 1 \ 6}^{\ 0 \ 1 \ 6}}}+{�� \ {|_{\ 1 \ 7}^{\ 0 \ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 0}^{\ 0 \ 2 \ 0}}}+{�� \ {|_{\ 2 \ 1}^{\ 0 \ 2 \ 1}}}+{�� \ {|_{\ 2 \ 2}^{\ 0 \ 2 \ 2}}}+{�� \ {|_{\ 2 \ 3}^{\ 0 \ 2 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 4}^{\ 0 \ 2 \ 4}}}+{�� \ {|_{\ 2 \ 5}^{\ 0 \ 2 \ 5}}}+{�� \ {|_{\ 2 \ 6}^{\ 0 \ 2 \ 6}}}+{�� \ {|_{\ 2 \ 7}^{\ 0 \ 2 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 0}^{\ 0 \ 3 \ 0}}}+{�� \ {|_{\ 3 \ 1}^{\ 0 \ 3 \ 1}}}+{�� \ {|_{\ 3 \ 2}^{\ 0 \ 3 \ 2}}}+{�� \ {|_{\ 3 \ 3}^{\ 0 \ 3 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 4}^{\ 0 \ 3 \ 4}}}+{�� \ {|_{\ 3 \ 5}^{\ 0 \ 3 \ 5}}}+{�� \ {|_{\ 3 \ 6}^{\ 0 \ 3 \ 6}}}+{�� \ {|_{\ 3 \ 7}^{\ 0 \ 3 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 0 \ 4 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 0 \ 4 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 0 \ 4 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 0 \ 4 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 0 \ 4 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 0 \ 4 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 0 \ 4 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 0 \ 4 \ 7}}}+{�� \ {|_{\ 5 \ 0}^{\ 0 \ 5 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 0 \ 5 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 0 \ 5 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 0 \ 5 \ 3}}}+{�� \ {|_{\ 5 \ 4}^{\ 0 \ 5 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 0 \ 5 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 0 \ 5 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 0 \ 5 \ 7}}}+{�� \ {|_{\ 6 \ 0}^{\ 0 \ 6 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 0 \ 6 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 0 \ 6 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 0 \ 6 \ 3}}}+{�� \ {|_{\ 6 \ 4}^{\ 0 \ 6 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 0 \ 6 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 0 \ 6 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 0 \ 6 \ 7}}}+{�� \ {|_{\ 7 \ 0}^{\ 0 \ 7 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 0 \ 7 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 0 \ 7 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 0 \ 7 \ 3}}}+{�� \ {|_{\ 7 \ 4}^{\ 0 \ 7 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 0 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 0 \ 7 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 0 \ 7 \ 7}}}+{�� \ {|_{\ 1 \ 0}^{\ 1 \ 0 \ 0}}}+{�� \ {|_{\ 1 \ 1}^{\ 1 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 2}^{\ 1 \ 0 \ 2}}}+{�� \ {|_{\ 1 \ 3}^{\ 1 \ 0 \ 3}}}+{�� \ {|_{\ 1 \ 4}^{\ 1 \ 0 \ 4}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 6}^{\ 1 \ 0 \ 6}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 0 \ 7}}}-{�� \ {|_{\ 0 \ 0}^{\ 1 \ 1 \ 0}}}-{�� \ {|_{\ 0 \ 1}^{\ 1 \ 1 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 2}^{\ 1 \ 1 \ 2}}}-{�� \ {|_{\ 0 \ 3}^{\ 1 \ 1 \ 3}}}-{�� \ {|_{\ 0 \ 4}^{\ 1 \ 1 \ 4}}}-{�� \ {|_{\ 0 \ 5}^{\ 1 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 1 \ 1 \ 6}}}-{�� \ {|_{\ 0 \ 7}^{\ 1 \ 1 \ 7}}}+{�� \ {|_{\ 3 \ 0}^{\ 1 \ 2 \ 0}}}+{�� \ {|_{\ 3 \ 1}^{\ 1 \ 2 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 2}^{\ 1 \ 2 \ 2}}}+{�� \ {|_{\ 3 \ 3}^{\ 1 \ 2 \ 3}}}+{�� \ {|_{\ 3 \ 4}^{\ 1 \ 2 \ 4}}}+{�� \ {|_{\ 3 \ 5}^{\ 1 \ 2 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 1 \ 2 \ 6}}}+{�� \ {|_{\ 3 \ 7}^{\ 1 \ 2 \ 7}}}-{�� \ {|_{\ 2 \ 0}^{\ 1 \ 3 \ 0}}}-{�� \ {|_{\ 2 \ 1}^{\ 1 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 2}^{\ 1 \ 3 \ 2}}}-{�� \ {|_{\ 2 \ 3}^{\ 1 \ 3 \ 3}}}-{�� \ {|_{\ 2 \ 4}^{\ 1 \ 3 \ 4}}}-{�� \ {|_{\ 2 \ 5}^{\ 1 \ 3 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 1 \ 3 \ 6}}}-{�� \ {|_{\ 2 \ 7}^{\ 1 \ 3 \ 7}}}+{�� \ {|_{\ 5 \ 0}^{\ 1 \ 4 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 1 \ 4 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 1 \ 4 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 1 \ 4 \ 3}}}+{�� \ {|_{\ 5 \ 4}^{\ 1 \ 4 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 1 \ 4 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 1 \ 4 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 1 \ 4 \ 7}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 1 \ 5 \ 0}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 1 \ 5 \ 1}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 1 \ 5 \ 2}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 1 \ 5 \ 3}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 1 \ 5 \ 4}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 1 \ 5 \ 5}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 1 \ 5 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 1 \ 5 \ 7}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 1 \ 6 \ 0}}}-{�� \ {|_{\ 7 \ 1}^{\ 1 \ 6 \ 1}}}-{�� \ {|_{\ 7 \ 2}^{\ 1 \ 6 \ 2}}}-{�� \ {|_{\ 7 \ 3}^{\ 1 \ 6 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 4}^{\ 1 \ 6 \ 4}}}-{�� \ {|_{\ 7 \ 5}^{\ 1 \ 6 \ 5}}}-{�� \ {|_{\ 7 \ 6}^{\ 1 \ 6 \ 6}}}-{�� \ {|_{\ 7 \ 7}^{\ 1 \ 6 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 1 \ 7 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 1 \ 7 \ 1}}}+{�� \ {|_{\ 6 \ 2}^{\ 1 \ 7 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 1 \ 7 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 4}^{\ 1 \ 7 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 1 \ 7 \ 5}}}+{�� \ {|_{\ 6 \ 6}^{\ 1 \ 7 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 1 \ 7 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 0}^{\ 2 \ 0 \ 0}}}+{�� \ {|_{\ 2 \ 1}^{\ 2 \ 0 \ 1}}}+{�� \ {|_{\ 2 \ 2}^{\ 2 \ 0 \ 2}}}+{�� \ {|_{\ 2 \ 3}^{\ 2 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 4}^{\ 2 \ 0 \ 4}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 0 \ 5}}}+{�� \ {|_{\ 2 \ 6}^{\ 2 \ 0 \ 6}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 0 \ 7}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 0}^{\ 2 \ 1 \ 0}}}-{�� \ {|_{\ 3 \ 1}^{\ 2 \ 1 \ 1}}}-{�� \ {|_{\ 3 \ 2}^{\ 2 \ 1 \ 2}}}-{�� \ {|_{\ 3 \ 3}^{\ 2 \ 1 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 4}^{\ 2 \ 1 \ 4}}}-{�� \ {|_{\ 3 \ 5}^{\ 2 \ 1 \ 5}}}-{�� \ {|_{\ 3 \ 6}^{\ 2 \ 1 \ 6}}}-{�� \ {|_{\ 3 \ 7}^{\ 2 \ 1 \ 7}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 0}^{\ 2 \ 2 \ 0}}}-{�� \ {|_{\ 0 \ 1}^{\ 2 \ 2 \ 1}}}-{�� \ {|_{\ 0 \ 2}^{\ 2 \ 2 \ 2}}}-{�� \ {|_{\ 0 \ 3}^{\ 2 \ 2 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 4}^{\ 2 \ 2 \ 4}}}-{�� \ {|_{\ 0 \ 5}^{\ 2 \ 2 \ 5}}}-{�� \ {|_{\ 0 \ 6}^{\ 2 \ 2 \ 6}}}-{�� \ {|_{\ 0 \ 7}^{\ 2 \ 2 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 0}^{\ 2 \ 3 \ 0}}}+{�� \ {|_{\ 1 \ 1}^{\ 2 \ 3 \ 1}}}+{�� \ {|_{\ 1 \ 2}^{\ 2 \ 3 \ 2}}}+{�� \ {|_{\ 1 \ 3}^{\ 2 \ 3 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 4}^{\ 2 \ 3 \ 4}}}+{�� \ {|_{\ 1 \ 5}^{\ 2 \ 3 \ 5}}}+{�� \ {|_{\ 1 \ 6}^{\ 2 \ 3 \ 6}}}+{�� \ {|_{\ 1 \ 7}^{\ 2 \ 3 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 2 \ 4 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 2 \ 4 \ 1}}}+{�� \ {|_{\ 6 \ 2}^{\ 2 \ 4 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 2 \ 4 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 4}^{\ 2 \ 4 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 2 \ 4 \ 5}}}+{�� \ {|_{\ 6 \ 6}^{\ 2 \ 4 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 2 \ 4 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 2 \ 5 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 2 \ 5 \ 1}}}+{�� \ {|_{\ 7 \ 2}^{\ 2 \ 5 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 2 \ 5 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 4}^{\ 2 \ 5 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 2 \ 5 \ 5}}}+{�� \ {|_{\ 7 \ 6}^{\ 2 \ 5 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 2 \ 5 \ 7}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 2 \ 6 \ 0}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 2 \ 6 \ 1}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 2 \ 6 \ 2}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 2 \ 6 \ 3}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 2 \ 6 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 2 \ 6 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 2 \ 6 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 2 \ 6 \ 7}}}-{�� \ {|_{\ 5 \ 0}^{\ 2 \ 7 \ 0}}}-{�� \ {|_{\ 5 \ 1}^{\ 2 \ 7 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 2 \ 7 \ 2}}}-{�� \ {|_{\ 5 \ 3}^{\ 2 \ 7 \ 3}}}-{�� \ {|_{\ 5 \ 4}^{\ 2 \ 7 \ 4}}}-{�� \ {|_{\ 5 \ 5}^{\ 2 \ 7 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 2 \ 7 \ 6}}}-{�� \ {|_{\ 5 \ 7}^{\ 2 \ 7 \ 7}}}+{�� \ {|_{\ 3 \ 0}^{\ 3 \ 0 \ 0}}}+{�� \ {|_{\ 3 \ 1}^{\ 3 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 2}^{\ 3 \ 0 \ 2}}}+{�� \ {|_{\ 3 \ 3}^{\ 3 \ 0 \ 3}}}+{�� \ {|_{\ 3 \ 4}^{\ 3 \ 0 \ 4}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 3 \ 0 \ 6}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 0 \ 7}}}+{�� \ {|_{\ 2 \ 0}^{\ 3 \ 1 \ 0}}}+{�� \ {|_{\ 2 \ 1}^{\ 3 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 2}^{\ 3 \ 1 \ 2}}}+{�� \ {|_{\ 2 \ 3}^{\ 3 \ 1 \ 3}}}+{�� \ {|_{\ 2 \ 4}^{\ 3 \ 1 \ 4}}}+{�� \ {|_{\ 2 \ 5}^{\ 3 \ 1 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 3 \ 1 \ 6}}}+{�� \ {|_{\ 2 \ 7}^{\ 3 \ 1 \ 7}}}-{�� \ {|_{\ 1 \ 0}^{\ 3 \ 2 \ 0}}}-{�� \ {|_{\ 1 \ 1}^{\ 3 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 2}^{\ 3 \ 2 \ 2}}}-{�� \ {|_{\ 1 \ 3}^{\ 3 \ 2 \ 3}}}-{�� \ {|_{\ 1 \ 4}^{\ 3 \ 2 \ 4}}}-{�� \ {|_{\ 1 \ 5}^{\ 3 \ 2 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 6}^{\ 3 \ 2 \ 6}}}-{�� \ {|_{\ 1 \ 7}^{\ 3 \ 2 \ 7}}}-{�� \ {|_{\ 0 \ 0}^{\ 3 \ 3 \ 0}}}-{�� \ {|_{\ 0 \ 1}^{\ 3 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 2}^{\ 3 \ 3 \ 2}}}-{�� \ {|_{\ 0 \ 3}^{\ 3 \ 3 \ 3}}}-{�� \ {|_{\ 0 \ 4}^{\ 3 \ 3 \ 4}}}-{�� \ {|_{\ 0 \ 5}^{\ 3 \ 3 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 3 \ 3 \ 6}}}-{�� \ {|_{\ 0 \ 7}^{\ 3 \ 3 \ 7}}}+{�� \ {|_{\ 7 \ 0}^{\ 3 \ 4 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 3 \ 4 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 3 \ 4 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 3 \ 4 \ 3}}}+{�� \ {|_{\ 7 \ 4}^{\ 3 \ 4 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 3 \ 4 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 3 \ 4 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 3 \ 4 \ 7}}}-{�� \ {|_{\ 6 \ 0}^{\ 3 \ 5 \ 0}}}-{�� \ {|_{\ 6 \ 1}^{\ 3 \ 5 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 3 \ 5 \ 2}}}-{�� \ {|_{\ 6 \ 3}^{\ 3 \ 5 \ 3}}}-{�� \ {|_{\ 6 \ 4}^{\ 3 \ 5 \ 4}}}-{�� \ {|_{\ 6 \ 5}^{\ 3 \ 5 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 3 \ 5 \ 6}}}-{�� \ {|_{\ 6 \ 7}^{\ 3 \ 5 \ 7}}}+{�� \ {|_{\ 5 \ 0}^{\ 3 \ 6 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 3 \ 6 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 3 \ 6 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 3 \ 6 \ 3}}}+{�� \ {|_{\ 5 \ 4}^{\ 3 \ 6 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 3 \ 6 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 3 \ 6 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 3 \ 6 \ 7}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 3 \ 7 \ 0}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 3 \ 7 \ 1}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 3 \ 7 \ 2}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 3 \ 7 \ 3}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 3 \ 7 \ 4}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 3 \ 7 \ 5}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 3 \ 7 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 3 \ 7 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 4 \ 0 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 4 \ 0 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 4 \ 0 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 4 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 0 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 4 \ 0 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 4 \ 0 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 4 \ 0 \ 7}}}-{�� \ {|_{\ 5 \ 0}^{\ 4 \ 1 \ 0}}}-{�� \ {|_{\ 5 \ 1}^{\ 4 \ 1 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 4 \ 1 \ 2}}}-{�� \ {|_{\ 5 \ 3}^{\ 4 \ 1 \ 3}}}-{�� \ {|_{\ 5 \ 4}^{\ 4 \ 1 \ 4}}}-{�� \ {|_{\ 5 \ 5}^{\ 4 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 4 \ 1 \ 6}}}-{�� \ {|_{\ 5 \ 7}^{\ 4 \ 1 \ 7}}}-{�� \ {|_{\ 6 \ 0}^{\ 4 \ 2 \ 0}}}-{�� \ {|_{\ 6 \ 1}^{\ 4 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 4 \ 2 \ 2}}}-{�� \ {|_{\ 6 \ 3}^{\ 4 \ 2 \ 3}}}-{�� \ {|_{\ 6 \ 4}^{\ 4 \ 2 \ 4}}}-{�� \ {|_{\ 6 \ 5}^{\ 4 \ 2 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 4 \ 2 \ 6}}}-{�� \ {|_{\ 6 \ 7}^{\ 4 \ 2 \ 7}}}-{�� \ {|_{\ 7 \ 0}^{\ 4 \ 3 \ 0}}}-{�� \ {|_{\ 7 \ 1}^{\ 4 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 4 \ 3 \ 2}}}-{�� \ {|_{\ 7 \ 3}^{\ 4 \ 3 \ 3}}}-{�� \ {|_{\ 7 \ 4}^{\ 4 \ 3 \ 4}}}-{�� \ {|_{\ 7 \ 5}^{\ 4 \ 3 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 4 \ 3 \ 6}}}-{�� \ {|_{\ 7 \ 7}^{\ 4 \ 3 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 4 \ 4 \ 0}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 4 \ 4 \ 1}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 4 \ 4 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 4 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 4}^{\ 4 \ 4 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 5}^{\ 4 \ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 6}^{\ 4 \ 4 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 7}^{\ 4 \ 4 \ 7}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 4 \ 5 \ 0}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 1}^{\ 4 \ 5 \ 1}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 2}^{\ 4 \ 5 \ 2}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 3}^{\ 4 \ 5 \ 3}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 4}^{\ 4 \ 5 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 5}^{\ 4 \ 5 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 6}^{\ 4 \ 5 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 7}^{\ 4 \ 5 \ 7}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 4 \ 6 \ 0}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 1}^{\ 4 \ 6 \ 1}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 2}^{\ 4 \ 6 \ 2}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 3}^{\ 4 \ 6 \ 3}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 4}^{\ 4 \ 6 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 5}^{\ 4 \ 6 \ 5}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 6}^{\ 4 \ 6 \ 6}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 7}^{\ 4 \ 6 \ 7}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 4 \ 7 \ 0}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 1}^{\ 4 \ 7 \ 1}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 2}^{\ 4 \ 7 \ 2}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 3}^{\ 4 \ 7 \ 3}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 4}^{\ 4 \ 7 \ 4}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 5}^{\ 4 \ 7 \ 5}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 6}^{\ 4 \ 7 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 7}^{\ 4 \ 7 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 0}^{\ 5 \ 0 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 5 \ 0 \ 1}}}+{�� \ {|_{\ 5 \ 2}^{\ 5 \ 0 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 5 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 4}^{\ 5 \ 0 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 0 \ 5}}}+{�� \ {|_{\ 5 \ 6}^{\ 5 \ 0 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 0 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 5 \ 1 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 5 \ 1 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 5 \ 1 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 5 \ 1 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 5 \ 1 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 5 \ 1 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 5 \ 1 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 5 \ 1 \ 7}}}-{�� \ {|_{\ 7 \ 0}^{\ 5 \ 2 \ 0}}}-{�� \ {|_{\ 7 \ 1}^{\ 5 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 5 \ 2 \ 2}}}-{�� \ {|_{\ 7 \ 3}^{\ 5 \ 2 \ 3}}}-{�� \ {|_{\ 7 \ 4}^{\ 5 \ 2 \ 4}}}-{�� \ {|_{\ 7 \ 5}^{\ 5 \ 2 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 5 \ 2 \ 6}}}-{�� \ {|_{\ 7 \ 7}^{\ 5 \ 2 \ 7}}}+{�� \ {|_{\ 6 \ 0}^{\ 5 \ 3 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 5 \ 3 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 5 \ 3 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 5 \ 3 \ 3}}}+{�� \ {|_{\ 6 \ 4}^{\ 5 \ 3 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 5 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 5 \ 3 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 5 \ 3 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 5 \ 4 \ 0}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 1}^{\ 5 \ 4 \ 1}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 2}^{\ 5 \ 4 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 3}^{\ 5 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 4}^{\ 5 \ 4 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 5}^{\ 5 \ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 6}^{\ 5 \ 4 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 7}^{\ 5 \ 4 \ 7}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 5 \ 5 \ 0}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 5 \ 5 \ 1}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 5 \ 5 \ 2}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 5 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 4}^{\ 5 \ 5 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 5}^{\ 5 \ 5 \ 5}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 6}^{\ 5 \ 5 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 7}^{\ 5 \ 5 \ 7}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 0}^{\ 5 \ 6 \ 0}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 1}^{\ 5 \ 6 \ 1}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 2}^{\ 5 \ 6 \ 2}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 3}^{\ 5 \ 6 \ 3}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 4}^{\ 5 \ 6 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 5}^{\ 5 \ 6 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 6}^{\ 5 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 7}^{\ 5 \ 6 \ 7}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 0}^{\ 5 \ 7 \ 0}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 1}^{\ 5 \ 7 \ 1}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 2}^{\ 5 \ 7 \ 2}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 3}^{\ 5 \ 7 \ 3}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 4}^{\ 5 \ 7 \ 4}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 5}^{\ 5 \ 7 \ 5}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 6}^{\ 5 \ 7 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 7}^{\ 5 \ 7 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 6 \ 0 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 6 \ 0 \ 1}}}+{�� \ {|_{\ 6 \ 2}^{\ 6 \ 0 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 6 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 4}^{\ 6 \ 0 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 0 \ 5}}}+{�� \ {|_{\ 6 \ 6}^{\ 6 \ 0 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 0 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 6 \ 1 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 6 \ 1 \ 1}}}+{�� \ {|_{\ 7 \ 2}^{\ 6 \ 1 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 6 \ 1 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 4}^{\ 6 \ 1 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 6 \ 1 \ 5}}}+{�� \ {|_{\ 7 \ 6}^{\ 6 \ 1 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 6 \ 1 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 6 \ 2 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 6 \ 2 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 6 \ 2 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 6 \ 2 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 6 \ 2 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 6 \ 2 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 6 \ 2 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 6 \ 2 \ 7}}}-{�� \ {|_{\ 5 \ 0}^{\ 6 \ 3 \ 0}}}-{�� \ {|_{\ 5 \ 1}^{\ 6 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 6 \ 3 \ 2}}}-{�� \ {|_{\ 5 \ 3}^{\ 6 \ 3 \ 3}}}-{�� \ {|_{\ 5 \ 4}^{\ 6 \ 3 \ 4}}}-{�� \ {|_{\ 5 \ 5}^{\ 6 \ 3 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 6 \ 3 \ 6}}}-{�� \ {|_{\ 5 \ 7}^{\ 6 \ 3 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 6 \ 4 \ 0}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 1}^{\ 6 \ 4 \ 1}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 2}^{\ 6 \ 4 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 3}^{\ 6 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 4}^{\ 6 \ 4 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 5}^{\ 6 \ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 6}^{\ 6 \ 4 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 7}^{\ 6 \ 4 \ 7}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 0}^{\ 6 \ 5 \ 0}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 1}^{\ 6 \ 5 \ 1}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 2}^{\ 6 \ 5 \ 2}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 3}^{\ 6 \ 5 \ 3}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 4}^{\ 6 \ 5 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 5}^{\ 6 \ 5 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 6}^{\ 6 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 7}^{\ 6 \ 5 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 6 \ 6 \ 0}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 6 \ 6 \ 1}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 6 \ 6 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 6 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 4}^{\ 6 \ 6 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 5}^{\ 6 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 6}^{\ 6 \ 6 \ 6}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 7}^{\ 6 \ 6 \ 7}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 0}^{\ 6 \ 7 \ 0}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 1}^{\ 6 \ 7 \ 1}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 2}^{\ 6 \ 7 \ 2}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 3}^{\ 6 \ 7 \ 3}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 4}^{\ 6 \ 7 \ 4}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 5}^{\ 6 \ 7 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 6}^{\ 6 \ 7 \ 6}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 7}^{\ 6 \ 7 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 7 \ 0 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 7 \ 0 \ 1}}}+{�� \ {|_{\ 7 \ 2}^{\ 7 \ 0 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 7 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 4}^{\ 7 \ 0 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 0 \ 5}}}+{�� \ {|_{\ 7 \ 6}^{\ 7 \ 0 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 0 \ 7}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 7 \ 1 \ 0}}}-{�� \ {|_{\ 6 \ 1}^{\ 7 \ 1 \ 1}}}-{�� \ {|_{\ 6 \ 2}^{\ 7 \ 1 \ 2}}}-{�� \ {|_{\ 6 \ 3}^{\ 7 \ 1 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 4}^{\ 7 \ 1 \ 4}}}-{�� \ {|_{\ 6 \ 5}^{\ 7 \ 1 \ 5}}}-{�� \ {|_{\ 6 \ 6}^{\ 7 \ 1 \ 6}}}-{�� \ {|_{\ 6 \ 7}^{\ 7 \ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 0}^{\ 7 \ 2 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 7 \ 2 \ 1}}}+{�� \ {|_{\ 5 \ 2}^{\ 7 \ 2 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 7 \ 2 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 4}^{\ 7 \ 2 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 7 \ 2 \ 5}}}+{�� \ {|_{\ 5 \ 6}^{\ 7 \ 2 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 7 \ 2 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 7 \ 3 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 7 \ 3 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 7 \ 3 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 7 \ 3 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 7 \ 3 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 7 \ 3 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 7 \ 3 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 7 \ 3 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 7 \ 4 \ 0}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 1}^{\ 7 \ 4 \ 1}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 2}^{\ 7 \ 4 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 3}^{\ 7 \ 4 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 4}^{\ 7 \ 4 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 5}^{\ 7 \ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 6}^{\ 7 \ 4 \ 6}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 7}^{\ 7 \ 4 \ 7}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 0}^{\ 7 \ 5 \ 0}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 1}^{\ 7 \ 5 \ 1}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 2}^{\ 7 \ 5 \ 2}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 3}^{\ 7 \ 5 \ 3}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 4}^{\ 7 \ 5 \ 4}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 5}^{\ 7 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 6}^{\ 7 \ 5 \ 6}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 7}^{\ 7 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 0}^{\ 7 \ 6 \ 0}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 1}^{\ 7 \ 6 \ 1}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 2}^{\ 7 \ 6 \ 2}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 3}^{\ 7 \ 6 \ 3}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 4}^{\ 7 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 5}^{\ 7 \ 6 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 6}^{\ 7 \ 6 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 7}^{\ 7 \ 6 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 7 \ 7 \ 0}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 7 \ 7 \ 1}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 7 \ 7 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 7 \ 7 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 4}^{\ 7 \ 7 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 5}^{\ 7 \ 7 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 6}^{\ 7 \ 7 \ 6}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 7}^{\ 7 \ 7 \ 7}}}](images/3499511309134498993-16.0px.png "\label{eq28}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0 \ 0}^{\ 0 \ 0 \ 0}}}+{�� \ {|_{\ 0 \ 1}^{\ 0 \ 0 \ 1}}}+{�� \ {|_{\ 0 \ 2}^{\ 0 \ 0 \ 2}}}+{�� \ {|_{\ 0 \ 3}^{\ 0 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 4}^{\ 0 \ 0 \ 4}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 0 \ 5}}}+{�� \ {|_{\ 0 \ 6}^{\ 0 \ 0 \ 6}}}+{�� \ {|_{\ 0 \ 7}^{\ 0 \ 0 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 0}^{\ 0 \ 1 \ 0}}}+{�� \ {|_{\ 1 \ 1}^{\ 0 \ 1 \ 1}}}+{�� \ {|_{\ 1 \ 2}^{\ 0 \ 1 \ 2}}}+{�� \ {|_{\ 1 \ 3}^{\ 0 \ 1 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 4}^{\ 0 \ 1 \ 4}}}+{�� \ {|_{\ 1 \ 5}^{\ 0 \ 1 \ 5}}}+{�� \ {|_{\ 1 \ 6}^{\ 0 \ 1 \ 6}}}+{�� \ {|_{\ 1 \ 7}^{\ 0 \ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 0}^{\ 0 \ 2 \ 0}}}+{�� \ {|_{\ 2 \ 1}^{\ 0 \ 2 \ 1}}}+{�� \ {|_{\ 2 \ 2}^{\ 0 \ 2 \ 2}}}+{�� \ {|_{\ 2 \ 3}^{\ 0 \ 2 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 4}^{\ 0 \ 2 \ 4}}}+{�� \ {|_{\ 2 \ 5}^{\ 0 \ 2 \ 5}}}+{�� \ {|_{\ 2 \ 6}^{\ 0 \ 2 \ 6}}}+{�� \ {|_{\ 2 \ 7}^{\ 0 \ 2 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 0}^{\ 0 \ 3 \ 0}}}+{�� \ {|_{\ 3 \ 1}^{\ 0 \ 3 \ 1}}}+{�� \ {|_{\ 3 \ 2}^{\ 0 \ 3 \ 2}}}+{�� \ {|_{\ 3 \ 3}^{\ 0 \ 3 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 4}^{\ 0 \ 3 \ 4}}}+{�� \ {|_{\ 3 \ 5}^{\ 0 \ 3 \ 5}}}+{�� \ {|_{\ 3 \ 6}^{\ 0 \ 3 \ 6}}}+{�� \ {|_{\ 3 \ 7}^{\ 0 \ 3 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 0 \ 4 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 0 \ 4 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 0 \ 4 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 0 \ 4 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 0 \ 4 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 0 \ 4 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 0 \ 4 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 0 \ 4 \ 7}}}+{�� \ {|_{\ 5 \ 0}^{\ 0 \ 5 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 0 \ 5 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 0 \ 5 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 0 \ 5 \ 3}}}+{�� \ {|_{\ 5 \ 4}^{\ 0 \ 5 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 0 \ 5 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 0 \ 5 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 0 \ 5 \ 7}}}+{�� \ {|_{\ 6 \ 0}^{\ 0 \ 6 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 0 \ 6 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 0 \ 6 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 0 \ 6 \ 3}}}+{�� \ {|_{\ 6 \ 4}^{\ 0 \ 6 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 0 \ 6 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 0 \ 6 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 0 \ 6 \ 7}}}+{�� \ {|_{\ 7 \ 0}^{\ 0 \ 7 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 0 \ 7 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 0 \ 7 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 0 \ 7 \ 3}}}+{�� \ {|_{\ 7 \ 4}^{\ 0 \ 7 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 0 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 0 \ 7 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 0 \ 7 \ 7}}}+{�� \ {|_{\ 1 \ 0}^{\ 1 \ 0 \ 0}}}+{�� \ {|_{\ 1 \ 1}^{\ 1 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 2}^{\ 1 \ 0 \ 2}}}+{�� \ {|_{\ 1 \ 3}^{\ 1 \ 0 \ 3}}}+{�� \ {|_{\ 1 \ 4}^{\ 1 \ 0 \ 4}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 6}^{\ 1 \ 0 \ 6}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 0 \ 7}}}-{�� \ {|_{\ 0 \ 0}^{\ 1 \ 1 \ 0}}}-{�� \ {|_{\ 0 \ 1}^{\ 1 \ 1 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 2}^{\ 1 \ 1 \ 2}}}-{�� \ {|_{\ 0 \ 3}^{\ 1 \ 1 \ 3}}}-{�� \ {|_{\ 0 \ 4}^{\ 1 \ 1 \ 4}}}-{�� \ {|_{\ 0 \ 5}^{\ 1 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 1 \ 1 \ 6}}}-{�� \ {|_{\ 0 \ 7}^{\ 1 \ 1 \ 7}}}+{�� \ {|_{\ 3 \ 0}^{\ 1 \ 2 \ 0}}}+{�� \ {|_{\ 3 \ 1}^{\ 1 \ 2 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 2}^{\ 1 \ 2 \ 2}}}+{�� \ {|_{\ 3 \ 3}^{\ 1 \ 2 \ 3}}}+{�� \ {|_{\ 3 \ 4}^{\ 1 \ 2 \ 4}}}+{�� \ {|_{\ 3 \ 5}^{\ 1 \ 2 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 1 \ 2 \ 6}}}+{�� \ {|_{\ 3 \ 7}^{\ 1 \ 2 \ 7}}}-{�� \ {|_{\ 2 \ 0}^{\ 1 \ 3 \ 0}}}-{�� \ {|_{\ 2 \ 1}^{\ 1 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 2}^{\ 1 \ 3 \ 2}}}-{�� \ {|_{\ 2 \ 3}^{\ 1 \ 3 \ 3}}}-{�� \ {|_{\ 2 \ 4}^{\ 1 \ 3 \ 4}}}-{�� \ {|_{\ 2 \ 5}^{\ 1 \ 3 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 1 \ 3 \ 6}}}-{�� \ {|_{\ 2 \ 7}^{\ 1 \ 3 \ 7}}}+{�� \ {|_{\ 5 \ 0}^{\ 1 \ 4 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 1 \ 4 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 1 \ 4 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 1 \ 4 \ 3}}}+{�� \ {|_{\ 5 \ 4}^{\ 1 \ 4 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 1 \ 4 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 1 \ 4 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 1 \ 4 \ 7}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 1 \ 5 \ 0}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 1 \ 5 \ 1}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 1 \ 5 \ 2}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 1 \ 5 \ 3}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 1 \ 5 \ 4}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 1 \ 5 \ 5}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 1 \ 5 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 1 \ 5 \ 7}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 1 \ 6 \ 0}}}-{�� \ {|_{\ 7 \ 1}^{\ 1 \ 6 \ 1}}}-{�� \ {|_{\ 7 \ 2}^{\ 1 \ 6 \ 2}}}-{�� \ {|_{\ 7 \ 3}^{\ 1 \ 6 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 4}^{\ 1 \ 6 \ 4}}}-{�� \ {|_{\ 7 \ 5}^{\ 1 \ 6 \ 5}}}-{�� \ {|_{\ 7 \ 6}^{\ 1 \ 6 \ 6}}}-{�� \ {|_{\ 7 \ 7}^{\ 1 \ 6 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 1 \ 7 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 1 \ 7 \ 1}}}+{�� \ {|_{\ 6 \ 2}^{\ 1 \ 7 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 1 \ 7 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 4}^{\ 1 \ 7 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 1 \ 7 \ 5}}}+{�� \ {|_{\ 6 \ 6}^{\ 1 \ 7 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 1 \ 7 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 0}^{\ 2 \ 0 \ 0}}}+{�� \ {|_{\ 2 \ 1}^{\ 2 \ 0 \ 1}}}+{�� \ {|_{\ 2 \ 2}^{\ 2 \ 0 \ 2}}}+{�� \ {|_{\ 2 \ 3}^{\ 2 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 4}^{\ 2 \ 0 \ 4}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 0 \ 5}}}+{�� \ {|_{\ 2 \ 6}^{\ 2 \ 0 \ 6}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 0 \ 7}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 0}^{\ 2 \ 1 \ 0}}}-{�� \ {|_{\ 3 \ 1}^{\ 2 \ 1 \ 1}}}-{�� \ {|_{\ 3 \ 2}^{\ 2 \ 1 \ 2}}}-{�� \ {|_{\ 3 \ 3}^{\ 2 \ 1 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 4}^{\ 2 \ 1 \ 4}}}-{�� \ {|_{\ 3 \ 5}^{\ 2 \ 1 \ 5}}}-{�� \ {|_{\ 3 \ 6}^{\ 2 \ 1 \ 6}}}-{�� \ {|_{\ 3 \ 7}^{\ 2 \ 1 \ 7}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 0}^{\ 2 \ 2 \ 0}}}-{�� \ {|_{\ 0 \ 1}^{\ 2 \ 2 \ 1}}}-{�� \ {|_{\ 0 \ 2}^{\ 2 \ 2 \ 2}}}-{�� \ {|_{\ 0 \ 3}^{\ 2 \ 2 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 4}^{\ 2 \ 2 \ 4}}}-{�� \ {|_{\ 0 \ 5}^{\ 2 \ 2 \ 5}}}-{�� \ {|_{\ 0 \ 6}^{\ 2 \ 2 \ 6}}}-{�� \ {|_{\ 0 \ 7}^{\ 2 \ 2 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 0}^{\ 2 \ 3 \ 0}}}+{�� \ {|_{\ 1 \ 1}^{\ 2 \ 3 \ 1}}}+{�� \ {|_{\ 1 \ 2}^{\ 2 \ 3 \ 2}}}+{�� \ {|_{\ 1 \ 3}^{\ 2 \ 3 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 4}^{\ 2 \ 3 \ 4}}}+{�� \ {|_{\ 1 \ 5}^{\ 2 \ 3 \ 5}}}+{�� \ {|_{\ 1 \ 6}^{\ 2 \ 3 \ 6}}}+{�� \ {|_{\ 1 \ 7}^{\ 2 \ 3 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 2 \ 4 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 2 \ 4 \ 1}}}+{�� \ {|_{\ 6 \ 2}^{\ 2 \ 4 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 2 \ 4 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 4}^{\ 2 \ 4 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 2 \ 4 \ 5}}}+{�� \ {|_{\ 6 \ 6}^{\ 2 \ 4 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 2 \ 4 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 2 \ 5 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 2 \ 5 \ 1}}}+{�� \ {|_{\ 7 \ 2}^{\ 2 \ 5 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 2 \ 5 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 4}^{\ 2 \ 5 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 2 \ 5 \ 5}}}+{�� \ {|_{\ 7 \ 6}^{\ 2 \ 5 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 2 \ 5 \ 7}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 2 \ 6 \ 0}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 2 \ 6 \ 1}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 2 \ 6 \ 2}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 2 \ 6 \ 3}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 2 \ 6 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 2 \ 6 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 2 \ 6 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 2 \ 6 \ 7}}}-{�� \ {|_{\ 5 \ 0}^{\ 2 \ 7 \ 0}}}-{�� \ {|_{\ 5 \ 1}^{\ 2 \ 7 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 2 \ 7 \ 2}}}-{�� \ {|_{\ 5 \ 3}^{\ 2 \ 7 \ 3}}}-{�� \ {|_{\ 5 \ 4}^{\ 2 \ 7 \ 4}}}-{�� \ {|_{\ 5 \ 5}^{\ 2 \ 7 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 2 \ 7 \ 6}}}-{�� \ {|_{\ 5 \ 7}^{\ 2 \ 7 \ 7}}}+{�� \ {|_{\ 3 \ 0}^{\ 3 \ 0 \ 0}}}+{�� \ {|_{\ 3 \ 1}^{\ 3 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 2}^{\ 3 \ 0 \ 2}}}+{�� \ {|_{\ 3 \ 3}^{\ 3 \ 0 \ 3}}}+{�� \ {|_{\ 3 \ 4}^{\ 3 \ 0 \ 4}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 3 \ 0 \ 6}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 0 \ 7}}}+{�� \ {|_{\ 2 \ 0}^{\ 3 \ 1 \ 0}}}+{�� \ {|_{\ 2 \ 1}^{\ 3 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 2}^{\ 3 \ 1 \ 2}}}+{�� \ {|_{\ 2 \ 3}^{\ 3 \ 1 \ 3}}}+{�� \ {|_{\ 2 \ 4}^{\ 3 \ 1 \ 4}}}+{�� \ {|_{\ 2 \ 5}^{\ 3 \ 1 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 3 \ 1 \ 6}}}+{�� \ {|_{\ 2 \ 7}^{\ 3 \ 1 \ 7}}}-{�� \ {|_{\ 1 \ 0}^{\ 3 \ 2 \ 0}}}-{�� \ {|_{\ 1 \ 1}^{\ 3 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 2}^{\ 3 \ 2 \ 2}}}-{�� \ {|_{\ 1 \ 3}^{\ 3 \ 2 \ 3}}}-{�� \ {|_{\ 1 \ 4}^{\ 3 \ 2 \ 4}}}-{�� \ {|_{\ 1 \ 5}^{\ 3 \ 2 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 6}^{\ 3 \ 2 \ 6}}}-{�� \ {|_{\ 1 \ 7}^{\ 3 \ 2 \ 7}}}-{�� \ {|_{\ 0 \ 0}^{\ 3 \ 3 \ 0}}}-{�� \ {|_{\ 0 \ 1}^{\ 3 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 2}^{\ 3 \ 3 \ 2}}}-{�� \ {|_{\ 0 \ 3}^{\ 3 \ 3 \ 3}}}-{�� \ {|_{\ 0 \ 4}^{\ 3 \ 3 \ 4}}}-{�� \ {|_{\ 0 \ 5}^{\ 3 \ 3 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 3 \ 3 \ 6}}}-{�� \ {|_{\ 0 \ 7}^{\ 3 \ 3 \ 7}}}+{�� \ {|_{\ 7 \ 0}^{\ 3 \ 4 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 3 \ 4 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 3 \ 4 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 3 \ 4 \ 3}}}+{�� \ {|_{\ 7 \ 4}^{\ 3 \ 4 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 3 \ 4 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 3 \ 4 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 3 \ 4 \ 7}}}-{�� \ {|_{\ 6 \ 0}^{\ 3 \ 5 \ 0}}}-{�� \ {|_{\ 6 \ 1}^{\ 3 \ 5 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 3 \ 5 \ 2}}}-{�� \ {|_{\ 6 \ 3}^{\ 3 \ 5 \ 3}}}-{�� \ {|_{\ 6 \ 4}^{\ 3 \ 5 \ 4}}}-{�� \ {|_{\ 6 \ 5}^{\ 3 \ 5 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 3 \ 5 \ 6}}}-{�� \ {|_{\ 6 \ 7}^{\ 3 \ 5 \ 7}}}+{�� \ {|_{\ 5 \ 0}^{\ 3 \ 6 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 3 \ 6 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 3 \ 6 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 3 \ 6 \ 3}}}+{�� \ {|_{\ 5 \ 4}^{\ 3 \ 6 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 3 \ 6 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 3 \ 6 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 3 \ 6 \ 7}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 3 \ 7 \ 0}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 3 \ 7 \ 1}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 3 \ 7 \ 2}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 3 \ 7 \ 3}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 3 \ 7 \ 4}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 3 \ 7 \ 5}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 3 \ 7 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 3 \ 7 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 4 \ 0 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 4 \ 0 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 4 \ 0 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 4 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 0 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 4 \ 0 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 4 \ 0 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 4 \ 0 \ 7}}}-{�� \ {|_{\ 5 \ 0}^{\ 4 \ 1 \ 0}}}-{�� \ {|_{\ 5 \ 1}^{\ 4 \ 1 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 4 \ 1 \ 2}}}-{�� \ {|_{\ 5 \ 3}^{\ 4 \ 1 \ 3}}}-{�� \ {|_{\ 5 \ 4}^{\ 4 \ 1 \ 4}}}-{�� \ {|_{\ 5 \ 5}^{\ 4 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 4 \ 1 \ 6}}}-{�� \ {|_{\ 5 \ 7}^{\ 4 \ 1 \ 7}}}-{�� \ {|_{\ 6 \ 0}^{\ 4 \ 2 \ 0}}}-{�� \ {|_{\ 6 \ 1}^{\ 4 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 4 \ 2 \ 2}}}-{�� \ {|_{\ 6 \ 3}^{\ 4 \ 2 \ 3}}}-{�� \ {|_{\ 6 \ 4}^{\ 4 \ 2 \ 4}}}-{�� \ {|_{\ 6 \ 5}^{\ 4 \ 2 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 4 \ 2 \ 6}}}-{�� \ {|_{\ 6 \ 7}^{\ 4 \ 2 \ 7}}}-{�� \ {|_{\ 7 \ 0}^{\ 4 \ 3 \ 0}}}-{�� \ {|_{\ 7 \ 1}^{\ 4 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 4 \ 3 \ 2}}}-{�� \ {|_{\ 7 \ 3}^{\ 4 \ 3 \ 3}}}-{�� \ {|_{\ 7 \ 4}^{\ 4 \ 3 \ 4}}}-{�� \ {|_{\ 7 \ 5}^{\ 4 \ 3 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 4 \ 3 \ 6}}}-{�� \ {|_{\ 7 \ 7}^{\ 4 \ 3 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 4 \ 4 \ 0}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 4 \ 4 \ 1}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 4 \ 4 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 4 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 4}^{\ 4 \ 4 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 5}^{\ 4 \ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 6}^{\ 4 \ 4 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 7}^{\ 4 \ 4 \ 7}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 4 \ 5 \ 0}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 1}^{\ 4 \ 5 \ 1}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 2}^{\ 4 \ 5 \ 2}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 3}^{\ 4 \ 5 \ 3}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 4}^{\ 4 \ 5 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 5}^{\ 4 \ 5 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 6}^{\ 4 \ 5 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 7}^{\ 4 \ 5 \ 7}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 4 \ 6 \ 0}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 1}^{\ 4 \ 6 \ 1}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 2}^{\ 4 \ 6 \ 2}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 3}^{\ 4 \ 6 \ 3}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 4}^{\ 4 \ 6 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 5}^{\ 4 \ 6 \ 5}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 6}^{\ 4 \ 6 \ 6}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 7}^{\ 4 \ 6 \ 7}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 4 \ 7 \ 0}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 1}^{\ 4 \ 7 \ 1}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 2}^{\ 4 \ 7 \ 2}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 3}^{\ 4 \ 7 \ 3}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 4}^{\ 4 \ 7 \ 4}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 5}^{\ 4 \ 7 \ 5}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 6}^{\ 4 \ 7 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 7}^{\ 4 \ 7 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 0}^{\ 5 \ 0 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 5 \ 0 \ 1}}}+{�� \ {|_{\ 5 \ 2}^{\ 5 \ 0 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 5 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 4}^{\ 5 \ 0 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 0 \ 5}}}+{�� \ {|_{\ 5 \ 6}^{\ 5 \ 0 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 0 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 5 \ 1 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 5 \ 1 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 5 \ 1 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 5 \ 1 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 5 \ 1 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 5 \ 1 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 5 \ 1 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 5 \ 1 \ 7}}}-{�� \ {|_{\ 7 \ 0}^{\ 5 \ 2 \ 0}}}-{�� \ {|_{\ 7 \ 1}^{\ 5 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 5 \ 2 \ 2}}}-{�� \ {|_{\ 7 \ 3}^{\ 5 \ 2 \ 3}}}-{�� \ {|_{\ 7 \ 4}^{\ 5 \ 2 \ 4}}}-{�� \ {|_{\ 7 \ 5}^{\ 5 \ 2 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 5 \ 2 \ 6}}}-{�� \ {|_{\ 7 \ 7}^{\ 5 \ 2 \ 7}}}+{�� \ {|_{\ 6 \ 0}^{\ 5 \ 3 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 5 \ 3 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 5 \ 3 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 5 \ 3 \ 3}}}+{�� \ {|_{\ 6 \ 4}^{\ 5 \ 3 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 5 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 5 \ 3 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 5 \ 3 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 5 \ 4 \ 0}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 1}^{\ 5 \ 4 \ 1}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 2}^{\ 5 \ 4 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 3}^{\ 5 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 4}^{\ 5 \ 4 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 5}^{\ 5 \ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 6}^{\ 5 \ 4 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 7}^{\ 5 \ 4 \ 7}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 5 \ 5 \ 0}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 5 \ 5 \ 1}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 5 \ 5 \ 2}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 5 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 4}^{\ 5 \ 5 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 5}^{\ 5 \ 5 \ 5}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 6}^{\ 5 \ 5 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 7}^{\ 5 \ 5 \ 7}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 0}^{\ 5 \ 6 \ 0}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 1}^{\ 5 \ 6 \ 1}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 2}^{\ 5 \ 6 \ 2}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 3}^{\ 5 \ 6 \ 3}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 4}^{\ 5 \ 6 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 5}^{\ 5 \ 6 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 6}^{\ 5 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 7}^{\ 5 \ 6 \ 7}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 0}^{\ 5 \ 7 \ 0}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 1}^{\ 5 \ 7 \ 1}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 2}^{\ 5 \ 7 \ 2}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 3}^{\ 5 \ 7 \ 3}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 4}^{\ 5 \ 7 \ 4}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 5}^{\ 5 \ 7 \ 5}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 6}^{\ 5 \ 7 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 7}^{\ 5 \ 7 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 6 \ 0 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 6 \ 0 \ 1}}}+{�� \ {|_{\ 6 \ 2}^{\ 6 \ 0 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 6 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 4}^{\ 6 \ 0 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 0 \ 5}}}+{�� \ {|_{\ 6 \ 6}^{\ 6 \ 0 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 0 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 6 \ 1 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 6 \ 1 \ 1}}}+{�� \ {|_{\ 7 \ 2}^{\ 6 \ 1 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 6 \ 1 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 4}^{\ 6 \ 1 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 6 \ 1 \ 5}}}+{�� \ {|_{\ 7 \ 6}^{\ 6 \ 1 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 6 \ 1 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 6 \ 2 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 6 \ 2 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 6 \ 2 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 6 \ 2 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 6 \ 2 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 6 \ 2 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 6 \ 2 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 6 \ 2 \ 7}}}-{�� \ {|_{\ 5 \ 0}^{\ 6 \ 3 \ 0}}}-{�� \ {|_{\ 5 \ 1}^{\ 6 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 6 \ 3 \ 2}}}-{�� \ {|_{\ 5 \ 3}^{\ 6 \ 3 \ 3}}}-{�� \ {|_{\ 5 \ 4}^{\ 6 \ 3 \ 4}}}-{�� \ {|_{\ 5 \ 5}^{\ 6 \ 3 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 6 \ 3 \ 6}}}-{�� \ {|_{\ 5 \ 7}^{\ 6 \ 3 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 6 \ 4 \ 0}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 1}^{\ 6 \ 4 \ 1}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 2}^{\ 6 \ 4 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 3}^{\ 6 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 4}^{\ 6 \ 4 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 5}^{\ 6 \ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 6}^{\ 6 \ 4 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 7}^{\ 6 \ 4 \ 7}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 0}^{\ 6 \ 5 \ 0}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 1}^{\ 6 \ 5 \ 1}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 2}^{\ 6 \ 5 \ 2}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 3}^{\ 6 \ 5 \ 3}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 4}^{\ 6 \ 5 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 5}^{\ 6 \ 5 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 6}^{\ 6 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 7}^{\ 6 \ 5 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 6 \ 6 \ 0}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 6 \ 6 \ 1}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 6 \ 6 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 6 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 4}^{\ 6 \ 6 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 5}^{\ 6 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 6}^{\ 6 \ 6 \ 6}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 7}^{\ 6 \ 6 \ 7}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 0}^{\ 6 \ 7 \ 0}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 1}^{\ 6 \ 7 \ 1}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 2}^{\ 6 \ 7 \ 2}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 3}^{\ 6 \ 7 \ 3}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 4}^{\ 6 \ 7 \ 4}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 5}^{\ 6 \ 7 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 6}^{\ 6 \ 7 \ 6}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 7}^{\ 6 \ 7 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 7 \ 0 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 7 \ 0 \ 1}}}+{�� \ {|_{\ 7 \ 2}^{\ 7 \ 0 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 7 \ 0 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 4}^{\ 7 \ 0 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 0 \ 5}}}+{�� \ {|_{\ 7 \ 6}^{\ 7 \ 0 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 0 \ 7}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 7 \ 1 \ 0}}}-{�� \ {|_{\ 6 \ 1}^{\ 7 \ 1 \ 1}}}-{�� \ {|_{\ 6 \ 2}^{\ 7 \ 1 \ 2}}}-{�� \ {|_{\ 6 \ 3}^{\ 7 \ 1 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 4}^{\ 7 \ 1 \ 4}}}-{�� \ {|_{\ 6 \ 5}^{\ 7 \ 1 \ 5}}}-{�� \ {|_{\ 6 \ 6}^{\ 7 \ 1 \ 6}}}-{�� \ {|_{\ 6 \ 7}^{\ 7 \ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 0}^{\ 7 \ 2 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 7 \ 2 \ 1}}}+{�� \ {|_{\ 5 \ 2}^{\ 7 \ 2 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 7 \ 2 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 4}^{\ 7 \ 2 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 7 \ 2 \ 5}}}+{�� \ {|_{\ 5 \ 6}^{\ 7 \ 2 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 7 \ 2 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 0}^{\ 7 \ 3 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 1}^{\ 7 \ 3 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 2}^{\ 7 \ 3 \ 2}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 3}^{\ 7 \ 3 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 7 \ 3 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 5}^{\ 7 \ 3 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 6}^{\ 7 \ 3 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 7}^{\ 7 \ 3 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 7 \ 4 \ 0}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 1}^{\ 7 \ 4 \ 1}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 2}^{\ 7 \ 4 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 3}^{\ 7 \ 4 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 4}^{\ 7 \ 4 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 5}^{\ 7 \ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 6}^{\ 7 \ 4 \ 6}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 7}^{\ 7 \ 4 \ 7}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 0}^{\ 7 \ 5 \ 0}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 1}^{\ 7 \ 5 \ 1}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 2}^{\ 7 \ 5 \ 2}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 3}^{\ 7 \ 5 \ 3}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 4}^{\ 7 \ 5 \ 4}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 5}^{\ 7 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 6}^{\ 7 \ 5 \ 6}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 7}^{\ 7 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 0}^{\ 7 \ 6 \ 0}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 1}^{\ 7 \ 6 \ 1}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 2}^{\ 7 \ 6 \ 2}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 3}^{\ 7 \ 6 \ 3}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 4}^{\ 7 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 5}^{\ 7 \ 6 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 6}^{\ 7 \ 6 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 7}^{\ 7 \ 6 \ 7}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 7 \ 7 \ 0}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 7 \ 7 \ 1}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 7 \ 7 \ 2}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 7 \ 7 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 4}^{\ 7 \ 7 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 5}^{\ 7 \ 7 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 6}^{\ 7 \ 7 \ 6}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 7}^{\ 7 \ 7 \ 7}}}") | (28) |
fricas
λ:=(Ω,I, Ω)/(I, W, I)
4 + + - -- + 4 + arity warning: ---- 5 0 + - -- 0 4 +
>> System error: Heap exhausted (no more space for allocation). There are still 807796736 bytes available; the request was for 1073741840 bytes.
PROCEED WITH CAUTION.
fricas
test
( I Ω ) /
( Y I ) = λ
2
+ +
- --
+ 2
+
arity warning: ----
3
0 +
- --
0 2
+
 | (29) |
Type: Boolean
fricas
test
( Ω I ) /
( I Y ) = λ
There are no library operations named Ω
Use HyperDoc Browse or issue
)what op Ω
to learn if there is any operation containing " Ω " in its name.
Cannot find a definition or applicable library operation named Ω
with argument type(s)
ClosedLinearOperator(OrderedVariableList([0,
Perhaps you should use "@" to indicate the required return type,
Frobenius Condition
Octonion algebra fails the Frobenius Condition!
fricas
H :=
Y /
λ
2
+ 0
-- -
+ 0
arity warning: ----
+ 0
- -
+ 0
![\label{eq30}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0}^{\ 0 \ 0}}}+{�� \ {|_{\ 1}^{\ 0 \ 1}}}+{�� \ {|_{\ 2}^{\ 0 \ 2}}}+{�� \ {|_{\ 3}^{\ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 0 \ 4}}}+
\
\
\displaystyle
{�� \ {|_{\ 5}^{\ 0 \ 5}}}+{�� \ {|_{\ 6}^{\ 0 \ 6}}}+{�� \ {|_{\ 7}^{\ 0 \ 7}}}+{�� \ {|_{\ 1}^{\ 1 \ 0}}}-{�� \ {|_{\ 0}^{\ 1 \ 1}}}+{�� \ {|_{\ 3}^{\ 1 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 2}^{\ 1 \ 3}}}+{�� \ {|_{\ 5}^{\ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 1 \ 5}}}-{�� \ {|_{\ 7}^{\ 1 \ 6}}}+{�� \ {|_{\ 6}^{\ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2}^{\ 2 \ 0}}}-{�� \ {|_{\ 3}^{\ 2 \ 1}}}-{�� \ {|_{\ 0}^{\ 2 \ 2}}}+{�� \ {|_{\ 1}^{\ 2 \ 3}}}+{�� \ {|_{\ 6}^{\ 2 \ 4}}}+{�� \ {|_{\ 7}^{\ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 2 \ 6}}}-{�� \ {|_{\ 5}^{\ 2 \ 7}}}+{�� \ {|_{\ 3}^{\ 3 \ 0}}}+{�� \ {|_{\ 2}^{\ 3 \ 1}}}-{�� \ {|_{\ 1}^{\ 3 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 3 \ 3}}}+{�� \ {|_{\ 7}^{\ 3 \ 4}}}-{�� \ {|_{\ 6}^{\ 3 \ 5}}}+{�� \ {|_{\ 5}^{\ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 3 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 4 \ 0}}}-{�� \ {|_{\ 5}^{\ 4 \ 1}}}-{�� \ {|_{\ 6}^{\ 4 \ 2}}}-{�� \ {|_{\ 7}^{\ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 4 \ 4}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1}^{\ 4 \ 5}}}+{{q_{2}}\ �� \ {|_{\ 2}^{\ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 3}^{\ 4 \ 7}}}+{�� \ {|_{\ 5}^{\ 5 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 5 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7}^{\ 5 \ 2}}}+{�� \ {|_{\ 6}^{\ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1}^{\ 5 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3}^{\ 5 \ 6}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2}^{\ 5 \ 7}}}+{�� \ {|_{\ 6}^{\ 6 \ 0}}}+{�� \ {|_{\ 7}^{\ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 6 \ 2}}}-{�� \ {|_{\ 5}^{\ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2}^{\ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3}^{\ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 6 \ 6}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1}^{\ 6 \ 7}}}+{�� \ {|_{\ 7}^{\ 7 \ 0}}}-
\
\
\displaystyle
{�� \ {|_{\ 6}^{\ 7 \ 1}}}+{�� \ {|_{\ 5}^{\ 7 \ 2}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 3}^{\ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2}^{\ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1}^{\ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 7 \ 7}}}](images/1522683786959960912-16.0px.png "\label{eq30}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0}^{\ 0 \ 0}}}+{�� \ {|_{\ 1}^{\ 0 \ 1}}}+{�� \ {|_{\ 2}^{\ 0 \ 2}}}+{�� \ {|_{\ 3}^{\ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 0 \ 4}}}+
\
\
\displaystyle
{�� \ {|_{\ 5}^{\ 0 \ 5}}}+{�� \ {|_{\ 6}^{\ 0 \ 6}}}+{�� \ {|_{\ 7}^{\ 0 \ 7}}}+{�� \ {|_{\ 1}^{\ 1 \ 0}}}-{�� \ {|_{\ 0}^{\ 1 \ 1}}}+{�� \ {|_{\ 3}^{\ 1 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 2}^{\ 1 \ 3}}}+{�� \ {|_{\ 5}^{\ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 1 \ 5}}}-{�� \ {|_{\ 7}^{\ 1 \ 6}}}+{�� \ {|_{\ 6}^{\ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2}^{\ 2 \ 0}}}-{�� \ {|_{\ 3}^{\ 2 \ 1}}}-{�� \ {|_{\ 0}^{\ 2 \ 2}}}+{�� \ {|_{\ 1}^{\ 2 \ 3}}}+{�� \ {|_{\ 6}^{\ 2 \ 4}}}+{�� \ {|_{\ 7}^{\ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 2 \ 6}}}-{�� \ {|_{\ 5}^{\ 2 \ 7}}}+{�� \ {|_{\ 3}^{\ 3 \ 0}}}+{�� \ {|_{\ 2}^{\ 3 \ 1}}}-{�� \ {|_{\ 1}^{\ 3 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 3 \ 3}}}+{�� \ {|_{\ 7}^{\ 3 \ 4}}}-{�� \ {|_{\ 6}^{\ 3 \ 5}}}+{�� \ {|_{\ 5}^{\ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 3 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 4 \ 0}}}-{�� \ {|_{\ 5}^{\ 4 \ 1}}}-{�� \ {|_{\ 6}^{\ 4 \ 2}}}-{�� \ {|_{\ 7}^{\ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 4 \ 4}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1}^{\ 4 \ 5}}}+{{q_{2}}\ �� \ {|_{\ 2}^{\ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 3}^{\ 4 \ 7}}}+{�� \ {|_{\ 5}^{\ 5 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 5 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7}^{\ 5 \ 2}}}+{�� \ {|_{\ 6}^{\ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1}^{\ 5 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3}^{\ 5 \ 6}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2}^{\ 5 \ 7}}}+{�� \ {|_{\ 6}^{\ 6 \ 0}}}+{�� \ {|_{\ 7}^{\ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 6 \ 2}}}-{�� \ {|_{\ 5}^{\ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2}^{\ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3}^{\ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 6 \ 6}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1}^{\ 6 \ 7}}}+{�� \ {|_{\ 7}^{\ 7 \ 0}}}-
\
\
\displaystyle
{�� \ {|_{\ 6}^{\ 7 \ 1}}}+{�� \ {|_{\ 5}^{\ 7 \ 2}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 3}^{\ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2}^{\ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1}^{\ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 7 \ 7}}}") | (30) |
fricas
Hr := (λ,I)/(I, Y)
2 + + - -- + 2 + arity warning: ---- 3 0 + - -- 0 2 +
![\label{eq31}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0 \ 0}^{\ 0 \ 0 \ 0}}}+{�� \ {|_{\ 0 \ 1}^{\ 0 \ 0 \ 1}}}+{�� \ {|_{\ 0 \ 2}^{\ 0 \ 0 \ 2}}}+{�� \ {|_{\ 0 \ 3}^{\ 0 \ 0 \ 3}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 0 \ 4}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 0 \ 5}}}+{�� \ {|_{\ 0 \ 6}^{\ 0 \ 0 \ 6}}}+{�� \ {|_{\ 0 \ 7}^{\ 0 \ 0 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 1}^{\ 0 \ 1 \ 0}}}-{�� \ {|_{\ 0 \ 0}^{\ 0 \ 1 \ 1}}}+{�� \ {|_{\ 0 \ 3}^{\ 0 \ 1 \ 2}}}-{�� \ {|_{\ 0 \ 2}^{\ 0 \ 1 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 5}^{\ 0 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 1 \ 5}}}-{�� \ {|_{\ 0 \ 7}^{\ 0 \ 1 \ 6}}}+{�� \ {|_{\ 0 \ 6}^{\ 0 \ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 2}^{\ 0 \ 2 \ 0}}}-{�� \ {|_{\ 0 \ 3}^{\ 0 \ 2 \ 1}}}-{�� \ {|_{\ 0 \ 0}^{\ 0 \ 2 \ 2}}}+{�� \ {|_{\ 0 \ 1}^{\ 0 \ 2 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 0 \ 2 \ 4}}}+{�� \ {|_{\ 0 \ 7}^{\ 0 \ 2 \ 5}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 2 \ 6}}}-{�� \ {|_{\ 0 \ 5}^{\ 0 \ 2 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 3}^{\ 0 \ 3 \ 0}}}+{�� \ {|_{\ 0 \ 2}^{\ 0 \ 3 \ 1}}}-{�� \ {|_{\ 0 \ 1}^{\ 0 \ 3 \ 2}}}-{�� \ {|_{\ 0 \ 0}^{\ 0 \ 3 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 7}^{\ 0 \ 3 \ 4}}}-{�� \ {|_{\ 0 \ 6}^{\ 0 \ 3 \ 5}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 3 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 4 \ 0}}}-{�� \ {|_{\ 0 \ 5}^{\ 0 \ 4 \ 1}}}-{�� \ {|_{\ 0 \ 6}^{\ 0 \ 4 \ 2}}}-{�� \ {|_{\ 0 \ 7}^{\ 0 \ 4 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 0 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 0 \ 4 \ 5}}}+{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 0 \ 4 \ 6}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 0 \ 4 \ 7}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 5 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 5 \ 1}}}-{�� \ {|_{\ 0 \ 7}^{\ 0 \ 5 \ 2}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 0 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 0 \ 5 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 0 \ 5 \ 5}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 3}^{\ 0 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 2}^{\ 0 \ 5 \ 7}}}+{�� \ {|_{\ 0 \ 6}^{\ 0 \ 6 \ 0}}}+{�� \ {|_{\ 0 \ 7}^{\ 0 \ 6 \ 1}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 6 \ 2}}}-{�� \ {|_{\ 0 \ 5}^{\ 0 \ 6 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 0 \ 6 \ 4}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 3}^{\ 0 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 0 \ 6 \ 6}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 1}^{\ 0 \ 6 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 7}^{\ 0 \ 7 \ 0}}}-{�� \ {|_{\ 0 \ 6}^{\ 0 \ 7 \ 1}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 7 \ 2}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 7 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 0 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 2}^{\ 0 \ 7 \ 5}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 1}^{\ 0 \ 7 \ 6}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 0 \ 7 \ 7}}}+{�� \ {|_{\ 1 \ 0}^{\ 1 \ 0 \ 0}}}+{�� \ {|_{\ 1 \ 1}^{\ 1 \ 0 \ 1}}}+{�� \ {|_{\ 1 \ 2}^{\ 1 \ 0 \ 2}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 3}^{\ 1 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 0 \ 4}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 0 \ 5}}}+{�� \ {|_{\ 1 \ 6}^{\ 1 \ 0 \ 6}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 7}^{\ 1 \ 0 \ 7}}}+{�� \ {|_{\ 1 \ 1}^{\ 1 \ 1 \ 0}}}-{�� \ {|_{\ 1 \ 0}^{\ 1 \ 1 \ 1}}}+{�� \ {|_{\ 1 \ 3}^{\ 1 \ 1 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 2}^{\ 1 \ 1 \ 3}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 1 \ 5}}}-{�� \ {|_{\ 1 \ 7}^{\ 1 \ 1 \ 6}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 6}^{\ 1 \ 1 \ 7}}}+{�� \ {|_{\ 1 \ 2}^{\ 1 \ 2 \ 0}}}-{�� \ {|_{\ 1 \ 3}^{\ 1 \ 2 \ 1}}}-{�� \ {|_{\ 1 \ 0}^{\ 1 \ 2 \ 2}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 1}^{\ 1 \ 2 \ 3}}}+{�� \ {|_{\ 1 \ 6}^{\ 1 \ 2 \ 4}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 2 \ 5}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 2 \ 6}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 5}^{\ 1 \ 2 \ 7}}}+{�� \ {|_{\ 1 \ 3}^{\ 1 \ 3 \ 0}}}+{�� \ {|_{\ 1 \ 2}^{\ 1 \ 3 \ 1}}}-{�� \ {|_{\ 1 \ 1}^{\ 1 \ 3 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 0}^{\ 1 \ 3 \ 3}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 3 \ 4}}}-{�� \ {|_{\ 1 \ 6}^{\ 1 \ 3 \ 5}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 3 \ 6}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 4 \ 0}}}-{�� \ {|_{\ 1 \ 5}^{\ 1 \ 4 \ 1}}}-{�� \ {|_{\ 1 \ 6}^{\ 1 \ 4 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 7}^{\ 1 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 1 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 1}^{\ 1 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 2}^{\ 1 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 3}^{\ 1 \ 4 \ 7}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 5 \ 1}}}-{�� \ {|_{\ 1 \ 7}^{\ 1 \ 5 \ 2}}}+{�� \ {|_{\ 1 \ 6}^{\ 1 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 1}^{\ 1 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 1 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 3}^{\ 1 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 2}^{\ 1 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 6}^{\ 1 \ 6 \ 0}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 6 \ 2}}}-{�� \ {|_{\ 1 \ 5}^{\ 1 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 2}^{\ 1 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 3}^{\ 1 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 1 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 1}^{\ 1 \ 6 \ 7}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 7 \ 0}}}-{�� \ {|_{\ 1 \ 6}^{\ 1 \ 7 \ 1}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 3}^{\ 1 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 2}^{\ 1 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 1}^{\ 1 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 1 \ 7 \ 7}}}+{�� \ {|_{\ 2 \ 0}^{\ 2 \ 0 \ 0}}}+{�� \ {|_{\ 2 \ 1}^{\ 2 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 2}^{\ 2 \ 0 \ 2}}}+{�� \ {|_{\ 2 \ 3}^{\ 2 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 0 \ 4}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 2 \ 0 \ 6}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 0 \ 7}}}+{�� \ {|_{\ 2 \ 1}^{\ 2 \ 1 \ 0}}}-{�� \ {|_{\ 2 \ 0}^{\ 2 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 3}^{\ 2 \ 1 \ 2}}}-{�� \ {|_{\ 2 \ 2}^{\ 2 \ 1 \ 3}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 7}^{\ 2 \ 1 \ 6}}}+{�� \ {|_{\ 2 \ 6}^{\ 2 \ 1 \ 7}}}+{�� \ {|_{\ 2 \ 2}^{\ 2 \ 2 \ 0}}}-{�� \ {|_{\ 2 \ 3}^{\ 2 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 0}^{\ 2 \ 2 \ 2}}}+{�� \ {|_{\ 2 \ 1}^{\ 2 \ 2 \ 3}}}+{�� \ {|_{\ 2 \ 6}^{\ 2 \ 2 \ 4}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 2 \ 6}}}-{�� \ {|_{\ 2 \ 5}^{\ 2 \ 2 \ 7}}}+{�� \ {|_{\ 2 \ 3}^{\ 2 \ 3 \ 0}}}+{�� \ {|_{\ 2 \ 2}^{\ 2 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 1}^{\ 2 \ 3 \ 2}}}-{�� \ {|_{\ 2 \ 0}^{\ 2 \ 3 \ 3}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 3 \ 4}}}-{�� \ {|_{\ 2 \ 6}^{\ 2 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 5}^{\ 2 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 4 \ 0}}}-{�� \ {|_{\ 2 \ 5}^{\ 2 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 2 \ 4 \ 2}}}-{�� \ {|_{\ 2 \ 7}^{\ 2 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 2 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 1}^{\ 2 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 2}^{\ 2 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 3}^{\ 2 \ 4 \ 7}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 5 \ 1}}}-{�� \ {|_{\ 2 \ 7}^{\ 2 \ 5 \ 2}}}+{�� \ {|_{\ 2 \ 6}^{\ 2 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 1}^{\ 2 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 2 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 3}^{\ 2 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 2}^{\ 2 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 2 \ 6 \ 0}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 6 \ 2}}}-{�� \ {|_{\ 2 \ 5}^{\ 2 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 2}^{\ 2 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 3}^{\ 2 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 2 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 1}^{\ 2 \ 6 \ 7}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 7 \ 0}}}-{�� \ {|_{\ 2 \ 6}^{\ 2 \ 7 \ 1}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 3}^{\ 2 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 2}^{\ 2 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 1}^{\ 2 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 2 \ 7 \ 7}}}+{�� \ {|_{\ 3 \ 0}^{\ 3 \ 0 \ 0}}}+{�� \ {|_{\ 3 \ 1}^{\ 3 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 2}^{\ 3 \ 0 \ 2}}}+{�� \ {|_{\ 3 \ 3}^{\ 3 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 0 \ 4}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 3 \ 0 \ 6}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 0 \ 7}}}+{�� \ {|_{\ 3 \ 1}^{\ 3 \ 1 \ 0}}}-{�� \ {|_{\ 3 \ 0}^{\ 3 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 3}^{\ 3 \ 1 \ 2}}}-{�� \ {|_{\ 3 \ 2}^{\ 3 \ 1 \ 3}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 7}^{\ 3 \ 1 \ 6}}}+{�� \ {|_{\ 3 \ 6}^{\ 3 \ 1 \ 7}}}+{�� \ {|_{\ 3 \ 2}^{\ 3 \ 2 \ 0}}}-{�� \ {|_{\ 3 \ 3}^{\ 3 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 0}^{\ 3 \ 2 \ 2}}}+{�� \ {|_{\ 3 \ 1}^{\ 3 \ 2 \ 3}}}+{�� \ {|_{\ 3 \ 6}^{\ 3 \ 2 \ 4}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 2 \ 6}}}-{�� \ {|_{\ 3 \ 5}^{\ 3 \ 2 \ 7}}}+{�� \ {|_{\ 3 \ 3}^{\ 3 \ 3 \ 0}}}+{�� \ {|_{\ 3 \ 2}^{\ 3 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 1}^{\ 3 \ 3 \ 2}}}-{�� \ {|_{\ 3 \ 0}^{\ 3 \ 3 \ 3}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 3 \ 4}}}-{�� \ {|_{\ 3 \ 6}^{\ 3 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 5}^{\ 3 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 4 \ 0}}}-{�� \ {|_{\ 3 \ 5}^{\ 3 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 3 \ 4 \ 2}}}-{�� \ {|_{\ 3 \ 7}^{\ 3 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 3 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 1}^{\ 3 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 2}^{\ 3 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 3}^{\ 3 \ 4 \ 7}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 5 \ 1}}}-{�� \ {|_{\ 3 \ 7}^{\ 3 \ 5 \ 2}}}+{�� \ {|_{\ 3 \ 6}^{\ 3 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 1}^{\ 3 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 3 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 3}^{\ 3 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 2}^{\ 3 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 3 \ 6 \ 0}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 6 \ 2}}}-{�� \ {|_{\ 3 \ 5}^{\ 3 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 2}^{\ 3 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 3}^{\ 3 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 3 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 1}^{\ 3 \ 6 \ 7}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 7 \ 0}}}-{�� \ {|_{\ 3 \ 6}^{\ 3 \ 7 \ 1}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 3}^{\ 3 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 2}^{\ 3 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 1}^{\ 3 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 3 \ 7 \ 7}}}+{�� \ {|_{\ 4 \ 0}^{\ 4 \ 0 \ 0}}}+{�� \ {|_{\ 4 \ 1}^{\ 4 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 2}^{\ 4 \ 0 \ 2}}}+{�� \ {|_{\ 4 \ 3}^{\ 4 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 0 \ 4}}}+{�� \ {|_{\ 4 \ 5}^{\ 4 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 6}^{\ 4 \ 0 \ 6}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 0 \ 7}}}+{�� \ {|_{\ 4 \ 1}^{\ 4 \ 1 \ 0}}}-{�� \ {|_{\ 4 \ 0}^{\ 4 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 3}^{\ 4 \ 1 \ 2}}}-{�� \ {|_{\ 4 \ 2}^{\ 4 \ 1 \ 3}}}+{�� \ {|_{\ 4 \ 5}^{\ 4 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 4 \ 7}^{\ 4 \ 1 \ 6}}}+{�� \ {|_{\ 4 \ 6}^{\ 4 \ 1 \ 7}}}+{�� \ {|_{\ 4 \ 2}^{\ 4 \ 2 \ 0}}}-{�� \ {|_{\ 4 \ 3}^{\ 4 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 4 \ 0}^{\ 4 \ 2 \ 2}}}+{�� \ {|_{\ 4 \ 1}^{\ 4 \ 2 \ 3}}}+{�� \ {|_{\ 4 \ 6}^{\ 4 \ 2 \ 4}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 2 \ 6}}}-{�� \ {|_{\ 4 \ 5}^{\ 4 \ 2 \ 7}}}+{�� \ {|_{\ 4 \ 3}^{\ 4 \ 3 \ 0}}}+{�� \ {|_{\ 4 \ 2}^{\ 4 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 4 \ 1}^{\ 4 \ 3 \ 2}}}-{�� \ {|_{\ 4 \ 0}^{\ 4 \ 3 \ 3}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 3 \ 4}}}-{�� \ {|_{\ 4 \ 6}^{\ 4 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 5}^{\ 4 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 4 \ 0}}}-{�� \ {|_{\ 4 \ 5}^{\ 4 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 4 \ 6}^{\ 4 \ 4 \ 2}}}-{�� \ {|_{\ 4 \ 7}^{\ 4 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 0}^{\ 4 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 4 \ 1}^{\ 4 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 4 \ 2}^{\ 4 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 4 \ 3}^{\ 4 \ 4 \ 7}}}+{�� \ {|_{\ 4 \ 5}^{\ 4 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 5 \ 1}}}-{�� \ {|_{\ 4 \ 7}^{\ 4 \ 5 \ 2}}}+{�� \ {|_{\ 4 \ 6}^{\ 4 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 1}^{\ 4 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 4 \ 0}^{\ 4 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 3}^{\ 4 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 2}^{\ 4 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 6}^{\ 4 \ 6 \ 0}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 6 \ 2}}}-{�� \ {|_{\ 4 \ 5}^{\ 4 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 4 \ 2}^{\ 4 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 3}^{\ 4 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 0}^{\ 4 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 1}^{\ 4 \ 6 \ 7}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 7 \ 0}}}-{�� \ {|_{\ 4 \ 6}^{\ 4 \ 7 \ 1}}}+{�� \ {|_{\ 4 \ 5}^{\ 4 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 3}^{\ 4 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 2}^{\ 4 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 1}^{\ 4 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 0}^{\ 4 \ 7 \ 7}}}+{�� \ {|_{\ 5 \ 0}^{\ 5 \ 0 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 5 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 5 \ 0 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 5 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 0 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 5 \ 0 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 0 \ 7}}}+{�� \ {|_{\ 5 \ 1}^{\ 5 \ 1 \ 0}}}-{�� \ {|_{\ 5 \ 0}^{\ 5 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 3}^{\ 5 \ 1 \ 2}}}-{�� \ {|_{\ 5 \ 2}^{\ 5 \ 1 \ 3}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 7}^{\ 5 \ 1 \ 6}}}+{�� \ {|_{\ 5 \ 6}^{\ 5 \ 1 \ 7}}}+{�� \ {|_{\ 5 \ 2}^{\ 5 \ 2 \ 0}}}-{�� \ {|_{\ 5 \ 3}^{\ 5 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 0}^{\ 5 \ 2 \ 2}}}+{�� \ {|_{\ 5 \ 1}^{\ 5 \ 2 \ 3}}}+{�� \ {|_{\ 5 \ 6}^{\ 5 \ 2 \ 4}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 2 \ 6}}}-{�� \ {|_{\ 5 \ 5}^{\ 5 \ 2 \ 7}}}+{�� \ {|_{\ 5 \ 3}^{\ 5 \ 3 \ 0}}}+{�� \ {|_{\ 5 \ 2}^{\ 5 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 1}^{\ 5 \ 3 \ 2}}}-{�� \ {|_{\ 5 \ 0}^{\ 5 \ 3 \ 3}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 3 \ 4}}}-{�� \ {|_{\ 5 \ 6}^{\ 5 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 5}^{\ 5 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 4 \ 0}}}-{�� \ {|_{\ 5 \ 5}^{\ 5 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 5 \ 4 \ 2}}}-{�� \ {|_{\ 5 \ 7}^{\ 5 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 0}^{\ 5 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 5 \ 1}^{\ 5 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 5 \ 2}^{\ 5 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 5 \ 3}^{\ 5 \ 4 \ 7}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 5 \ 1}}}-{�� \ {|_{\ 5 \ 7}^{\ 5 \ 5 \ 2}}}+{�� \ {|_{\ 5 \ 6}^{\ 5 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 1}^{\ 5 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 5 \ 0}^{\ 5 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 3}^{\ 5 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 2}^{\ 5 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 5 \ 6 \ 0}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 6 \ 2}}}-{�� \ {|_{\ 5 \ 5}^{\ 5 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 5 \ 2}^{\ 5 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 3}^{\ 5 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 0}^{\ 5 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 1}^{\ 5 \ 6 \ 7}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 7 \ 0}}}-{�� \ {|_{\ 5 \ 6}^{\ 5 \ 7 \ 1}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 3}^{\ 5 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 2}^{\ 5 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 1}^{\ 5 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 0}^{\ 5 \ 7 \ 7}}}+{�� \ {|_{\ 6 \ 0}^{\ 6 \ 0 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 6 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 6 \ 0 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 6 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 0 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 6 \ 0 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 0 \ 7}}}+{�� \ {|_{\ 6 \ 1}^{\ 6 \ 1 \ 0}}}-{�� \ {|_{\ 6 \ 0}^{\ 6 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 3}^{\ 6 \ 1 \ 2}}}-{�� \ {|_{\ 6 \ 2}^{\ 6 \ 1 \ 3}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 7}^{\ 6 \ 1 \ 6}}}+{�� \ {|_{\ 6 \ 6}^{\ 6 \ 1 \ 7}}}+{�� \ {|_{\ 6 \ 2}^{\ 6 \ 2 \ 0}}}-{�� \ {|_{\ 6 \ 3}^{\ 6 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 6 \ 2 \ 2}}}+{�� \ {|_{\ 6 \ 1}^{\ 6 \ 2 \ 3}}}+{�� \ {|_{\ 6 \ 6}^{\ 6 \ 2 \ 4}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 2 \ 6}}}-{�� \ {|_{\ 6 \ 5}^{\ 6 \ 2 \ 7}}}+{�� \ {|_{\ 6 \ 3}^{\ 6 \ 3 \ 0}}}+{�� \ {|_{\ 6 \ 2}^{\ 6 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 1}^{\ 6 \ 3 \ 2}}}-{�� \ {|_{\ 6 \ 0}^{\ 6 \ 3 \ 3}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 3 \ 4}}}-{�� \ {|_{\ 6 \ 6}^{\ 6 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 5}^{\ 6 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 4 \ 0}}}-{�� \ {|_{\ 6 \ 5}^{\ 6 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 6 \ 4 \ 2}}}-{�� \ {|_{\ 6 \ 7}^{\ 6 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 0}^{\ 6 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 6 \ 1}^{\ 6 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 6 \ 2}^{\ 6 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 6 \ 3}^{\ 6 \ 4 \ 7}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 5 \ 1}}}-{�� \ {|_{\ 6 \ 7}^{\ 6 \ 5 \ 2}}}+{�� \ {|_{\ 6 \ 6}^{\ 6 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 1}^{\ 6 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 6 \ 0}^{\ 6 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 3}^{\ 6 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 2}^{\ 6 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 6 \ 6 \ 0}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 6 \ 2}}}-{�� \ {|_{\ 6 \ 5}^{\ 6 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 6 \ 2}^{\ 6 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 3}^{\ 6 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 0}^{\ 6 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 1}^{\ 6 \ 6 \ 7}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 7 \ 0}}}-{�� \ {|_{\ 6 \ 6}^{\ 6 \ 7 \ 1}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 3}^{\ 6 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 2}^{\ 6 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 1}^{\ 6 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 0}^{\ 6 \ 7 \ 7}}}+{�� \ {|_{\ 7 \ 0}^{\ 7 \ 0 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 7 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 7 \ 0 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 7 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 0 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 7 \ 0 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 0 \ 7}}}+{�� \ {|_{\ 7 \ 1}^{\ 7 \ 1 \ 0}}}-{�� \ {|_{\ 7 \ 0}^{\ 7 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 3}^{\ 7 \ 1 \ 2}}}-{�� \ {|_{\ 7 \ 2}^{\ 7 \ 1 \ 3}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 7}^{\ 7 \ 1 \ 6}}}+{�� \ {|_{\ 7 \ 6}^{\ 7 \ 1 \ 7}}}+{�� \ {|_{\ 7 \ 2}^{\ 7 \ 2 \ 0}}}-{�� \ {|_{\ 7 \ 3}^{\ 7 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 7 \ 2 \ 2}}}+{�� \ {|_{\ 7 \ 1}^{\ 7 \ 2 \ 3}}}+{�� \ {|_{\ 7 \ 6}^{\ 7 \ 2 \ 4}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 2 \ 6}}}-{�� \ {|_{\ 7 \ 5}^{\ 7 \ 2 \ 7}}}+{�� \ {|_{\ 7 \ 3}^{\ 7 \ 3 \ 0}}}+{�� \ {|_{\ 7 \ 2}^{\ 7 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 1}^{\ 7 \ 3 \ 2}}}-{�� \ {|_{\ 7 \ 0}^{\ 7 \ 3 \ 3}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 3 \ 4}}}-{�� \ {|_{\ 7 \ 6}^{\ 7 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 5}^{\ 7 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 4 \ 0}}}-{�� \ {|_{\ 7 \ 5}^{\ 7 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 7 \ 4 \ 2}}}-{�� \ {|_{\ 7 \ 7}^{\ 7 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 0}^{\ 7 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 7 \ 1}^{\ 7 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 7 \ 2}^{\ 7 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 7 \ 3}^{\ 7 \ 4 \ 7}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 5 \ 1}}}-{�� \ {|_{\ 7 \ 7}^{\ 7 \ 5 \ 2}}}+{�� \ {|_{\ 7 \ 6}^{\ 7 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 1}^{\ 7 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 7 \ 0}^{\ 7 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 3}^{\ 7 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 2}^{\ 7 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 7 \ 6 \ 0}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 6 \ 2}}}-{�� \ {|_{\ 7 \ 5}^{\ 7 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 7 \ 2}^{\ 7 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 3}^{\ 7 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 0}^{\ 7 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 1}^{\ 7 \ 6 \ 7}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 7 \ 0}}}-{�� \ {|_{\ 7 \ 6}^{\ 7 \ 7 \ 1}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 3}^{\ 7 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 2}^{\ 7 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 1}^{\ 7 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 0}^{\ 7 \ 7 \ 7}}}](images/3544465937922858046-16.0px.png "\label{eq31}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0 \ 0}^{\ 0 \ 0 \ 0}}}+{�� \ {|_{\ 0 \ 1}^{\ 0 \ 0 \ 1}}}+{�� \ {|_{\ 0 \ 2}^{\ 0 \ 0 \ 2}}}+{�� \ {|_{\ 0 \ 3}^{\ 0 \ 0 \ 3}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 0 \ 4}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 0 \ 5}}}+{�� \ {|_{\ 0 \ 6}^{\ 0 \ 0 \ 6}}}+{�� \ {|_{\ 0 \ 7}^{\ 0 \ 0 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 1}^{\ 0 \ 1 \ 0}}}-{�� \ {|_{\ 0 \ 0}^{\ 0 \ 1 \ 1}}}+{�� \ {|_{\ 0 \ 3}^{\ 0 \ 1 \ 2}}}-{�� \ {|_{\ 0 \ 2}^{\ 0 \ 1 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 5}^{\ 0 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 1 \ 5}}}-{�� \ {|_{\ 0 \ 7}^{\ 0 \ 1 \ 6}}}+{�� \ {|_{\ 0 \ 6}^{\ 0 \ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 2}^{\ 0 \ 2 \ 0}}}-{�� \ {|_{\ 0 \ 3}^{\ 0 \ 2 \ 1}}}-{�� \ {|_{\ 0 \ 0}^{\ 0 \ 2 \ 2}}}+{�� \ {|_{\ 0 \ 1}^{\ 0 \ 2 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 0 \ 2 \ 4}}}+{�� \ {|_{\ 0 \ 7}^{\ 0 \ 2 \ 5}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 2 \ 6}}}-{�� \ {|_{\ 0 \ 5}^{\ 0 \ 2 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 3}^{\ 0 \ 3 \ 0}}}+{�� \ {|_{\ 0 \ 2}^{\ 0 \ 3 \ 1}}}-{�� \ {|_{\ 0 \ 1}^{\ 0 \ 3 \ 2}}}-{�� \ {|_{\ 0 \ 0}^{\ 0 \ 3 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 7}^{\ 0 \ 3 \ 4}}}-{�� \ {|_{\ 0 \ 6}^{\ 0 \ 3 \ 5}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 3 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 4 \ 0}}}-{�� \ {|_{\ 0 \ 5}^{\ 0 \ 4 \ 1}}}-{�� \ {|_{\ 0 \ 6}^{\ 0 \ 4 \ 2}}}-{�� \ {|_{\ 0 \ 7}^{\ 0 \ 4 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 0 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 0 \ 4 \ 5}}}+{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 0 \ 4 \ 6}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 0 \ 4 \ 7}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 5 \ 0}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 5 \ 1}}}-{�� \ {|_{\ 0 \ 7}^{\ 0 \ 5 \ 2}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 0 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 1}^{\ 0 \ 5 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 0 \ 5 \ 5}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 3}^{\ 0 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 2}^{\ 0 \ 5 \ 7}}}+{�� \ {|_{\ 0 \ 6}^{\ 0 \ 6 \ 0}}}+{�� \ {|_{\ 0 \ 7}^{\ 0 \ 6 \ 1}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 6 \ 2}}}-{�� \ {|_{\ 0 \ 5}^{\ 0 \ 6 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 2}^{\ 0 \ 6 \ 4}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 3}^{\ 0 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 0 \ 6 \ 6}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 1}^{\ 0 \ 6 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 7}^{\ 0 \ 7 \ 0}}}-{�� \ {|_{\ 0 \ 6}^{\ 0 \ 7 \ 1}}}+{�� \ {|_{\ 0 \ 5}^{\ 0 \ 7 \ 2}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 0 \ 4}^{\ 0 \ 7 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 3}^{\ 0 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 2}^{\ 0 \ 7 \ 5}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 0 \ 1}^{\ 0 \ 7 \ 6}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 0 \ 0}^{\ 0 \ 7 \ 7}}}+{�� \ {|_{\ 1 \ 0}^{\ 1 \ 0 \ 0}}}+{�� \ {|_{\ 1 \ 1}^{\ 1 \ 0 \ 1}}}+{�� \ {|_{\ 1 \ 2}^{\ 1 \ 0 \ 2}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 3}^{\ 1 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 0 \ 4}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 0 \ 5}}}+{�� \ {|_{\ 1 \ 6}^{\ 1 \ 0 \ 6}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 7}^{\ 1 \ 0 \ 7}}}+{�� \ {|_{\ 1 \ 1}^{\ 1 \ 1 \ 0}}}-{�� \ {|_{\ 1 \ 0}^{\ 1 \ 1 \ 1}}}+{�� \ {|_{\ 1 \ 3}^{\ 1 \ 1 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 2}^{\ 1 \ 1 \ 3}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 1 \ 5}}}-{�� \ {|_{\ 1 \ 7}^{\ 1 \ 1 \ 6}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 6}^{\ 1 \ 1 \ 7}}}+{�� \ {|_{\ 1 \ 2}^{\ 1 \ 2 \ 0}}}-{�� \ {|_{\ 1 \ 3}^{\ 1 \ 2 \ 1}}}-{�� \ {|_{\ 1 \ 0}^{\ 1 \ 2 \ 2}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 1}^{\ 1 \ 2 \ 3}}}+{�� \ {|_{\ 1 \ 6}^{\ 1 \ 2 \ 4}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 2 \ 5}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 2 \ 6}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 5}^{\ 1 \ 2 \ 7}}}+{�� \ {|_{\ 1 \ 3}^{\ 1 \ 3 \ 0}}}+{�� \ {|_{\ 1 \ 2}^{\ 1 \ 3 \ 1}}}-{�� \ {|_{\ 1 \ 1}^{\ 1 \ 3 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 0}^{\ 1 \ 3 \ 3}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 3 \ 4}}}-{�� \ {|_{\ 1 \ 6}^{\ 1 \ 3 \ 5}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 3 \ 6}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 4 \ 0}}}-{�� \ {|_{\ 1 \ 5}^{\ 1 \ 4 \ 1}}}-{�� \ {|_{\ 1 \ 6}^{\ 1 \ 4 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 1 \ 7}^{\ 1 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 1 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 1}^{\ 1 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 2}^{\ 1 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 1 \ 3}^{\ 1 \ 4 \ 7}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 5 \ 1}}}-{�� \ {|_{\ 1 \ 7}^{\ 1 \ 5 \ 2}}}+{�� \ {|_{\ 1 \ 6}^{\ 1 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 1}^{\ 1 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 1 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 3}^{\ 1 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 2}^{\ 1 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 1 \ 6}^{\ 1 \ 6 \ 0}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 6 \ 2}}}-{�� \ {|_{\ 1 \ 5}^{\ 1 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1 \ 2}^{\ 1 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 3}^{\ 1 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 1 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 1}^{\ 1 \ 6 \ 7}}}+{�� \ {|_{\ 1 \ 7}^{\ 1 \ 7 \ 0}}}-{�� \ {|_{\ 1 \ 6}^{\ 1 \ 7 \ 1}}}+{�� \ {|_{\ 1 \ 5}^{\ 1 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 1 \ 4}^{\ 1 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 3}^{\ 1 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 2}^{\ 1 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1 \ 1}^{\ 1 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 1 \ 0}^{\ 1 \ 7 \ 7}}}+{�� \ {|_{\ 2 \ 0}^{\ 2 \ 0 \ 0}}}+{�� \ {|_{\ 2 \ 1}^{\ 2 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 2}^{\ 2 \ 0 \ 2}}}+{�� \ {|_{\ 2 \ 3}^{\ 2 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 0 \ 4}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 2 \ 0 \ 6}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 0 \ 7}}}+{�� \ {|_{\ 2 \ 1}^{\ 2 \ 1 \ 0}}}-{�� \ {|_{\ 2 \ 0}^{\ 2 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 3}^{\ 2 \ 1 \ 2}}}-{�� \ {|_{\ 2 \ 2}^{\ 2 \ 1 \ 3}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 7}^{\ 2 \ 1 \ 6}}}+{�� \ {|_{\ 2 \ 6}^{\ 2 \ 1 \ 7}}}+{�� \ {|_{\ 2 \ 2}^{\ 2 \ 2 \ 0}}}-{�� \ {|_{\ 2 \ 3}^{\ 2 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 0}^{\ 2 \ 2 \ 2}}}+{�� \ {|_{\ 2 \ 1}^{\ 2 \ 2 \ 3}}}+{�� \ {|_{\ 2 \ 6}^{\ 2 \ 2 \ 4}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 2 \ 6}}}-{�� \ {|_{\ 2 \ 5}^{\ 2 \ 2 \ 7}}}+{�� \ {|_{\ 2 \ 3}^{\ 2 \ 3 \ 0}}}+{�� \ {|_{\ 2 \ 2}^{\ 2 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 1}^{\ 2 \ 3 \ 2}}}-{�� \ {|_{\ 2 \ 0}^{\ 2 \ 3 \ 3}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 3 \ 4}}}-{�� \ {|_{\ 2 \ 6}^{\ 2 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 5}^{\ 2 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 4 \ 0}}}-{�� \ {|_{\ 2 \ 5}^{\ 2 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 2 \ 4 \ 2}}}-{�� \ {|_{\ 2 \ 7}^{\ 2 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 2 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 1}^{\ 2 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 2}^{\ 2 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 2 \ 3}^{\ 2 \ 4 \ 7}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 5 \ 1}}}-{�� \ {|_{\ 2 \ 7}^{\ 2 \ 5 \ 2}}}+{�� \ {|_{\ 2 \ 6}^{\ 2 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 1}^{\ 2 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 2 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 3}^{\ 2 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 2}^{\ 2 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2 \ 6}^{\ 2 \ 6 \ 0}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 6 \ 2}}}-{�� \ {|_{\ 2 \ 5}^{\ 2 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2 \ 2}^{\ 2 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 3}^{\ 2 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 2 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 1}^{\ 2 \ 6 \ 7}}}+{�� \ {|_{\ 2 \ 7}^{\ 2 \ 7 \ 0}}}-{�� \ {|_{\ 2 \ 6}^{\ 2 \ 7 \ 1}}}+{�� \ {|_{\ 2 \ 5}^{\ 2 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 2 \ 4}^{\ 2 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 3}^{\ 2 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 2}^{\ 2 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2 \ 1}^{\ 2 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 2 \ 0}^{\ 2 \ 7 \ 7}}}+{�� \ {|_{\ 3 \ 0}^{\ 3 \ 0 \ 0}}}+{�� \ {|_{\ 3 \ 1}^{\ 3 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 2}^{\ 3 \ 0 \ 2}}}+{�� \ {|_{\ 3 \ 3}^{\ 3 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 0 \ 4}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 3 \ 0 \ 6}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 0 \ 7}}}+{�� \ {|_{\ 3 \ 1}^{\ 3 \ 1 \ 0}}}-{�� \ {|_{\ 3 \ 0}^{\ 3 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 3}^{\ 3 \ 1 \ 2}}}-{�� \ {|_{\ 3 \ 2}^{\ 3 \ 1 \ 3}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 7}^{\ 3 \ 1 \ 6}}}+{�� \ {|_{\ 3 \ 6}^{\ 3 \ 1 \ 7}}}+{�� \ {|_{\ 3 \ 2}^{\ 3 \ 2 \ 0}}}-{�� \ {|_{\ 3 \ 3}^{\ 3 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 0}^{\ 3 \ 2 \ 2}}}+{�� \ {|_{\ 3 \ 1}^{\ 3 \ 2 \ 3}}}+{�� \ {|_{\ 3 \ 6}^{\ 3 \ 2 \ 4}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 2 \ 6}}}-{�� \ {|_{\ 3 \ 5}^{\ 3 \ 2 \ 7}}}+{�� \ {|_{\ 3 \ 3}^{\ 3 \ 3 \ 0}}}+{�� \ {|_{\ 3 \ 2}^{\ 3 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 1}^{\ 3 \ 3 \ 2}}}-{�� \ {|_{\ 3 \ 0}^{\ 3 \ 3 \ 3}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 3 \ 4}}}-{�� \ {|_{\ 3 \ 6}^{\ 3 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 5}^{\ 3 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 4 \ 0}}}-{�� \ {|_{\ 3 \ 5}^{\ 3 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 3 \ 4 \ 2}}}-{�� \ {|_{\ 3 \ 7}^{\ 3 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 3 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 1}^{\ 3 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 2}^{\ 3 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 3 \ 3}^{\ 3 \ 4 \ 7}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 5 \ 1}}}-{�� \ {|_{\ 3 \ 7}^{\ 3 \ 5 \ 2}}}+{�� \ {|_{\ 3 \ 6}^{\ 3 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 1}^{\ 3 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 3 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 3}^{\ 3 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 2}^{\ 3 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 3 \ 6}^{\ 3 \ 6 \ 0}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 6 \ 2}}}-{�� \ {|_{\ 3 \ 5}^{\ 3 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 3 \ 2}^{\ 3 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 3}^{\ 3 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 3 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 1}^{\ 3 \ 6 \ 7}}}+{�� \ {|_{\ 3 \ 7}^{\ 3 \ 7 \ 0}}}-{�� \ {|_{\ 3 \ 6}^{\ 3 \ 7 \ 1}}}+{�� \ {|_{\ 3 \ 5}^{\ 3 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 3 \ 4}^{\ 3 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 3}^{\ 3 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 2}^{\ 3 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 3 \ 1}^{\ 3 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 3 \ 0}^{\ 3 \ 7 \ 7}}}+{�� \ {|_{\ 4 \ 0}^{\ 4 \ 0 \ 0}}}+{�� \ {|_{\ 4 \ 1}^{\ 4 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 2}^{\ 4 \ 0 \ 2}}}+{�� \ {|_{\ 4 \ 3}^{\ 4 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 0 \ 4}}}+{�� \ {|_{\ 4 \ 5}^{\ 4 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 6}^{\ 4 \ 0 \ 6}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 0 \ 7}}}+{�� \ {|_{\ 4 \ 1}^{\ 4 \ 1 \ 0}}}-{�� \ {|_{\ 4 \ 0}^{\ 4 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 3}^{\ 4 \ 1 \ 2}}}-{�� \ {|_{\ 4 \ 2}^{\ 4 \ 1 \ 3}}}+{�� \ {|_{\ 4 \ 5}^{\ 4 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 4 \ 7}^{\ 4 \ 1 \ 6}}}+{�� \ {|_{\ 4 \ 6}^{\ 4 \ 1 \ 7}}}+{�� \ {|_{\ 4 \ 2}^{\ 4 \ 2 \ 0}}}-{�� \ {|_{\ 4 \ 3}^{\ 4 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 4 \ 0}^{\ 4 \ 2 \ 2}}}+{�� \ {|_{\ 4 \ 1}^{\ 4 \ 2 \ 3}}}+{�� \ {|_{\ 4 \ 6}^{\ 4 \ 2 \ 4}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 2 \ 6}}}-{�� \ {|_{\ 4 \ 5}^{\ 4 \ 2 \ 7}}}+{�� \ {|_{\ 4 \ 3}^{\ 4 \ 3 \ 0}}}+{�� \ {|_{\ 4 \ 2}^{\ 4 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 4 \ 1}^{\ 4 \ 3 \ 2}}}-{�� \ {|_{\ 4 \ 0}^{\ 4 \ 3 \ 3}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 3 \ 4}}}-{�� \ {|_{\ 4 \ 6}^{\ 4 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 5}^{\ 4 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 4 \ 0}}}-{�� \ {|_{\ 4 \ 5}^{\ 4 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 4 \ 6}^{\ 4 \ 4 \ 2}}}-{�� \ {|_{\ 4 \ 7}^{\ 4 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 0}^{\ 4 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 4 \ 1}^{\ 4 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 4 \ 2}^{\ 4 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 4 \ 3}^{\ 4 \ 4 \ 7}}}+{�� \ {|_{\ 4 \ 5}^{\ 4 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 5 \ 1}}}-{�� \ {|_{\ 4 \ 7}^{\ 4 \ 5 \ 2}}}+{�� \ {|_{\ 4 \ 6}^{\ 4 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 1}^{\ 4 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 4 \ 0}^{\ 4 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 3}^{\ 4 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 2}^{\ 4 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 4 \ 6}^{\ 4 \ 6 \ 0}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 6 \ 2}}}-{�� \ {|_{\ 4 \ 5}^{\ 4 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 4 \ 2}^{\ 4 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 3}^{\ 4 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 0}^{\ 4 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 1}^{\ 4 \ 6 \ 7}}}+{�� \ {|_{\ 4 \ 7}^{\ 4 \ 7 \ 0}}}-{�� \ {|_{\ 4 \ 6}^{\ 4 \ 7 \ 1}}}+{�� \ {|_{\ 4 \ 5}^{\ 4 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4 \ 4}^{\ 4 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 3}^{\ 4 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 2}^{\ 4 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 4 \ 1}^{\ 4 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 4 \ 0}^{\ 4 \ 7 \ 7}}}+{�� \ {|_{\ 5 \ 0}^{\ 5 \ 0 \ 0}}}+{�� \ {|_{\ 5 \ 1}^{\ 5 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 2}^{\ 5 \ 0 \ 2}}}+{�� \ {|_{\ 5 \ 3}^{\ 5 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 0 \ 4}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 5 \ 0 \ 6}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 0 \ 7}}}+{�� \ {|_{\ 5 \ 1}^{\ 5 \ 1 \ 0}}}-{�� \ {|_{\ 5 \ 0}^{\ 5 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 3}^{\ 5 \ 1 \ 2}}}-{�� \ {|_{\ 5 \ 2}^{\ 5 \ 1 \ 3}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 7}^{\ 5 \ 1 \ 6}}}+{�� \ {|_{\ 5 \ 6}^{\ 5 \ 1 \ 7}}}+{�� \ {|_{\ 5 \ 2}^{\ 5 \ 2 \ 0}}}-{�� \ {|_{\ 5 \ 3}^{\ 5 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 0}^{\ 5 \ 2 \ 2}}}+{�� \ {|_{\ 5 \ 1}^{\ 5 \ 2 \ 3}}}+{�� \ {|_{\ 5 \ 6}^{\ 5 \ 2 \ 4}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 2 \ 6}}}-{�� \ {|_{\ 5 \ 5}^{\ 5 \ 2 \ 7}}}+{�� \ {|_{\ 5 \ 3}^{\ 5 \ 3 \ 0}}}+{�� \ {|_{\ 5 \ 2}^{\ 5 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 1}^{\ 5 \ 3 \ 2}}}-{�� \ {|_{\ 5 \ 0}^{\ 5 \ 3 \ 3}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 3 \ 4}}}-{�� \ {|_{\ 5 \ 6}^{\ 5 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 5}^{\ 5 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 4 \ 0}}}-{�� \ {|_{\ 5 \ 5}^{\ 5 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 5 \ 4 \ 2}}}-{�� \ {|_{\ 5 \ 7}^{\ 5 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 0}^{\ 5 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 5 \ 1}^{\ 5 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 5 \ 2}^{\ 5 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 5 \ 3}^{\ 5 \ 4 \ 7}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 5 \ 1}}}-{�� \ {|_{\ 5 \ 7}^{\ 5 \ 5 \ 2}}}+{�� \ {|_{\ 5 \ 6}^{\ 5 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 1}^{\ 5 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 5 \ 0}^{\ 5 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 3}^{\ 5 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 2}^{\ 5 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 5 \ 6}^{\ 5 \ 6 \ 0}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 6 \ 2}}}-{�� \ {|_{\ 5 \ 5}^{\ 5 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 5 \ 2}^{\ 5 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 3}^{\ 5 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 0}^{\ 5 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 1}^{\ 5 \ 6 \ 7}}}+{�� \ {|_{\ 5 \ 7}^{\ 5 \ 7 \ 0}}}-{�� \ {|_{\ 5 \ 6}^{\ 5 \ 7 \ 1}}}+{�� \ {|_{\ 5 \ 5}^{\ 5 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 5 \ 4}^{\ 5 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 3}^{\ 5 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 2}^{\ 5 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 5 \ 1}^{\ 5 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 5 \ 0}^{\ 5 \ 7 \ 7}}}+{�� \ {|_{\ 6 \ 0}^{\ 6 \ 0 \ 0}}}+{�� \ {|_{\ 6 \ 1}^{\ 6 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 2}^{\ 6 \ 0 \ 2}}}+{�� \ {|_{\ 6 \ 3}^{\ 6 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 0 \ 4}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 6 \ 0 \ 6}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 0 \ 7}}}+{�� \ {|_{\ 6 \ 1}^{\ 6 \ 1 \ 0}}}-{�� \ {|_{\ 6 \ 0}^{\ 6 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 3}^{\ 6 \ 1 \ 2}}}-{�� \ {|_{\ 6 \ 2}^{\ 6 \ 1 \ 3}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 7}^{\ 6 \ 1 \ 6}}}+{�� \ {|_{\ 6 \ 6}^{\ 6 \ 1 \ 7}}}+{�� \ {|_{\ 6 \ 2}^{\ 6 \ 2 \ 0}}}-{�� \ {|_{\ 6 \ 3}^{\ 6 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 0}^{\ 6 \ 2 \ 2}}}+{�� \ {|_{\ 6 \ 1}^{\ 6 \ 2 \ 3}}}+{�� \ {|_{\ 6 \ 6}^{\ 6 \ 2 \ 4}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 2 \ 6}}}-{�� \ {|_{\ 6 \ 5}^{\ 6 \ 2 \ 7}}}+{�� \ {|_{\ 6 \ 3}^{\ 6 \ 3 \ 0}}}+{�� \ {|_{\ 6 \ 2}^{\ 6 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 1}^{\ 6 \ 3 \ 2}}}-{�� \ {|_{\ 6 \ 0}^{\ 6 \ 3 \ 3}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 3 \ 4}}}-{�� \ {|_{\ 6 \ 6}^{\ 6 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 5}^{\ 6 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 4 \ 0}}}-{�� \ {|_{\ 6 \ 5}^{\ 6 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 6 \ 4 \ 2}}}-{�� \ {|_{\ 6 \ 7}^{\ 6 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 0}^{\ 6 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 6 \ 1}^{\ 6 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 6 \ 2}^{\ 6 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 6 \ 3}^{\ 6 \ 4 \ 7}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 5 \ 1}}}-{�� \ {|_{\ 6 \ 7}^{\ 6 \ 5 \ 2}}}+{�� \ {|_{\ 6 \ 6}^{\ 6 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 1}^{\ 6 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 6 \ 0}^{\ 6 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 3}^{\ 6 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 2}^{\ 6 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6 \ 6}^{\ 6 \ 6 \ 0}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 6 \ 2}}}-{�� \ {|_{\ 6 \ 5}^{\ 6 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 6 \ 2}^{\ 6 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 3}^{\ 6 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 0}^{\ 6 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 1}^{\ 6 \ 6 \ 7}}}+{�� \ {|_{\ 6 \ 7}^{\ 6 \ 7 \ 0}}}-{�� \ {|_{\ 6 \ 6}^{\ 6 \ 7 \ 1}}}+{�� \ {|_{\ 6 \ 5}^{\ 6 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 6 \ 4}^{\ 6 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 3}^{\ 6 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 2}^{\ 6 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 6 \ 1}^{\ 6 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 6 \ 0}^{\ 6 \ 7 \ 7}}}+{�� \ {|_{\ 7 \ 0}^{\ 7 \ 0 \ 0}}}+{�� \ {|_{\ 7 \ 1}^{\ 7 \ 0 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 2}^{\ 7 \ 0 \ 2}}}+{�� \ {|_{\ 7 \ 3}^{\ 7 \ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 0 \ 4}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 7 \ 0 \ 6}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 0 \ 7}}}+{�� \ {|_{\ 7 \ 1}^{\ 7 \ 1 \ 0}}}-{�� \ {|_{\ 7 \ 0}^{\ 7 \ 1 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 3}^{\ 7 \ 1 \ 2}}}-{�� \ {|_{\ 7 \ 2}^{\ 7 \ 1 \ 3}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 1 \ 4}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 1 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 7}^{\ 7 \ 1 \ 6}}}+{�� \ {|_{\ 7 \ 6}^{\ 7 \ 1 \ 7}}}+{�� \ {|_{\ 7 \ 2}^{\ 7 \ 2 \ 0}}}-{�� \ {|_{\ 7 \ 3}^{\ 7 \ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 0}^{\ 7 \ 2 \ 2}}}+{�� \ {|_{\ 7 \ 1}^{\ 7 \ 2 \ 3}}}+{�� \ {|_{\ 7 \ 6}^{\ 7 \ 2 \ 4}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 2 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 2 \ 6}}}-{�� \ {|_{\ 7 \ 5}^{\ 7 \ 2 \ 7}}}+{�� \ {|_{\ 7 \ 3}^{\ 7 \ 3 \ 0}}}+{�� \ {|_{\ 7 \ 2}^{\ 7 \ 3 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 1}^{\ 7 \ 3 \ 2}}}-{�� \ {|_{\ 7 \ 0}^{\ 7 \ 3 \ 3}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 3 \ 4}}}-{�� \ {|_{\ 7 \ 6}^{\ 7 \ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 5}^{\ 7 \ 3 \ 6}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 3 \ 7}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 4 \ 0}}}-{�� \ {|_{\ 7 \ 5}^{\ 7 \ 4 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 7 \ 4 \ 2}}}-{�� \ {|_{\ 7 \ 7}^{\ 7 \ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 0}^{\ 7 \ 4 \ 4}}}+{{q_{2}}\ �� \ {|_{\ 7 \ 1}^{\ 7 \ 4 \ 5}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 7 \ 2}^{\ 7 \ 4 \ 6}}}+{{q_{2}}\ �� \ {|_{\ 7 \ 3}^{\ 7 \ 4 \ 7}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 5 \ 0}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 5 \ 1}}}-{�� \ {|_{\ 7 \ 7}^{\ 7 \ 5 \ 2}}}+{�� \ {|_{\ 7 \ 6}^{\ 7 \ 5 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 1}^{\ 7 \ 5 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 7 \ 0}^{\ 7 \ 5 \ 5}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 3}^{\ 7 \ 5 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 2}^{\ 7 \ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 7 \ 6}^{\ 7 \ 6 \ 0}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 6 \ 1}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 6 \ 2}}}-{�� \ {|_{\ 7 \ 5}^{\ 7 \ 6 \ 3}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 7 \ 2}^{\ 7 \ 6 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 3}^{\ 7 \ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 0}^{\ 7 \ 6 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 1}^{\ 7 \ 6 \ 7}}}+{�� \ {|_{\ 7 \ 7}^{\ 7 \ 7 \ 0}}}-{�� \ {|_{\ 7 \ 6}^{\ 7 \ 7 \ 1}}}+{�� \ {|_{\ 7 \ 5}^{\ 7 \ 7 \ 2}}}+ \
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 7 \ 4}^{\ 7 \ 7 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 3}^{\ 7 \ 7 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 2}^{\ 7 \ 7 \ 5}}}+
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 7 \ 1}^{\ 7 \ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 7 \ 0}^{\ 7 \ 7 \ 7}}}") | (31) |
fricas
test(Hr = H )
 | (32) |
Type: Boolean
fricas
Hl := (I,λ)/(Y, I);
2 + + - -- + 2 + arity warning: ---- 3 0 + - -- 0 2 +
fricas
test( Hl = H )
 | (33) |
Type: Boolean
fricas
test( Hr = Hl )
 | (34) |
Type: Boolean
Perhaps this is not too surprising since Octonion algebra is non-associative. Nevertheless Octonions are "Frobenius" in a more general sense because there is a non-degenerate associative pairing.
i = Unit of the algebra
fricas
i:=ⅇ.1
 | (35) |
fricas
test
i /
λ = Ω
0 0
- -
+ 0
arity warning: ---
+ 0
- -
+ 0
 | (36) |
Type: Boolean
Handle and handle element
fricas
Φ:ℒ :=
λ /
X /
Y
2
0 +
- --
0 2
+
arity warning: ----
2
0 +
- --
0 2
+
![\label{eq37}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0}^{\ 0 \ 0}}}+{�� \ {|_{\ 1}^{\ 0 \ 1}}}+{�� \ {|_{\ 2}^{\ 0 \ 2}}}+{�� \ {|_{\ 3}^{\ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 0 \ 4}}}+
\
\
\displaystyle
{�� \ {|_{\ 5}^{\ 0 \ 5}}}+{�� \ {|_{\ 6}^{\ 0 \ 6}}}+{�� \ {|_{\ 7}^{\ 0 \ 7}}}+{�� \ {|_{\ 1}^{\ 1 \ 0}}}-{�� \ {|_{\ 0}^{\ 1 \ 1}}}-{�� \ {|_{\ 3}^{\ 1 \ 2}}}+
\
\
\displaystyle
{�� \ {|_{\ 2}^{\ 1 \ 3}}}-{�� \ {|_{\ 5}^{\ 1 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 1 \ 5}}}+{�� \ {|_{\ 7}^{\ 1 \ 6}}}-{�� \ {|_{\ 6}^{\ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2}^{\ 2 \ 0}}}+{�� \ {|_{\ 3}^{\ 2 \ 1}}}-{�� \ {|_{\ 0}^{\ 2 \ 2}}}-{�� \ {|_{\ 1}^{\ 2 \ 3}}}-{�� \ {|_{\ 6}^{\ 2 \ 4}}}-{�� \ {|_{\ 7}^{\ 2 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 2 \ 6}}}+{�� \ {|_{\ 5}^{\ 2 \ 7}}}+{�� \ {|_{\ 3}^{\ 3 \ 0}}}-{�� \ {|_{\ 2}^{\ 3 \ 1}}}+{�� \ {|_{\ 1}^{\ 3 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 3 \ 3}}}-{�� \ {|_{\ 7}^{\ 3 \ 4}}}+{�� \ {|_{\ 6}^{\ 3 \ 5}}}-{�� \ {|_{\ 5}^{\ 3 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 3 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 4 \ 0}}}+{�� \ {|_{\ 5}^{\ 4 \ 1}}}+{�� \ {|_{\ 6}^{\ 4 \ 2}}}+{�� \ {|_{\ 7}^{\ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 4 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1}^{\ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 2}^{\ 4 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 3}^{\ 4 \ 7}}}+{�� \ {|_{\ 5}^{\ 5 \ 0}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 5 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7}^{\ 5 \ 2}}}-{�� \ {|_{\ 6}^{\ 5 \ 3}}}+{{q_{2}}\ �� \ {|_{\ 1}^{\ 5 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 5 \ 5}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3}^{\ 5 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2}^{\ 5 \ 7}}}+{�� \ {|_{\ 6}^{\ 6 \ 0}}}-{�� \ {|_{\ 7}^{\ 6 \ 1}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 6 \ 2}}}+{�� \ {|_{\ 5}^{\ 6 \ 3}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2}^{\ 6 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3}^{\ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 6 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1}^{\ 6 \ 7}}}+{�� \ {|_{\ 7}^{\ 7 \ 0}}}+
\
\
\displaystyle
{�� \ {|_{\ 6}^{\ 7 \ 1}}}-{�� \ {|_{\ 5}^{\ 7 \ 2}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 7 \ 3}}}+{{q_{2}}\ �� \ {|_{\ 3}^{\ 7 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2}^{\ 7 \ 5}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1}^{\ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 7 \ 7}}}](images/470934212364296485-16.0px.png "\label{eq37}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0}^{\ 0 \ 0}}}+{�� \ {|_{\ 1}^{\ 0 \ 1}}}+{�� \ {|_{\ 2}^{\ 0 \ 2}}}+{�� \ {|_{\ 3}^{\ 0 \ 3}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 0 \ 4}}}+
\
\
\displaystyle
{�� \ {|_{\ 5}^{\ 0 \ 5}}}+{�� \ {|_{\ 6}^{\ 0 \ 6}}}+{�� \ {|_{\ 7}^{\ 0 \ 7}}}+{�� \ {|_{\ 1}^{\ 1 \ 0}}}-{�� \ {|_{\ 0}^{\ 1 \ 1}}}-{�� \ {|_{\ 3}^{\ 1 \ 2}}}+
\
\
\displaystyle
{�� \ {|_{\ 2}^{\ 1 \ 3}}}-{�� \ {|_{\ 5}^{\ 1 \ 4}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 1 \ 5}}}+{�� \ {|_{\ 7}^{\ 1 \ 6}}}-{�� \ {|_{\ 6}^{\ 1 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 2}^{\ 2 \ 0}}}+{�� \ {|_{\ 3}^{\ 2 \ 1}}}-{�� \ {|_{\ 0}^{\ 2 \ 2}}}-{�� \ {|_{\ 1}^{\ 2 \ 3}}}-{�� \ {|_{\ 6}^{\ 2 \ 4}}}-{�� \ {|_{\ 7}^{\ 2 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 2 \ 6}}}+{�� \ {|_{\ 5}^{\ 2 \ 7}}}+{�� \ {|_{\ 3}^{\ 3 \ 0}}}-{�� \ {|_{\ 2}^{\ 3 \ 1}}}+{�� \ {|_{\ 1}^{\ 3 \ 2}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 3 \ 3}}}-{�� \ {|_{\ 7}^{\ 3 \ 4}}}+{�� \ {|_{\ 6}^{\ 3 \ 5}}}-{�� \ {|_{\ 5}^{\ 3 \ 6}}}+{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 3 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 4 \ 0}}}+{�� \ {|_{\ 5}^{\ 4 \ 1}}}+{�� \ {|_{\ 6}^{\ 4 \ 2}}}+{�� \ {|_{\ 7}^{\ 4 \ 3}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 4 \ 4}}}-
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 1}^{\ 4 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 2}^{\ 4 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 3}^{\ 4 \ 7}}}+{�� \ {|_{\ 5}^{\ 5 \ 0}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 5 \ 1}}}+
\
\
\displaystyle
{�� \ {|_{\ 7}^{\ 5 \ 2}}}-{�� \ {|_{\ 6}^{\ 5 \ 3}}}+{{q_{2}}\ �� \ {|_{\ 1}^{\ 5 \ 4}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 5 \ 5}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 3}^{\ 5 \ 6}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 2}^{\ 5 \ 7}}}+{�� \ {|_{\ 6}^{\ 6 \ 0}}}-{�� \ {|_{\ 7}^{\ 6 \ 1}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 6 \ 2}}}+{�� \ {|_{\ 5}^{\ 6 \ 3}}}+
\
\
\displaystyle
{{q_{2}}\ �� \ {|_{\ 2}^{\ 6 \ 4}}}-{�� \ {\overline{q_{2}}}\ {|_{\ 3}^{\ 6 \ 5}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 6 \ 6}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 1}^{\ 6 \ 7}}}+{�� \ {|_{\ 7}^{\ 7 \ 0}}}+
\
\
\displaystyle
{�� \ {|_{\ 6}^{\ 7 \ 1}}}-{�� \ {|_{\ 5}^{\ 7 \ 2}}}-{{{{q_{2}}\ ��}\over{\overline{q_{2}}}}\ {|_{\ 4}^{\ 7 \ 3}}}+{{q_{2}}\ �� \ {|_{\ 3}^{\ 7 \ 4}}}+{�� \ {\overline{q_{2}}}\ {|_{\ 2}^{\ 7 \ 5}}}-
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\ 1}^{\ 7 \ 6}}}-{{q_{2}}\ �� \ {|_{\ 0}^{\ 7 \ 7}}}") | (37) |
fricas
Φ1:= i /
Φ
0 +
- -
+ +
arity warning: ----
2
0 +
- --
0 +
 | (38) |
Definition 5
i U
fricas
ι:ℒ:=
( i I ) /
( Ų )
+ 0
-- -
2 0
+
arity warning: ----
2
+ 0
-- -
2 0
+
![\label{eq39}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0 \ 0}^{\ 0}}}+{�� \ {|_{\ 0 \ 1}^{\ 1}}}+{�� \ {|_{\ 0 \ 2}^{\ 2}}}+{�� \ {|_{\ 0 \ 3}^{\ 3}}}+{�� \ {|_{\ 0 \ 4}^{\ 4}}}+{�� \ {|_{\ 0 \ 5}^{\ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 6}}}+{�� \ {|_{\ 0 \ 7}^{\ 7}}}](images/4320441085660920122-16.0px.png "\label{eq39}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0 \ 0}^{\ 0}}}+{�� \ {|_{\ 0 \ 1}^{\ 1}}}+{�� \ {|_{\ 0 \ 2}^{\ 2}}}+{�� \ {|_{\ 0 \ 3}^{\ 3}}}+{�� \ {|_{\ 0 \ 4}^{\ 4}}}+{�� \ {|_{\ 0 \ 5}^{\ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 0 \ 6}^{\ 6}}}+{�� \ {|_{\ 0 \ 7}^{\ 7}}}") | (39) |
Y=U ι
fricas
test
Y /
ι = Ų
 | (40) |
Type: Boolean
For example:
fricas
ex1:=[q[2]=1,p[1]=1]
![\label{eq41}\left[{{q_{2}}= 1}, \:{{p_{1}}= 1}\right]](images/2285156300645507468-16.0px.png "\label{eq41}\left[{{q_{2}}= 1}, \:{{p_{1}}= 1}\right]") | (41) |
Type: List(Equation(Polynomial(Integer)))
fricas
Ų0:ℒ :=eval(Ų,ex1)
 | (42) |
fricas
Ω0:ℒ :=eval(Ω,ex1)$ℒ
 | (43) |
fricas
λ0:ℒ :=eval(λ,ex1)$ℒ
 | (44) |
fricas
H0:ℒ :=eval(H,ex1)$ℒ
![\label{eq45}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0}^{\ 0 \ 0}}}+{�� \ {|_{\ 1}^{\ 0 \ 1}}}+{�� \ {|_{\ 2}^{\ 0 \ 2}}}+{�� \ {|_{\ 3}^{\ 0 \ 3}}}+{�� \ {|_{\ 4}^{\ 0 \ 4}}}+{�� \ {|_{\ 5}^{\ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6}^{\ 0 \ 6}}}+{�� \ {|_{\ 7}^{\ 0 \ 7}}}+{�� \ {|_{\ 1}^{\ 1 \ 0}}}-{�� \ {|_{\ 0}^{\ 1 \ 1}}}+{�� \ {|_{\ 3}^{\ 1 \ 2}}}-{�� \ {|_{\ 2}^{\ 1 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 5}^{\ 1 \ 4}}}-{�� \ {|_{\ 4}^{\ 1 \ 5}}}-{�� \ {|_{\ 7}^{\ 1 \ 6}}}+{�� \ {|_{\ 6}^{\ 1 \ 7}}}+{�� \ {|_{\ 2}^{\ 2 \ 0}}}-{�� \ {|_{\ 3}^{\ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 2 \ 2}}}+{�� \ {|_{\ 1}^{\ 2 \ 3}}}+{�� \ {|_{\ 6}^{\ 2 \ 4}}}+{�� \ {|_{\ 7}^{\ 2 \ 5}}}-{�� \ {|_{\ 4}^{\ 2 \ 6}}}-{�� \ {|_{\ 5}^{\ 2 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 3}^{\ 3 \ 0}}}+{�� \ {|_{\ 2}^{\ 3 \ 1}}}-{�� \ {|_{\ 1}^{\ 3 \ 2}}}-{�� \ {|_{\ 0}^{\ 3 \ 3}}}+{�� \ {|_{\ 7}^{\ 3 \ 4}}}-{�� \ {|_{\ 6}^{\ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5}^{\ 3 \ 6}}}-{�� \ {|_{\ 4}^{\ 3 \ 7}}}+{�� \ {|_{\ 4}^{\ 4 \ 0}}}-{�� \ {|_{\ 5}^{\ 4 \ 1}}}-{�� \ {|_{\ 6}^{\ 4 \ 2}}}-{�� \ {|_{\ 7}^{\ 4 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 4 \ 4}}}+{�� \ {|_{\ 1}^{\ 4 \ 5}}}+{�� \ {|_{\ 2}^{\ 4 \ 6}}}+{�� \ {|_{\ 3}^{\ 4 \ 7}}}+{�� \ {|_{\ 5}^{\ 5 \ 0}}}+{�� \ {|_{\ 4}^{\ 5 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7}^{\ 5 \ 2}}}+{�� \ {|_{\ 6}^{\ 5 \ 3}}}-{�� \ {|_{\ 1}^{\ 5 \ 4}}}-{�� \ {|_{\ 0}^{\ 5 \ 5}}}-{�� \ {|_{\ 3}^{\ 5 \ 6}}}+{�� \ {|_{\ 2}^{\ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6}^{\ 6 \ 0}}}+{�� \ {|_{\ 7}^{\ 6 \ 1}}}+{�� \ {|_{\ 4}^{\ 6 \ 2}}}-{�� \ {|_{\ 5}^{\ 6 \ 3}}}-{�� \ {|_{\ 2}^{\ 6 \ 4}}}+{�� \ {|_{\ 3}^{\ 6 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 6 \ 6}}}-{�� \ {|_{\ 1}^{\ 6 \ 7}}}+{�� \ {|_{\ 7}^{\ 7 \ 0}}}-{�� \ {|_{\ 6}^{\ 7 \ 1}}}+{�� \ {|_{\ 5}^{\ 7 \ 2}}}+{�� \ {|_{\ 4}^{\ 7 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 3}^{\ 7 \ 4}}}-{�� \ {|_{\ 2}^{\ 7 \ 5}}}+{�� \ {|_{\ 1}^{\ 7 \ 6}}}-{�� \ {|_{\ 0}^{\ 7 \ 7}}}](images/780995624430917616-16.0px.png "\label{eq45}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\ 0}^{\ 0 \ 0}}}+{�� \ {|_{\ 1}^{\ 0 \ 1}}}+{�� \ {|_{\ 2}^{\ 0 \ 2}}}+{�� \ {|_{\ 3}^{\ 0 \ 3}}}+{�� \ {|_{\ 4}^{\ 0 \ 4}}}+{�� \ {|_{\ 5}^{\ 0 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 6}^{\ 0 \ 6}}}+{�� \ {|_{\ 7}^{\ 0 \ 7}}}+{�� \ {|_{\ 1}^{\ 1 \ 0}}}-{�� \ {|_{\ 0}^{\ 1 \ 1}}}+{�� \ {|_{\ 3}^{\ 1 \ 2}}}-{�� \ {|_{\ 2}^{\ 1 \ 3}}}+
\
\
\displaystyle
{�� \ {|_{\ 5}^{\ 1 \ 4}}}-{�� \ {|_{\ 4}^{\ 1 \ 5}}}-{�� \ {|_{\ 7}^{\ 1 \ 6}}}+{�� \ {|_{\ 6}^{\ 1 \ 7}}}+{�� \ {|_{\ 2}^{\ 2 \ 0}}}-{�� \ {|_{\ 3}^{\ 2 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 2 \ 2}}}+{�� \ {|_{\ 1}^{\ 2 \ 3}}}+{�� \ {|_{\ 6}^{\ 2 \ 4}}}+{�� \ {|_{\ 7}^{\ 2 \ 5}}}-{�� \ {|_{\ 4}^{\ 2 \ 6}}}-{�� \ {|_{\ 5}^{\ 2 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 3}^{\ 3 \ 0}}}+{�� \ {|_{\ 2}^{\ 3 \ 1}}}-{�� \ {|_{\ 1}^{\ 3 \ 2}}}-{�� \ {|_{\ 0}^{\ 3 \ 3}}}+{�� \ {|_{\ 7}^{\ 3 \ 4}}}-{�� \ {|_{\ 6}^{\ 3 \ 5}}}+
\
\
\displaystyle
{�� \ {|_{\ 5}^{\ 3 \ 6}}}-{�� \ {|_{\ 4}^{\ 3 \ 7}}}+{�� \ {|_{\ 4}^{\ 4 \ 0}}}-{�� \ {|_{\ 5}^{\ 4 \ 1}}}-{�� \ {|_{\ 6}^{\ 4 \ 2}}}-{�� \ {|_{\ 7}^{\ 4 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 4 \ 4}}}+{�� \ {|_{\ 1}^{\ 4 \ 5}}}+{�� \ {|_{\ 2}^{\ 4 \ 6}}}+{�� \ {|_{\ 3}^{\ 4 \ 7}}}+{�� \ {|_{\ 5}^{\ 5 \ 0}}}+{�� \ {|_{\ 4}^{\ 5 \ 1}}}-
\
\
\displaystyle
{�� \ {|_{\ 7}^{\ 5 \ 2}}}+{�� \ {|_{\ 6}^{\ 5 \ 3}}}-{�� \ {|_{\ 1}^{\ 5 \ 4}}}-{�� \ {|_{\ 0}^{\ 5 \ 5}}}-{�� \ {|_{\ 3}^{\ 5 \ 6}}}+{�� \ {|_{\ 2}^{\ 5 \ 7}}}+
\
\
\displaystyle
{�� \ {|_{\ 6}^{\ 6 \ 0}}}+{�� \ {|_{\ 7}^{\ 6 \ 1}}}+{�� \ {|_{\ 4}^{\ 6 \ 2}}}-{�� \ {|_{\ 5}^{\ 6 \ 3}}}-{�� \ {|_{\ 2}^{\ 6 \ 4}}}+{�� \ {|_{\ 3}^{\ 6 \ 5}}}-
\
\
\displaystyle
{�� \ {|_{\ 0}^{\ 6 \ 6}}}-{�� \ {|_{\ 1}^{\ 6 \ 7}}}+{�� \ {|_{\ 7}^{\ 7 \ 0}}}-{�� \ {|_{\ 6}^{\ 7 \ 1}}}+{�� \ {|_{\ 5}^{\ 7 \ 2}}}+{�� \ {|_{\ 4}^{\ 7 \ 3}}}-
\
\
\displaystyle
{�� \ {|_{\ 3}^{\ 7 \ 4}}}-{�� \ {|_{\ 2}^{\ 7 \ 5}}}+{�� \ {|_{\ 1}^{\ 7 \ 6}}}-{�� \ {|_{\ 0}^{\ 7 \ 7}}}") | (45) |