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Edit detail for Octonion Algebra is Frobenius in Just One Way revision 7 of 8

1 2 3 4 5 6 7 8
Editor: Bill Page
Time: 2011/05/06 10:17:07 GMT-7
Note: use new LOP

changed:
-)library CARTEN MONAL PROP LIN CALEY
-\end{axiom}
)library CARTEN MONAL PROP LOP CALEY
\end{axiom}

changed:
-𝐋 := LinearOperator(dim, OVAR [], ℚ)
-𝐞:ℒ 𝐋      := basisVectors()
-𝐝:ℒ 𝐋      := basisForms()
𝐋 := LinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7], ℚ)
𝐞:ℒ 𝐋      := basisOut()
𝐝:ℒ 𝐋      := basisIn()

Octonion Algebra Is Frobenius In Just One Way

Linear operators over a 8-dimensional vector space representing octonnion algebra

Ref:

We need the Axiom LinearOperator? library.

axiom
)library CARTEN MONAL PROP LOP CALEY
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/zope2/var/LatexWiki/CARTEN.NRLIB/CARTEN Monoidal is now explicitly exposed in frame initial Monoidal will be automatically loaded when needed from /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL Prop is now explicitly exposed in frame initial Prop will be automatically loaded when needed from /var/zope2/var/LatexWiki/PROP.NRLIB/PROP LinearOperator is now explicitly exposed in frame initial LinearOperator will be automatically loaded when needed from /var/zope2/var/LatexWiki/LOP.NRLIB/LOP CaleyDickson is now explicitly exposed in frame initial CaleyDickson will be automatically loaded when needed from /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY

Use the following macros for convenient notation

axiom
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
axiom
-- list
macro Ξ(f,i,n)==[f for i in n]
Type: Void
axiom
-- 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.

axiom
dim:=8

\label{eq1}8(1)
Type: PositiveInteger?
axiom
macro ℒ == List
Type: Void
axiom
macro ℂ == CaleyDickson
Type: Void
axiom
macro ℚ == Expression Integer
Type: Void
axiom
𝐋 := LinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7], ℚ)

\label{eq2}\hbox{\axiomType{LinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 0, 1, 2, 3, 4, 5, 6, 7 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(2)
Type: Type
axiom
𝐞:ℒ 𝐋      := basisOut()

\label{eq3}\left[{|_{0}}, \:{|_{1}}, \:{|_{2}}, \:{|_{3}}, \:{|_{4}}, \:{|_{5}}, \:{|_{6}}, \:{|_{7}}\right](3)
Type: List(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))
axiom
𝐝:ℒ 𝐋      := basisIn()

\label{eq4}\left[{|_{\ }^{0}}, \:{|_{\ }^{1}}, \:{|_{\ }^{2}}, \:{|_{\ }^{3}}, \:{|_{\ }^{4}}, \:{|_{\ }^{5}}, \:{|_{\ }^{6}}, \:{|_{\ }^{7}}\right](4)
Type: List(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))
axiom
I:𝐋:=[1];   -- identity for composition
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
X:𝐋:=[2,1]; -- twist
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

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

axiom
--q0:=sb('q,[0])
q0:=1  -- not split-complex

\label{eq5}1(5)
Type: PositiveInteger?
axiom
--q1:=sb('q,[1])
q1:=1  -- not co-quaternion

\label{eq6}1(6)
Type: PositiveInteger?
axiom
q2:=sb('q,[2])

\label{eq7}q_{2}(7)
Type: Symbol
axiom
--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

axiom
B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ)

\label{eq8}\left[ 1, \: i , \: j , \:{ij}, \: k , \:{ik}, \:{jk}, \:{{ij}k}\right](8)
Type: List(CaleyDickson?(CaleyDickson?(CaleyDickson?(Expression(Integer),i,1),j,1),k,*01q(2)))
axiom
-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

\label{eq9}\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}&{{-{q_{2}}i}j}
\
{ik}& - k &{{ij}k}& -{jk}&{{q_{2}}i}& -{q_{2}}&{{{q_{2}}i}j}&{-{q_{2}}j}
\
{jk}&{-{ij}k}& - k &{ik}&{{q_{2}}j}&{{-{q_{2}}i}j}& -{q_{2}}&{{q_{2}}i}
\
{{ij}k}&{jk}&{- ik}& - k &{{{q_{2}}i}j}&{{q_{2}}j}&{-{q_{2}}i}& -{q_{2}}
(9)
Type: Matrix(CaleyDickson?(CaleyDickson?(CaleyDickson?(Expression(Integer),i,1),j,1),k,*01q(2)))
axiom
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real real(x/y),M)
Type: Void
axiom
-- The result is a nested list
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ;
axiom
Compiling function S with type CaleyDickson(CaleyDickson(
      CaleyDickson(Expression(Integer),i,1),j,1),k,*01q(2)) -> Matrix(
      Expression(Integer))
Type: List(List(List(Expression(Integer))))
axiom
-- 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{eq10}\begin{array}{@{}l}
\displaystyle
{|_{0}^{0 \  0}}+{|_{1}^{0 \  1}}+{|_{2}^{0 \  2}}+{|_{3}^{0 \  3}}+{|_{4}^{0 \  4}}+{|_{5}^{0 \  5}}+{|_{6}^{0 \  6}}+{|_{7}^{0 \  7}}+ 
\
\
\displaystyle
{|_{1}^{1 \  0}}-{|_{0}^{1 \  1}}+{|_{3}^{1 \  2}}-{|_{2}^{1 \  3}}+{|_{5}^{1 \  4}}-{|_{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}}-{|_{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}}+{|_{5}^{3 \  6}}-{|_{4}^{3 \  7}}+ 
\
\
\displaystyle
{|_{4}^{4 \  0}}-{|_{5}^{4 \  1}}-{|_{6}^{4 \  2}}-{|_{7}^{4 \  3}}-{{q_{2}}\ {|_{0}^{4 \  4}}}+{{q_{2}}\ {|_{1}^{4 \  5}}}+{{q_{2}}\ {|_{2}^{4 \  6}}}+ 
\
\
\displaystyle
{{q_{2}}\ {|_{3}^{4 \  7}}}+{|_{5}^{5 \  0}}+{|_{4}^{5 \  1}}-{|_{7}^{5 \  2}}+{|_{6}^{5 \  3}}-{{q_{2}}\ {|_{1}^{5 \  4}}}-{{q_{2}}\ {|_{0}^{5 \  5}}}- 
\
\
\displaystyle
{{q_{2}}\ {|_{3}^{5 \  6}}}+{{q_{2}}\ {|_{2}^{5 \  7}}}+{|_{6}^{6 \  0}}+{|_{7}^{6 \  1}}+{|_{4}^{6 \  2}}-{|_{5}^{6 \  3}}-{{q_{2}}\ {|_{2}^{6 \  4}}}+ 
\
\
\displaystyle
{{q_{2}}\ {|_{3}^{6 \  5}}}-{{q_{2}}\ {|_{0}^{6 \  6}}}-{{q_{2}}\ {|_{1}^{6 \  7}}}+{|_{7}^{7 \  0}}-{|_{6}^{7 \  1}}+{|_{5}^{7 \  2}}+{|_{4}^{7 \  3}}- 
\
\
\displaystyle
{{q_{2}}\ {|_{3}^{7 \  4}}}-{{q_{2}}\ {|_{2}^{7 \  5}}}+{{q_{2}}\ {|_{1}^{7 \  6}}}-{{q_{2}}\ {|_{0}^{7 \  7}}}
(10)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
arity Y

\label{eq11}2 \over 1(11)
Type: Prop(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))
axiom
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)

\label{eq12}\left[ 
\begin{array}{cccccccc}
{|_{0}}&{|_{1}}&{|_{2}}&{|_{3}}&{|_{4}}&{|_{5}}&{|_{6}}&{|_{7}}
\
{|_{1}}& -{|_{0}}& -{|_{3}}&{|_{2}}& -{|_{5}}&{|_{4}}&{|_{7}}& -{|_{6}}
\
{|_{2}}&{|_{3}}& -{|_{0}}& -{|_{1}}& -{|_{6}}& -{|_{7}}&{|_{4}}&{|_{5}}
\
{|_{3}}& -{|_{2}}&{|_{1}}& -{|_{0}}& -{|_{7}}&{|_{6}}& -{|_{5}}&{|_{4}}
\
{|_{4}}&{|_{5}}&{|_{6}}&{|_{7}}& -{{q_{2}}\ {|_{0}}}& -{{q_{2}}\ {|_{1}}}& -{{q_{2}}\ {|_{2}}}& -{{q_{2}}\ {|_{3}}}
\
{|_{5}}& -{|_{4}}&{|_{7}}& -{|_{6}}&{{q_{2}}\ {|_{1}}}& -{{q_{2}}\ {|_{0}}}&{{q_{2}}\ {|_{3}}}& -{{q_{2}}\ {|_{2}}}
\
{|_{6}}& -{|_{7}}& -{|_{4}}&{|_{5}}&{{q_{2}}\ {|_{2}}}& -{{q_{2}}\ {|_{3}}}& -{{q_{2}}\ {|_{0}}}&{{q_{2}}\ {|_{1}}}
\
{|_{7}}&{|_{6}}& -{|_{5}}& -{|_{4}}&{{q_{2}}\ {|_{3}}}&{{q_{2}}\ {|_{2}}}& -{{q_{2}}\ {|_{1}}}& -{{q_{2}}\ {|_{0}}}
(12)
Type: Matrix(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))

A scalar product is denoted by the (2,0)-tensor U = \{ u_{ij} \}

axiom
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

\label{eq13}\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}}}
(13)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

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:


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\label{eq14}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (14)
(three-point function) is zero.

Using the LinearOperator? domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.

axiom
ω:𝐋 :=(Y*I)/U  - (I*Y)/U;
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

Definition 2

An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.

We may consider the problem where multiplication Y is given, and look for all associative scalar products U = U(Y)

This problem can be solved using linear algebra.

axiom
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);
Type: Matrix(Expression(Integer))
axiom
u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];
Type: Matrix(Polynomial(Integer))
axiom
J::OutputForm * u::OutputForm = 0

\label{eq15}(15)
Type: Equation(OutputForm?)
axiom
nrows(J),ncols(J)

\label{eq16}\left[{512}, \:{64}\right](16)
Type: Tuple(PositiveInteger?)

The matrix J transforms the coefficients of the tensor U into coefficients of the tensor \Phi. We are looking for the general linear family of tensors U=U(Y,p_i) such that J transforms U into \Phi=0 for any such U.

If the null space of the J matrix is not empty we can use the basis to find all non-trivial solutions for U:

axiom
Ñ:=nullSpace(J);
Type: List(Vector(Expression(Integer)))
axiom
ℰ:=map((x,y)+->x=y, concat
       map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) )

\label{eq17}\begin{array}{@{}l}
\displaystyle
\left[{{u^{1, \: 1}}= -{{p_{1}}\over{q_{2}}}}, \:{{u^{1, \: 2}}= 0}, \:{{u^{1, \: 3}}= 0}, \:{{u^{1, \: 4}}= 0}, \:{{u^{1, \: 5}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{1, \: 6}}= 0}, \:{{u^{1, \: 7}}= 0}, \:{{u^{1, \: 8}}= 0}, \:{{u^{2, \: 1}}= 0}, \:{{u^{2, \: 2}}={{p_{1}}\over{q_{2}}}}, \: \right.
\
\
\displaystyle
\left.{{u^{2, \: 3}}= 0}, \:{{u^{2, \: 4}}= 0}, \:{{u^{2, \: 5}}= 0}, \:{{u^{2, \: 6}}= 0}, \:{{u^{2, \: 7}}= 0}, \:{{u^{2, \: 8}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{3, \: 1}}= 0}, \:{{u^{3, \: 2}}= 0}, \:{{u^{3, \: 3}}={{p_{1}}\over{q_{2}}}}, \:{{u^{3, \: 4}}= 0}, \:{{u^{3, \: 5}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{3, \: 6}}= 0}, \:{{u^{3, \: 7}}= 0}, \:{{u^{3, \: 8}}= 0}, \:{{u^{4, \: 1}}= 0}, \:{{u^{4, \: 2}}= 0}, \:{{u^{4, \: 3}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{4, \: 4}}={{p_{1}}\over{q_{2}}}}, \:{{u^{4, \: 5}}= 0}, \:{{u^{4, \: 6}}= 0}, \:{{u^{4, \: 7}}= 0}, \:{{u^{4, \: 8}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{5, \: 1}}= 0}, \:{{u^{5, \: 2}}= 0}, \:{{u^{5, \: 3}}= 0}, \:{{u^{5, \: 4}}= 0}, \:{{u^{5, \: 5}}={p_{1}}}, \: \right.
\
\
\displaystyle
\left.{{u^{5, \: 6}}= 0}, \:{{u^{5, \: 7}}= 0}, \:{{u^{5, \: 8}}= 0}, \:{{u^{6, \: 1}}= 0}, \:{{u^{6, \: 2}}= 0}, \:{{u^{6, \: 3}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{6, \: 4}}= 0}, \:{{u^{6, \: 5}}= 0}, \:{{u^{6, \: 6}}={p_{1}}}, \:{{u^{6, \: 7}}= 0}, \:{{u^{6, \: 8}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{7, \: 1}}= 0}, \:{{u^{7, \: 2}}= 0}, \:{{u^{7, \: 3}}= 0}, \:{{u^{7, \: 4}}= 0}, \:{{u^{7, \: 5}}= 0}, \:{{u^{7, \: 6}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{7, \: 7}}={p_{1}}}, \:{{u^{7, \: 8}}= 0}, \:{{u^{8, \: 1}}= 0}, \:{{u^{8, \: 2}}= 0}, \:{{u^{8, \: 3}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{8, \: 4}}= 0}, \:{{u^{8, \: 5}}= 0}, \:{{u^{8, \: 6}}= 0}, \:{{u^{8, \: 7}}= 0}, \:{{u^{8, \: 8}}={p_{1}}}\right] (17)
Type: List(Equation(Expression(Integer)))

This defines a family of Frobenius algebras:

axiom
zero? eval(ω,ℰ)

\label{eq18} \mbox{\rm true} (18)
Type: Boolean

The pairing is necessarily diagonal!

axiom
Ų:𝐋 := eval(U,ℰ)

\label{eq19}\begin{array}{@{}l}
\displaystyle
-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{0 \  0}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \  1}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \  2}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \  3}}}+{{p_{1}}\ {|_{\ }^{4 \  4}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{5 \  5}}}+{{p_{1}}\ {|_{\ }^{6 \  6}}}+{{p_{1}}\ {|_{\ }^{7 \  7}}}
(19)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų, i,1..dim), j,1..dim)

\label{eq20}\left[ 
\begin{array}{cccccccc}
-{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}
(20)
Type: Matrix(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))

The scalar product must be non-degenerate:

axiom
Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,1..dim)

\label{eq21}-{{{p_{1}}^8}\over{{q_{2}}^4}}(21)
Type: Expression(Integer)
axiom
factor Ů

\label{eq22}-{{{p_{1}}^8}\over{{q_{2}}^4}}(22)
Type: Factored(Expression(Integer))

Definition 3

Co-pairing

Solve the [Snake Relation]? as a system of linear equations.

axiom
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/Ų, i,1..dim), j,1..dim)

\label{eq23}\left[ 
\begin{array}{cccccccc}
-{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}
(23)
Type: Matrix(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))
axiom
mU:=transpose inverse map(retract,Um)

\label{eq24}\left[ 
\begin{array}{cccccccc}
-{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}
(24)
Type: Matrix(Expression(Integer))
axiom
Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim)

\label{eq25}\begin{array}{@{}l}
\displaystyle
-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}}}+{{1 \over{p_{1}}}\ {|_{4 \  4}}}+{{1 \over{p_{1}}}\ {|_{5 \  5}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{6 \  6}}}+{{1 \over{p_{1}}}\ {|_{7 \  7}}}
(25)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)

\label{eq26}\left[ 
\begin{array}{cccccccc}
-{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}
(26)
Type: Matrix(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))

Check "dimension" and the snake relations.

axiom
d:𝐋:=
       Ω    /
       X    /
       Ų

\label{eq27}8(27)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
test
    (    I Ω     )  /
    (     Ų I    )  =  I

\label{eq28} \mbox{\rm true} (28)
Type: Boolean
axiom
test
    (     Ω I    )  /
    (    I Ų     )  =  I

\label{eq29} \mbox{\rm true} (29)
Type: Boolean

Definition 4

Co-algebra

Compute the "three-point" function and use it to define co-multiplication.

axiom
W:=(Y,I)/Ų

\label{eq30}\begin{array}{@{}l}
\displaystyle
-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{0 \  0 \  0}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{0 \  1 \  1}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{0 \  2 \  2}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{0 \  3 \  3}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{0 \  4 \  4}}}+{{p_{1}}\ {|_{\ }^{0 \  5 \  5}}}+{{p_{1}}\ {|_{\ }^{0 \  6 \  6}}}+{{p_{1}}\ {|_{\ }^{0 \  7 \  7}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \  0 \  1}}}+ 
\
\
\displaystyle
{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \  1 \  0}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \  2 \  3}}}-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \  3 \  2}}}+{{p_{1}}\ {|_{\ }^{1 \  4 \  5}}}-{{p_{1}}\ {|_{\ }^{1 \  5 \  4}}}- 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{1 \  6 \  7}}}+{{p_{1}}\ {|_{\ }^{1 \  7 \  6}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \  0 \  2}}}-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \  1 \  3}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \  2 \  0}}}+ 
\
\
\displaystyle
{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \  3 \  1}}}+{{p_{1}}\ {|_{\ }^{2 \  4 \  6}}}+{{p_{1}}\ {|_{\ }^{2 \  5 \  7}}}-{{p_{1}}\ {|_{\ }^{2 \  6 \  4}}}-{{p_{1}}\ {|_{\ }^{2 \  7 \  5}}}+ 
\
\
\displaystyle
{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \  0 \  3}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \  1 \  2}}}-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \  2 \  1}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \  3 \  0}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{3 \  4 \  7}}}-{{p_{1}}\ {|_{\ }^{3 \  5 \  6}}}+{{p_{1}}\ {|_{\ }^{3 \  6 \  5}}}-{{p_{1}}\ {|_{\ }^{3 \  7 \  4}}}+{{p_{1}}\ {|_{\ }^{4 \  0 \  4}}}- 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{4 \  1 \  5}}}-{{p_{1}}\ {|_{\ }^{4 \  2 \  6}}}-{{p_{1}}\ {|_{\ }^{4 \  3 \  7}}}+{{p_{1}}\ {|_{\ }^{4 \  4 \  0}}}+{{p_{1}}\ {|_{\ }^{4 \  5 \  1}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{4 \  6 \  2}}}+{{p_{1}}\ {|_{\ }^{4 \  7 \  3}}}+{{p_{1}}\ {|_{\ }^{5 \  0 \  5}}}+{{p_{1}}\ {|_{\ }^{5 \  1 \  4}}}-{{p_{1}}\ {|_{\ }^{5 \  2 \  7}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{5 \  3 \  6}}}-{{p_{1}}\ {|_{\ }^{5 \  4 \  1}}}+{{p_{1}}\ {|_{\ }^{5 \  5 \  0}}}-{{p_{1}}\ {|_{\ }^{5 \  6 \  3}}}+{{p_{1}}\ {|_{\ }^{5 \  7 \  2}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{6 \  0 \  6}}}+{{p_{1}}\ {|_{\ }^{6 \  1 \  7}}}+{{p_{1}}\ {|_{\ }^{6 \  2 \  4}}}-{{p_{1}}\ {|_{\ }^{6 \  3 \  5}}}-{{p_{1}}\ {|_{\ }^{6 \  4 \  2}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{6 \  5 \  3}}}+{{p_{1}}\ {|_{\ }^{6 \  6 \  0}}}-{{p_{1}}\ {|_{\ }^{6 \  7 \  1}}}+{{p_{1}}\ {|_{\ }^{7 \  0 \  7}}}-{{p_{1}}\ {|_{\ }^{7 \  1 \  6}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{7 \  2 \  5}}}+{{p_{1}}\ {|_{\ }^{7 \  3 \  4}}}-{{p_{1}}\ {|_{\ }^{7 \  4 \  3}}}-{{p_{1}}\ {|_{\ }^{7 \  5 \  2}}}+{{p_{1}}\ {|_{\ }^{7 \  6 \  1}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{7 \  7 \  0}}}
(30)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
λ:=(Ω,I,Ω)/(I,W,I)

\label{eq31}\begin{array}{@{}l}
\displaystyle
-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}^{0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}^{0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}^{0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}^{0}}}+{{1 \over{p_{1}}}\ {|_{4 \  4}^{0}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \  5}^{0}}}+{{1 \over{p_{1}}}\ {|_{6 \  6}^{0}}}+{{1 \over{p_{1}}}\ {|_{7 \  7}^{0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  1}^{1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  0}^{1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{2 \  3}^{1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  2}^{1}}}-{{1 \over{p_{1}}}\ {|_{4 \  5}^{1}}}+{{1 \over{p_{1}}}\ {|_{5 \  4}^{1}}}+{{1 \over{p_{1}}}\ {|_{6 \  7}^{1}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{7 \  6}^{1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  2}^{2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  3}^{2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  0}^{2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  1}^{2}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  6}^{2}}}-{{1 \over{p_{1}}}\ {|_{5 \  7}^{2}}}+{{1 \over{p_{1}}}\ {|_{6 \  4}^{2}}}+{{1 \over{p_{1}}}\ {|_{7 \  5}^{2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  3}^{3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \  2}^{3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  1}^{3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  0}^{3}}}-{{1 \over{p_{1}}}\ {|_{4 \  7}^{3}}}+{{1 \over{p_{1}}}\ {|_{5 \  6}^{3}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{6 \  5}^{3}}}+{{1 \over{p_{1}}}\ {|_{7 \  4}^{3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{4}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{7}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{7}}}
(31)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

axiom
test
     (    I Ω     )  /
     (     Y I    )  =  λ

\label{eq32} \mbox{\rm true} (32)
Type: Boolean
axiom
test
     (     Ω I    )  /
     (    I Y     )  =  λ

\label{eq33} \mbox{\rm true} (33)
Type: Boolean

Frobenius Condition

Octonion algebra fails the Frobenius Condition!

axiom
Χ :=
       Y    /
       λ

\label{eq34}\begin{array}{@{}l}
\displaystyle
-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}^{0 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}^{0 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}^{0 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}^{0 \  0}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  4}^{0 \  0}}}+{{1 \over{p_{1}}}\ {|_{5 \  5}^{0 \  0}}}+{{1 \over{p_{1}}}\ {|_{6 \  6}^{0 \  0}}}+{{1 \over{p_{1}}}\ {|_{7 \  7}^{0 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  1}^{0 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  0}^{0 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  3}^{0 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  2}^{0 \  1}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  5}^{0 \  1}}}+{{1 \over{p_{1}}}\ {|_{5 \  4}^{0 \  1}}}+{{1 \over{p_{1}}}\ {|_{6 \  7}^{0 \  1}}}-{{1 \over{p_{1}}}\ {|_{7 \  6}^{0 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  2}^{0 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  3}^{0 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  0}^{0 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  1}^{0 \  2}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  6}^{0 \  2}}}-{{1 \over{p_{1}}}\ {|_{5 \  7}^{0 \  2}}}+{{1 \over{p_{1}}}\ {|_{6 \  4}^{0 \  2}}}+{{1 \over{p_{1}}}\ {|_{7 \  5}^{0 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  3}^{0 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  2}^{0 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  1}^{0 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  0}^{0 \  3}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  7}^{0 \  3}}}+{{1 \over{p_{1}}}\ {|_{5 \  6}^{0 \  3}}}-{{1 \over{p_{1}}}\ {|_{6 \  5}^{0 \  3}}}+{{1 \over{p_{1}}}\ {|_{7 \  4}^{0 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{0 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{0 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{0 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{0 \  4}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{0 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{0 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{0 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{0 \  4}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{0 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{0 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{0 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{0 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{0 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{0 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{0 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{0 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{0 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{0 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{0 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{0 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{0 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{0 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{0 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{0 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{0 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{0 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{0 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{0 \  7}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{0 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{0 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{0 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{0 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  1}^{1 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  0}^{1 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  3}^{1 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  2}^{1 \  0}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  5}^{1 \  0}}}+{{1 \over{p_{1}}}\ {|_{5 \  4}^{1 \  0}}}+{{1 \over{p_{1}}}\ {|_{6 \  7}^{1 \  0}}}-{{1 \over{p_{1}}}\ {|_{7 \  6}^{1 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}^{1 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}^{1 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}^{1 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}^{1 \  1}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  4}^{1 \  1}}}-{{1 \over{p_{1}}}\ {|_{5 \  5}^{1 \  1}}}-{{1 \over{p_{1}}}\ {|_{6 \  6}^{1 \  1}}}-{{1 \over{p_{1}}}\ {|_{7 \  7}^{1 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  3}^{1 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  2}^{1 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  1}^{1 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  0}^{1 \  2}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  7}^{1 \  2}}}+{{1 \over{p_{1}}}\ {|_{5 \  6}^{1 \  2}}}-{{1 \over{p_{1}}}\ {|_{6 \  5}^{1 \  2}}}+{{1 \over{p_{1}}}\ {|_{7 \  4}^{1 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  2}^{1 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  3}^{1 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  0}^{1 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  1}^{1 \  3}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  6}^{1 \  3}}}+{{1 \over{p_{1}}}\ {|_{5 \  7}^{1 \  3}}}-{{1 \over{p_{1}}}\ {|_{6 \  4}^{1 \  3}}}-{{1 \over{p_{1}}}\ {|_{7 \  5}^{1 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{1 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{1 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{1 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{1 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{1 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{1 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{1 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{1 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{1 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{1 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{1 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{1 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{1 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{1 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{1 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{1 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{1 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{1 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{1 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{1 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{1 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{1 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{1 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{1 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{1 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{1 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{1 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{1 \  7}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{1 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{1 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{1 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{1 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  2}^{2 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  3}^{2 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  0}^{2 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  1}^{2 \  0}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  6}^{2 \  0}}}-{{1 \over{p_{1}}}\ {|_{5 \  7}^{2 \  0}}}+{{1 \over{p_{1}}}\ {|_{6 \  4}^{2 \  0}}}+{{1 \over{p_{1}}}\ {|_{7 \  5}^{2 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  3}^{2 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  2}^{2 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  1}^{2 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  0}^{2 \  1}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  7}^{2 \  1}}}-{{1 \over{p_{1}}}\ {|_{5 \  6}^{2 \  1}}}+{{1 \over{p_{1}}}\ {|_{6 \  5}^{2 \  1}}}-{{1 \over{p_{1}}}\ {|_{7 \  4}^{2 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}^{2 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}^{2 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}^{2 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}^{2 \  2}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  4}^{2 \  2}}}-{{1 \over{p_{1}}}\ {|_{5 \  5}^{2 \  2}}}-{{1 \over{p_{1}}}\ {|_{6 \  6}^{2 \  2}}}-{{1 \over{p_{1}}}\ {|_{7 \  7}^{2 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  1}^{2 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  0}^{2 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  3}^{2 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  2}^{2 \  3}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  5}^{2 \  3}}}+{{1 \over{p_{1}}}\ {|_{5 \  4}^{2 \  3}}}+{{1 \over{p_{1}}}\ {|_{6 \  7}^{2 \  3}}}-{{1 \over{p_{1}}}\ {|_{7 \  6}^{2 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{2 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{2 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{2 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{2 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{2 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{2 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{2 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{2 \  4}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{2 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{2 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{2 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{2 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{2 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{2 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{2 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{2 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{2 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{2 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{2 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{2 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{2 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{2 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{2 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{2 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{2 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{2 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{2 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{2 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{2 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{2 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{2 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{2 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  3}^{3 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  2}^{3 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  1}^{3 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  0}^{3 \  0}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  7}^{3 \  0}}}+{{1 \over{p_{1}}}\ {|_{5 \  6}^{3 \  0}}}-{{1 \over{p_{1}}}\ {|_{6 \  5}^{3 \  0}}}+{{1 \over{p_{1}}}\ {|_{7 \  4}^{3 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  2}^{3 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  3}^{3 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  0}^{3 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  1}^{3 \  1}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  6}^{3 \  1}}}-{{1 \over{p_{1}}}\ {|_{5 \  7}^{3 \  1}}}+{{1 \over{p_{1}}}\ {|_{6 \  4}^{3 \  1}}}+{{1 \over{p_{1}}}\ {|_{7 \  5}^{3 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  1}^{3 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  0}^{3 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  3}^{3 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  2}^{3 \  2}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  5}^{3 \  2}}}-{{1 \over{p_{1}}}\ {|_{5 \  4}^{3 \  2}}}-{{1 \over{p_{1}}}\ {|_{6 \  7}^{3 \  2}}}+{{1 \over{p_{1}}}\ {|_{7 \  6}^{3 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}^{3 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}^{3 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}^{3 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}^{3 \  3}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  4}^{3 \  3}}}-{{1 \over{p_{1}}}\ {|_{5 \  5}^{3 \  3}}}-{{1 \over{p_{1}}}\ {|_{6 \  6}^{3 \  3}}}-{{1 \over{p_{1}}}\ {|_{7 \  7}^{3 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{3 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{3 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{3 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{3 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{3 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{3 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{3 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{3 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{3 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{3 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{3 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{3 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{3 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{3 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{3 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{3 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{3 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{3 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{3 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{3 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{3 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{3 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{3 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{3 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{3 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{3 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{3 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{3 \  7}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{3 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{3 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{3 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{3 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{4 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{4 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{4 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{4 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{4 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{4 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{4 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{4 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{4 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{4 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{4 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{4 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{4 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{4 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{4 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{4 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{4 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{4 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{4 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{4 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{4 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{4 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{4 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{4 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{4 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{4 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{4 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{4 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{4 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{4 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{4 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{4 \  3}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  0}^{4 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  1}^{4 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  2}^{4 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  3}^{4 \  4}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  4}^{4 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  5}^{4 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  6}^{4 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  7}^{4 \  4}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  1}^{4 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  0}^{4 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  3}^{4 \  5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  2}^{4 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  5}^{4 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  4}^{4 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  7}^{4 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  6}^{4 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  2}^{4 \  6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  3}^{4 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  0}^{4 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  1}^{4 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  6}^{4 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  7}^{4 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  4}^{4 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  5}^{4 \  6}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  3}^{4 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  2}^{4 \  7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  1}^{4 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  0}^{4 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  7}^{4 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  6}^{4 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  5}^{4 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  4}^{4 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{5 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{5 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{5 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{5 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{5 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{5 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{5 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{5 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{5 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{5 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{5 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{5 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{5 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{5 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{5 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{5 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{5 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{5 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{5 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{5 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{5 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{5 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{5 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{5 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{5 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{5 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{5 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{5 \  3}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{5 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{5 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{5 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{5 \  3}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  1}^{5 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  0}^{5 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  3}^{5 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  2}^{5 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  5}^{5 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  4}^{5 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  7}^{5 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  6}^{5 \  4}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  0}^{5 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  1}^{5 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  2}^{5 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  3}^{5 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  4}^{5 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  5}^{5 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  6}^{5 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  7}^{5 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  3}^{5 \  6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  2}^{5 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  1}^{5 \  6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  0}^{5 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  7}^{5 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  6}^{5 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  5}^{5 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  4}^{5 \  6}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  2}^{5 \  7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  3}^{5 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  0}^{5 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  1}^{5 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  6}^{5 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  7}^{5 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  4}^{5 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  5}^{5 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{6 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{6 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{6 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{6 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{6 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{6 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{6 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{6 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{6 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{6 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{6 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{6 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{6 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{6 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{6 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{6 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{6 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{6 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{6 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{6 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{6 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{6 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{6 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{6 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{6 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{6 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{6 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{6 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{6 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{6 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{6 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{6 \  3}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  2}^{6 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  3}^{6 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  0}^{6 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  1}^{6 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  6}^{6 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  7}^{6 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  4}^{6 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  5}^{6 \  4}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  3}^{6 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  2}^{6 \  5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  1}^{6 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  0}^{6 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  7}^{6 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  6}^{6 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  5}^{6 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  4}^{6 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  0}^{6 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  1}^{6 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  2}^{6 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  3}^{6 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  4}^{6 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  5}^{6 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  6}^{6 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  7}^{6 \  6}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  1}^{6 \  7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  0}^{6 \  7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  3}^{6 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  2}^{6 \  7}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  5}^{6 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  4}^{6 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  7}^{6 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  6}^{6 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{7 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{7 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{7 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{7 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{7 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{7 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{7 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{7 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{7 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{7 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{7 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{7 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{7 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{7 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{7 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{7 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{7 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{7 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{7 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{7 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{7 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{7 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{7 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{7 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{7 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{7 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{7 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{7 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{7 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{7 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{7 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{7 \  3}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  3}^{7 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  2}^{7 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  1}^{7 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  0}^{7 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  7}^{7 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  6}^{7 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  5}^{7 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  4}^{7 \  4}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  2}^{7 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  3}^{7 \  5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  0}^{7 \  5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  1}^{7 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  6}^{7 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  7}^{7 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  4}^{7 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  5}^{7 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  1}^{7 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  0}^{7 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  3}^{7 \  6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  2}^{7 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  5}^{7 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  4}^{7 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  7}^{7 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  6}^{7 \  6}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  0}^{7 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  1}^{7 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  2}^{7 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  3}^{7 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  4}^{7 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  5}^{7 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  6}^{7 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  7}^{7 \  7}}}
(34)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
Χr := (λ,I)/(I,Y)

\label{eq35}\begin{array}{@{}l}
\displaystyle
-{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}^{0 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}^{0 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}^{0 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}^{0 \  0}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  4}^{0 \  0}}}+{{1 \over{p_{1}}}\ {|_{5 \  5}^{0 \  0}}}+{{1 \over{p_{1}}}\ {|_{6 \  6}^{0 \  0}}}+{{1 \over{p_{1}}}\ {|_{7 \  7}^{0 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  1}^{0 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  0}^{0 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  3}^{0 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  2}^{0 \  1}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  5}^{0 \  1}}}+{{1 \over{p_{1}}}\ {|_{5 \  4}^{0 \  1}}}+{{1 \over{p_{1}}}\ {|_{6 \  7}^{0 \  1}}}-{{1 \over{p_{1}}}\ {|_{7 \  6}^{0 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  2}^{0 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  3}^{0 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  0}^{0 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  1}^{0 \  2}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  6}^{0 \  2}}}-{{1 \over{p_{1}}}\ {|_{5 \  7}^{0 \  2}}}+{{1 \over{p_{1}}}\ {|_{6 \  4}^{0 \  2}}}+{{1 \over{p_{1}}}\ {|_{7 \  5}^{0 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  3}^{0 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  2}^{0 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  1}^{0 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  0}^{0 \  3}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  7}^{0 \  3}}}+{{1 \over{p_{1}}}\ {|_{5 \  6}^{0 \  3}}}-{{1 \over{p_{1}}}\ {|_{6 \  5}^{0 \  3}}}+{{1 \over{p_{1}}}\ {|_{7 \  4}^{0 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{0 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{0 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{0 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{0 \  4}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{0 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{0 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{0 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{0 \  4}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{0 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{0 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{0 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{0 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{0 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{0 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{0 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{0 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{0 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{0 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{0 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{0 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{0 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{0 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{0 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{0 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{0 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{0 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{0 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{0 \  7}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{0 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{0 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{0 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{0 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  1}^{1 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  0}^{1 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  3}^{1 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  2}^{1 \  0}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  5}^{1 \  0}}}+{{1 \over{p_{1}}}\ {|_{5 \  4}^{1 \  0}}}+{{1 \over{p_{1}}}\ {|_{6 \  7}^{1 \  0}}}-{{1 \over{p_{1}}}\ {|_{7 \  6}^{1 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}^{1 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}^{1 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}^{1 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}^{1 \  1}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  4}^{1 \  1}}}-{{1 \over{p_{1}}}\ {|_{5 \  5}^{1 \  1}}}-{{1 \over{p_{1}}}\ {|_{6 \  6}^{1 \  1}}}-{{1 \over{p_{1}}}\ {|_{7 \  7}^{1 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  3}^{1 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  2}^{1 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  1}^{1 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  0}^{1 \  2}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  7}^{1 \  2}}}-{{1 \over{p_{1}}}\ {|_{5 \  6}^{1 \  2}}}+{{1 \over{p_{1}}}\ {|_{6 \  5}^{1 \  2}}}-{{1 \over{p_{1}}}\ {|_{7 \  4}^{1 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  2}^{1 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  3}^{1 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  0}^{1 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  1}^{1 \  3}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  6}^{1 \  3}}}-{{1 \over{p_{1}}}\ {|_{5 \  7}^{1 \  3}}}+{{1 \over{p_{1}}}\ {|_{6 \  4}^{1 \  3}}}+{{1 \over{p_{1}}}\ {|_{7 \  5}^{1 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{1 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{1 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{1 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{1 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{1 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{1 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{1 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{1 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{1 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{1 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{1 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{1 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{1 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{1 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{1 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{1 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{1 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{1 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{1 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{1 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{1 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{1 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{1 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{1 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{1 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{1 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{1 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{1 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{1 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{1 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{1 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{1 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  2}^{2 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  3}^{2 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  0}^{2 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  1}^{2 \  0}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  6}^{2 \  0}}}-{{1 \over{p_{1}}}\ {|_{5 \  7}^{2 \  0}}}+{{1 \over{p_{1}}}\ {|_{6 \  4}^{2 \  0}}}+{{1 \over{p_{1}}}\ {|_{7 \  5}^{2 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  3}^{2 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  2}^{2 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  1}^{2 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  0}^{2 \  1}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  7}^{2 \  1}}}+{{1 \over{p_{1}}}\ {|_{5 \  6}^{2 \  1}}}-{{1 \over{p_{1}}}\ {|_{6 \  5}^{2 \  1}}}+{{1 \over{p_{1}}}\ {|_{7 \  4}^{2 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}^{2 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}^{2 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}^{2 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}^{2 \  2}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  4}^{2 \  2}}}-{{1 \over{p_{1}}}\ {|_{5 \  5}^{2 \  2}}}-{{1 \over{p_{1}}}\ {|_{6 \  6}^{2 \  2}}}-{{1 \over{p_{1}}}\ {|_{7 \  7}^{2 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  1}^{2 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  0}^{2 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  3}^{2 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  2}^{2 \  3}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  5}^{2 \  3}}}-{{1 \over{p_{1}}}\ {|_{5 \  4}^{2 \  3}}}-{{1 \over{p_{1}}}\ {|_{6 \  7}^{2 \  3}}}+{{1 \over{p_{1}}}\ {|_{7 \  6}^{2 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{2 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{2 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{2 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{2 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{2 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{2 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{2 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{2 \  4}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{2 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{2 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{2 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{2 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{2 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{2 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{2 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{2 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{2 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{2 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{2 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{2 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{2 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{2 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{2 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{2 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{2 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{2 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{2 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{2 \  7}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{2 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{2 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{2 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{2 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  3}^{3 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  2}^{3 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  1}^{3 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  0}^{3 \  0}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  7}^{3 \  0}}}+{{1 \over{p_{1}}}\ {|_{5 \  6}^{3 \  0}}}-{{1 \over{p_{1}}}\ {|_{6 \  5}^{3 \  0}}}+{{1 \over{p_{1}}}\ {|_{7 \  4}^{3 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  2}^{3 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  3}^{3 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  0}^{3 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  1}^{3 \  1}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  6}^{3 \  1}}}+{{1 \over{p_{1}}}\ {|_{5 \  7}^{3 \  1}}}-{{1 \over{p_{1}}}\ {|_{6 \  4}^{3 \  1}}}-{{1 \over{p_{1}}}\ {|_{7 \  5}^{3 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  1}^{3 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  0}^{3 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  3}^{3 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  2}^{3 \  2}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  5}^{3 \  2}}}+{{1 \over{p_{1}}}\ {|_{5 \  4}^{3 \  2}}}+{{1 \over{p_{1}}}\ {|_{6 \  7}^{3 \  2}}}-{{1 \over{p_{1}}}\ {|_{7 \  6}^{3 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  0}^{3 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  1}^{3 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  2}^{3 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  3}^{3 \  3}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  4}^{3 \  3}}}-{{1 \over{p_{1}}}\ {|_{5 \  5}^{3 \  3}}}-{{1 \over{p_{1}}}\ {|_{6 \  6}^{3 \  3}}}-{{1 \over{p_{1}}}\ {|_{7 \  7}^{3 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{3 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{3 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{3 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{3 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{3 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{3 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{3 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{3 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{3 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{3 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{3 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{3 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{3 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{3 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{3 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{3 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{3 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{3 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{3 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{3 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{3 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{3 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{3 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{3 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{3 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{3 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{3 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{3 \  7}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{3 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{3 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{3 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{3 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{4 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{4 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{4 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{4 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{4 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{4 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{4 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{4 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{4 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{4 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{4 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{4 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{4 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{4 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{4 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{4 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{4 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{4 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{4 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{4 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{4 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{4 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{4 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{4 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{4 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{4 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{4 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{4 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{4 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{4 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{4 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{4 \  3}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  0}^{4 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  1}^{4 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  2}^{4 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  3}^{4 \  4}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  4}^{4 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  5}^{4 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  6}^{4 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  7}^{4 \  4}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  1}^{4 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  0}^{4 \  5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  3}^{4 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  2}^{4 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  5}^{4 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  4}^{4 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  7}^{4 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  6}^{4 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  2}^{4 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  3}^{4 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  0}^{4 \  6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  1}^{4 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  6}^{4 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  7}^{4 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  4}^{4 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  5}^{4 \  6}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  3}^{4 \  7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  2}^{4 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  1}^{4 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  0}^{4 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  7}^{4 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  6}^{4 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  5}^{4 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  4}^{4 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{5 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{5 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{5 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{5 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{5 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{5 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{5 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{5 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{5 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{5 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{5 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{5 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{5 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{5 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{5 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{5 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{5 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{5 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{5 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{5 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{5 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{5 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{5 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{5 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{5 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{5 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{5 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{5 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{5 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{5 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{5 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{5 \  3}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  1}^{5 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  0}^{5 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  3}^{5 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  2}^{5 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  5}^{5 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  4}^{5 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  7}^{5 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  6}^{5 \  4}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  0}^{5 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  1}^{5 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  2}^{5 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  3}^{5 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  4}^{5 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  5}^{5 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  6}^{5 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  7}^{5 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  3}^{5 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  2}^{5 \  6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  1}^{5 \  6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  0}^{5 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  7}^{5 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  6}^{5 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  5}^{5 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  4}^{5 \  6}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  2}^{5 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  3}^{5 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  0}^{5 \  7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  1}^{5 \  7}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  6}^{5 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  7}^{5 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  4}^{5 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  5}^{5 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{6 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{6 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{6 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{6 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{6 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{6 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{6 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{6 \  0}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{6 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{6 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{6 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{6 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{6 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{6 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{6 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{6 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{6 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{6 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{6 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{6 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{6 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{6 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{6 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{6 \  2}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{6 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{6 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{6 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{6 \  3}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{6 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{6 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{6 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{6 \  3}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  2}^{6 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  3}^{6 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  0}^{6 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  1}^{6 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  6}^{6 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  7}^{6 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  4}^{6 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  5}^{6 \  4}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  3}^{6 \  5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  2}^{6 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  1}^{6 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  0}^{6 \  5}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  7}^{6 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  6}^{6 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  5}^{6 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  4}^{6 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  0}^{6 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  1}^{6 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  2}^{6 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  3}^{6 \  6}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  4}^{6 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  5}^{6 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  6}^{6 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  7}^{6 \  6}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  1}^{6 \  7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  0}^{6 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  3}^{6 \  7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  2}^{6 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  5}^{6 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  4}^{6 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  7}^{6 \  7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \  6}^{6 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  7}^{7 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  6}^{7 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  5}^{7 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \  4}^{7 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  3}^{7 \  0}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  2}^{7 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  1}^{7 \  0}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  0}^{7 \  0}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  6}^{7 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  7}^{7 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  4}^{7 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  5}^{7 \  1}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  2}^{7 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  3}^{7 \  1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  0}^{7 \  1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  1}^{7 \  1}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  5}^{7 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{1 \  4}^{7 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \  7}^{7 \  2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  6}^{7 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  1}^{7 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  0}^{7 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  3}^{7 \  2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  2}^{7 \  2}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{0 \  4}^{7 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \  5}^{7 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \  6}^{7 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \  7}^{7 \  3}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  0}^{7 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  1}^{7 \  3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  2}^{7 \  3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  3}^{7 \  3}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  3}^{7 \  4}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  2}^{7 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  1}^{7 \  4}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  0}^{7 \  4}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  7}^{7 \  4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  6}^{7 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  5}^{7 \  4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  4}^{7 \  4}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  2}^{7 \  5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  3}^{7 \  5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  0}^{7 \  5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  1}^{7 \  5}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  6}^{7 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \  7}^{7 \  5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  4}^{7 \  5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  5}^{7 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  1}^{7 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  0}^{7 \  6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  3}^{7 \  6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  2}^{7 \  6}}}+ 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  5}^{7 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  4}^{7 \  6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \  7}^{7 \  6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  6}^{7 \  6}}}+ 
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{0 \  0}^{7 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \  1}^{7 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \  2}^{7 \  7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \  3}^{7 \  7}}}- 
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{4 \  4}^{7 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \  5}^{7 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \  6}^{7 \  7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \  7}^{7 \  7}}}
(35)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
test(Χr = Χ )

\label{eq36} \mbox{\rm false} (36)
Type: Boolean
axiom
Χl := (I,λ)/(Y,I);
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
test( Χl = Χ )

\label{eq37} \mbox{\rm false} (37)
Type: Boolean
axiom
test( Χr = Χl )

\label{eq38} \mbox{\rm true} (38)
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

axiom
i:=𝐞.1

\label{eq39}|_{0}(39)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
test
         i     /
         λ     =    Ω

\label{eq40} \mbox{\rm true} (40)
Type: Boolean

Handle

axiom
H:𝐋 :=
         λ     /
         X     /
         Y

\label{eq41}\begin{array}{@{}l}
\displaystyle
-{{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{0}^{0}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{1}^{1}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{2}^{2}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{3}^{3}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{4}^{4}}}+ 
\
\
\displaystyle
{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{5}^{5}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{6}^{6}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{7}^{7}}}
(41)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

Definition 5

Co-unit
  i 
  U
  

axiom
ι:𝐋:=
    (    i I    ) /
    (     Ų     )

\label{eq42}-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{0}}}(42)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

Y=U
ι  
axiom
test
        Y    /
        ι       = Ų

\label{eq43} \mbox{\rm true} (43)
Type: Boolean

For example:

axiom
ex1:=[q[2]=1,p[1]=1]

\label{eq44}\left[{{q_{2}}= 1}, \:{{p_{1}}= 1}\right](44)
Type: List(Equation(Polynomial(Integer)))
axiom
Ų0:𝐋  :=eval(Ų,ex1)

\label{eq45}-{|_{\ }^{0 \  0}}+{|_{\ }^{1 \  1}}+{|_{\ }^{2 \  2}}+{|_{\ }^{3 \  3}}+{|_{\ }^{4 \  4}}+{|_{\ }^{5 \  5}}+{|_{\ }^{6 \  6}}+{|_{\ }^{7 \  7}}(45)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
Ω0:𝐋  :=eval(Ω,ex1)$𝐋

\label{eq46}-{|_{0 \  0}}+{|_{1 \  1}}+{|_{2 \  2}}+{|_{3 \  3}}+{|_{4 \  4}}+{|_{5 \  5}}+{|_{6 \  6}}+{|_{7 \  7}}(46)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
λ0:𝐋  :=eval(λ,ex1)$𝐋

\label{eq47}\begin{array}{@{}l}
\displaystyle
-{|_{0 \  0}^{0}}+{|_{1 \  1}^{0}}+{|_{2 \  2}^{0}}+{|_{3 \  3}^{0}}+{|_{4 \  4}^{0}}+{|_{5 \  5}^{0}}+{|_{6 \  6}^{0}}+ 
\
\
\displaystyle
{|_{7 \  7}^{0}}-{|_{0 \  1}^{1}}-{|_{1 \  0}^{1}}-{|_{2 \  3}^{1}}+{|_{3 \  2}^{1}}-{|_{4 \  5}^{1}}+{|_{5 \  4}^{1}}+{|_{6 \  7}^{1}}- 
\
\
\displaystyle
{|_{7 \  6}^{1}}-{|_{0 \  2}^{2}}+{|_{1 \  3}^{2}}-{|_{2 \  0}^{2}}-{|_{3 \  1}^{2}}-{|_{4 \  6}^{2}}-{|_{5 \  7}^{2}}+{|_{6 \  4}^{2}}+ 
\
\
\displaystyle
{|_{7 \  5}^{2}}-{|_{0 \  3}^{3}}-{|_{1 \  2}^{3}}+{|_{2 \  1}^{3}}-{|_{3 \  0}^{3}}-{|_{4 \  7}^{3}}+{|_{5 \  6}^{3}}-{|_{6 \  5}^{3}}+ 
\
\
\displaystyle
{|_{7 \  4}^{3}}-{|_{0 \  4}^{4}}+{|_{1 \  5}^{4}}+{|_{2 \  6}^{4}}+{|_{3 \  7}^{4}}-{|_{4 \  0}^{4}}-{|_{5 \  1}^{4}}-{|_{6 \  2}^{4}}- 
\
\
\displaystyle
{|_{7 \  3}^{4}}-{|_{0 \  5}^{5}}-{|_{1 \  4}^{5}}+{|_{2 \  7}^{5}}-{|_{3 \  6}^{5}}+{|_{4 \  1}^{5}}-{|_{5 \  0}^{5}}+{|_{6 \  3}^{5}}- 
\
\
\displaystyle
{|_{7 \  2}^{5}}-{|_{0 \  6}^{6}}-{|_{1 \  7}^{6}}-{|_{2 \  4}^{6}}+{|_{3 \  5}^{6}}+{|_{4 \  2}^{6}}-{|_{5 \  3}^{6}}-{|_{6 \  0}^{6}}+ 
\
\
\displaystyle
{|_{7 \  1}^{6}}-{|_{0 \  7}^{7}}+{|_{1 \  6}^{7}}-{|_{2 \  5}^{7}}-{|_{3 \  4}^{7}}+{|_{4 \  3}^{7}}+{|_{5 \  2}^{7}}-{|_{6 \  1}^{7}}- 
\
\
\displaystyle
{|_{7 \  0}^{7}}
(47)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
axiom
H0:𝐋 :=eval(H,ex1)$𝐋

\label{eq48}-{8 \ {|_{0}^{0}}}+{4 \ {|_{1}^{1}}}+{4 \ {|_{2}^{2}}}+{4 \ {|_{3}^{3}}}+{4 \ {|_{4}^{4}}}+{4 \ {|_{5}^{5}}}+{4 \ {|_{6}^{6}}}+{4 \ {|_{7}^{7}}}(48)
Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))