help links subscribe changes refresh edit

	Frobenius Algebra, Vector Spaces and Polynomial Ideals
	↑	→ SandBox Grassmann Algebra Is Frobenius In Many Ways

Octonion Algebra is Frobenius in Just One Way

Edit detail for Octonion Algebra is Frobenius in Just One Way revision 6 of 8

	1 2 3 4 5 6 7 8
Editor: Bill Page Time: 2011/04/27 20:39:52 GMT-7
Note: Frobenius condition

added:
- http://en.wikipedia.org/wiki/Octonion


added:
         X     /

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 Algebras

Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)
http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories

Joachim Kock
http://en.wikipedia.org/wiki/Frobenius_algebra
http://en.wikipedia.org/wiki/Octonion

We need the Axiom LinearOperator? library.

axiom

)library CARTEN MONAL PROP LIN 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/LIN.NRLIB/LIN
   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(dim, OVAR [], ℚ)

$\label{eq2}\hbox{\axiomType{LinearOperator}\ } (8, \hbox{\axiomType{OrderedVariableList}\ } ([ ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))$

(2)

Type: Type

axiom

𝐞:ℒ 𝐋      := basisVectors()

$\label{eq3}\left[{|_{1}}, \:{|_{2}}, \:{|_{3}}, \:{|_{4}}, \:{|_{5}}, \:{|_{6}}, \:{|_{7}}, \:{|_{8}}\right]$

(3)

Type: List(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

axiom

𝐝:ℒ 𝐋      := basisForms()

$\label{eq4}\left[{|_{\ }^{1}}, \:{|_{\ }^{2}}, \:{|_{\ }^{3}}, \:{|_{\ }^{4}}, \:{|_{\ }^{5}}, \:{|_{\ }^{6}}, \:{|_{\ }^{7}}, \:{|_{\ }^{8}}\right]$

(4)

Type: List(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

axiom

I:𝐋:=[1];   -- identity for composition

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

X:𝐋:=[2,1]; -- twist

Type: LinearOperator?(8,OrderedVariableList?([]),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)

$\label{eq10}\begin{array}{@{}l} \displaystyle {|_{1}^{1 \ 1}}+{|_{2}^{1 \ 2}}+{|_{3}^{1 \ 3}}+{|_{4}^{1 \ 4}}+{|_{5}^{1 \ 5}}+{|_{6}^{1 \ 6}}+{|_{7}^{1 \ 7}}+{|_{8}^{1 \ 8}}+ \ \ \displaystyle {|_{2}^{2 \ 1}}-{|_{1}^{2 \ 2}}+{|_{4}^{2 \ 3}}-{|_{3}^{2 \ 4}}+{|_{6}^{2 \ 5}}-{|_{5}^{2 \ 6}}-{|_{8}^{2 \ 7}}+{|_{7}^{2 \ 8}}+ \ \ \displaystyle {|_{3}^{3 \ 1}}-{|_{4}^{3 \ 2}}-{|_{1}^{3 \ 3}}+{|_{2}^{3 \ 4}}+{|_{7}^{3 \ 5}}+{|_{8}^{3 \ 6}}-{|_{5}^{3 \ 7}}-{|_{6}^{3 \ 8}}+ \ \ \displaystyle {|_{4}^{4 \ 1}}+{|_{3}^{4 \ 2}}-{|_{2}^{4 \ 3}}-{|_{1}^{4 \ 4}}+{|_{8}^{4 \ 5}}-{|_{7}^{4 \ 6}}+{|_{6}^{4 \ 7}}-{|_{5}^{4 \ 8}}+ \ \ \displaystyle {|_{5}^{5 \ 1}}-{|_{6}^{5 \ 2}}-{|_{7}^{5 \ 3}}-{|_{8}^{5 \ 4}}-{{q_{2}}\ {|_{1}^{5 \ 5}}}+{{q_{2}}\ {|_{2}^{5 \ 6}}}+{{q_{2}}\ {|_{3}^{5 \ 7}}}+ \ \ \displaystyle {{q_{2}}\ {|_{4}^{5 \ 8}}}+{|_{6}^{6 \ 1}}+{|_{5}^{6 \ 2}}-{|_{8}^{6 \ 3}}+{|_{7}^{6 \ 4}}-{{q_{2}}\ {|_{2}^{6 \ 5}}}-{{q_{2}}\ {|_{1}^{6 \ 6}}}- \ \ \displaystyle {{q_{2}}\ {|_{4}^{6 \ 7}}}+{{q_{2}}\ {|_{3}^{6 \ 8}}}+{|_{7}^{7 \ 1}}+{|_{8}^{7 \ 2}}+{|_{5}^{7 \ 3}}-{|_{6}^{7 \ 4}}-{{q_{2}}\ {|_{3}^{7 \ 5}}}+ \ \ \displaystyle {{q_{2}}\ {|_{4}^{7 \ 6}}}-{{q_{2}}\ {|_{1}^{7 \ 7}}}-{{q_{2}}\ {|_{2}^{7 \ 8}}}+{|_{8}^{8 \ 1}}-{|_{7}^{8 \ 2}}+{|_{6}^{8 \ 3}}+{|_{5}^{8 \ 4}}- \ \ \displaystyle {{q_{2}}\ {|_{4}^{8 \ 5}}}-{{q_{2}}\ {|_{3}^{8 \ 6}}}+{{q_{2}}\ {|_{2}^{8 \ 7}}}-{{q_{2}}\ {|_{1}^{8 \ 8}}}$

(10)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

arity Y

$\label{eq11}2 \over 1$

(11)

Type: Prop(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

axiom

matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)

$\label{eq12}\left[ \begin{array}{cccccccc} {|_{1}}&{|_{2}}&{|_{3}}&{|_{4}}&{|_{5}}&{|_{6}}&{|_{7}}&{|_{8}} \ {|_{2}}& -{|_{1}}& -{|_{4}}&{|_{3}}& -{|_{6}}&{|_{5}}&{|_{8}}& -{|_{7}} \ {|_{3}}&{|_{4}}& -{|_{1}}& -{|_{2}}& -{|_{7}}& -{|_{8}}&{|_{5}}&{|_{6}} \ {|_{4}}& -{|_{3}}&{|_{2}}& -{|_{1}}& -{|_{8}}&{|_{7}}& -{|_{6}}&{|_{5}} \ {|_{5}}&{|_{6}}&{|_{7}}&{|_{8}}& -{{q_{2}}\ {|_{1}}}& -{{q_{2}}\ {|_{2}}}& -{{q_{2}}\ {|_{3}}}& -{{q_{2}}\ {|_{4}}} \ {|_{6}}& -{|_{5}}&{|_{8}}& -{|_{7}}&{{q_{2}}\ {|_{2}}}& -{{q_{2}}\ {|_{1}}}&{{q_{2}}\ {|_{4}}}& -{{q_{2}}\ {|_{3}}} \ {|_{7}}& -{|_{8}}& -{|_{5}}&{|_{6}}&{{q_{2}}\ {|_{3}}}& -{{q_{2}}\ {|_{4}}}& -{{q_{2}}\ {|_{1}}}&{{q_{2}}\ {|_{2}}} \ {|_{8}}&{|_{7}}& -{|_{6}}& -{|_{5}}&{{q_{2}}\ {|_{4}}}&{{q_{2}}\ {|_{3}}}& -{{q_{2}}\ {|_{2}}}& -{{q_{2}}\ {|_{1}}}$

(12)

Type: Matrix(LinearOperator?(8,OrderedVariableList?([]),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}}\ {|_{\ }^{1 \ 1}}}+{{u^{1, \: 2}}\ {|_{\ }^{1 \ 2}}}+{{u^{1, \: 3}}\ {|_{\ }^{1 \ 3}}}+{{u^{1, \: 4}}\ {|_{\ }^{1 \ 4}}}+ \ \ \displaystyle {{u^{1, \: 5}}\ {|_{\ }^{1 \ 5}}}+{{u^{1, \: 6}}\ {|_{\ }^{1 \ 6}}}+{{u^{1, \: 7}}\ {|_{\ }^{1 \ 7}}}+{{u^{1, \: 8}}\ {|_{\ }^{1 \ 8}}}+ \ \ \displaystyle {{u^{2, \: 1}}\ {|_{\ }^{2 \ 1}}}+{{u^{2, \: 2}}\ {|_{\ }^{2 \ 2}}}+{{u^{2, \: 3}}\ {|_{\ }^{2 \ 3}}}+{{u^{2, \: 4}}\ {|_{\ }^{2 \ 4}}}+ \ \ \displaystyle {{u^{2, \: 5}}\ {|_{\ }^{2 \ 5}}}+{{u^{2, \: 6}}\ {|_{\ }^{2 \ 6}}}+{{u^{2, \: 7}}\ {|_{\ }^{2 \ 7}}}+{{u^{2, \: 8}}\ {|_{\ }^{2 \ 8}}}+ \ \ \displaystyle {{u^{3, \: 1}}\ {|_{\ }^{3 \ 1}}}+{{u^{3, \: 2}}\ {|_{\ }^{3 \ 2}}}+{{u^{3, \: 3}}\ {|_{\ }^{3 \ 3}}}+{{u^{3, \: 4}}\ {|_{\ }^{3 \ 4}}}+ \ \ \displaystyle {{u^{3, \: 5}}\ {|_{\ }^{3 \ 5}}}+{{u^{3, \: 6}}\ {|_{\ }^{3 \ 6}}}+{{u^{3, \: 7}}\ {|_{\ }^{3 \ 7}}}+{{u^{3, \: 8}}\ {|_{\ }^{3 \ 8}}}+ \ \ \displaystyle {{u^{4, \: 1}}\ {|_{\ }^{4 \ 1}}}+{{u^{4, \: 2}}\ {|_{\ }^{4 \ 2}}}+{{u^{4, \: 3}}\ {|_{\ }^{4 \ 3}}}+{{u^{4, \: 4}}\ {|_{\ }^{4 \ 4}}}+ \ \ \displaystyle {{u^{4, \: 5}}\ {|_{\ }^{4 \ 5}}}+{{u^{4, \: 6}}\ {|_{\ }^{4 \ 6}}}+{{u^{4, \: 7}}\ {|_{\ }^{4 \ 7}}}+{{u^{4, \: 8}}\ {|_{\ }^{4 \ 8}}}+ \ \ \displaystyle {{u^{5, \: 1}}\ {|_{\ }^{5 \ 1}}}+{{u^{5, \: 2}}\ {|_{\ }^{5 \ 2}}}+{{u^{5, \: 3}}\ {|_{\ }^{5 \ 3}}}+{{u^{5, \: 4}}\ {|_{\ }^{5 \ 4}}}+ \ \ \displaystyle {{u^{5, \: 5}}\ {|_{\ }^{5 \ 5}}}+{{u^{5, \: 6}}\ {|_{\ }^{5 \ 6}}}+{{u^{5, \: 7}}\ {|_{\ }^{5 \ 7}}}+{{u^{5, \: 8}}\ {|_{\ }^{5 \ 8}}}+ \ \ \displaystyle {{u^{6, \: 1}}\ {|_{\ }^{6 \ 1}}}+{{u^{6, \: 2}}\ {|_{\ }^{6 \ 2}}}+{{u^{6, \: 3}}\ {|_{\ }^{6 \ 3}}}+{{u^{6, \: 4}}\ {|_{\ }^{6 \ 4}}}+ \ \ \displaystyle {{u^{6, \: 5}}\ {|_{\ }^{6 \ 5}}}+{{u^{6, \: 6}}\ {|_{\ }^{6 \ 6}}}+{{u^{6, \: 7}}\ {|_{\ }^{6 \ 7}}}+{{u^{6, \: 8}}\ {|_{\ }^{6 \ 8}}}+ \ \ \displaystyle {{u^{7, \: 1}}\ {|_{\ }^{7 \ 1}}}+{{u^{7, \: 2}}\ {|_{\ }^{7 \ 2}}}+{{u^{7, \: 3}}\ {|_{\ }^{7 \ 3}}}+{{u^{7, \: 4}}\ {|_{\ }^{7 \ 4}}}+ \ \ \displaystyle {{u^{7, \: 5}}\ {|_{\ }^{7 \ 5}}}+{{u^{7, \: 6}}\ {|_{\ }^{7 \ 6}}}+{{u^{7, \: 7}}\ {|_{\ }^{7 \ 7}}}+{{u^{7, \: 8}}\ {|_{\ }^{7 \ 8}}}+ \ \ \displaystyle {{u^{8, \: 1}}\ {|_{\ }^{8 \ 1}}}+{{u^{8, \: 2}}\ {|_{\ }^{8 \ 2}}}+{{u^{8, \: 3}}\ {|_{\ }^{8 \ 3}}}+{{u^{8, \: 4}}\ {|_{\ }^{8 \ 4}}}+ \ \ \displaystyle {{u^{8, \: 5}}\ {|_{\ }^{8 \ 5}}}+{{u^{8, \: 6}}\ {|_{\ }^{8 \ 6}}}+{{u^{8, \: 7}}\ {|_{\ }^{8 \ 7}}}+{{u^{8, \: 8}}\ {|_{\ }^{8 \ 8}}}$

(13)

Type: LinearOperator?(8,OrderedVariableList?([]),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:

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(4.82,0.92) \psbezier[linewidth=0.04](2.2,0.9)(2.2,0.1)(2.6,0.1)(2.6,0.9) \psline[linewidth=0.04cm](2.4,0.3)(2.4,-0.1) \psbezier[linewidth=0.04](2.4,-0.1)(2.4,-0.9)(3.0,-0.9)(3.0,-0.1) \psline[linewidth=0.04cm](3.0,-0.1)(3.0,0.9) \psbezier[linewidth=0.04](4.8,0.9)(4.8,0.1)(4.4,0.1)(4.4,0.9) \psline[linewidth=0.04cm](4.6,0.3)(4.6,-0.1) \psbezier[linewidth=0.04](4.6,-0.1)(4.6,-0.9)(4.0,-0.9)(4.0,-0.1) \psline[linewidth=0.04cm](4.0,-0.1)(4.0,0.9) \usefont{T1}{ptm}{m}{n} \rput(3.4948437,0.205){-} \psline[linewidth=0.04cm](0.6,-0.7)(0.6,0.9) \psbezier[linewidth=0.04](0.0,-0.1)(0.0,-0.9)(1.2,-0.9)(1.2,-0.1) \psline[linewidth=0.04cm](0.0,-0.1)(0.0,0.9) \psline[linewidth=0.04cm](1.2,-0.1)(1.2,0.9) \usefont{T1}{ptm}{m}{n} \rput(1.6948438,0.205){=} \end{pspicture} }$

$\label{eq14} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}$

(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?(8,OrderedVariableList?([]),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

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 into coefficients of the tensor $\Phi$ . We are looking for the general linear family of tensors such that J transforms into $\Phi=0$ 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:

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)

$\label{eq19}\begin{array}{@{}l} \displaystyle -{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \ 1}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \ 2}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \ 3}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{4 \ 4}}}+{{p_{1}}\ {|_{\ }^{5 \ 5}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{6 \ 6}}}+{{p_{1}}\ {|_{\ }^{7 \ 7}}}+{{p_{1}}\ {|_{\ }^{8 \ 8}}}$

(19)

Type: LinearOperator?(8,OrderedVariableList?([]),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?(8,OrderedVariableList?([]),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?(8,OrderedVariableList?([]),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)

$\label{eq25}\begin{array}{@{}l} \displaystyle -{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}}}+{{1 \over{p_{1}}}\ {|_{5 \ 5}}}+{{1 \over{p_{1}}}\ {|_{6 \ 6}}}+ \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{7 \ 7}}}+{{1 \over{p_{1}}}\ {|_{8 \ 8}}}$

(25)

Type: LinearOperator?(8,OrderedVariableList?([]),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?(8,OrderedVariableList?([]),Expression(Integer)))

Check "dimension" and the snake relations.

axiom

d:𝐋:=
       Ω    /
       X    /
       Ų

$\label{eq27}8$

(27)

Type: LinearOperator?(8,OrderedVariableList?([]),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}}}\ {|_{\ }^{1 \ 1 \ 1}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \ 2 \ 2}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \ 3 \ 3}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \ 4 \ 4}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{1 \ 5 \ 5}}}+{{p_{1}}\ {|_{\ }^{1 \ 6 \ 6}}}+{{p_{1}}\ {|_{\ }^{1 \ 7 \ 7}}}+{{p_{1}}\ {|_{\ }^{1 \ 8 \ 8}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \ 1 \ 2}}}+ \ \ \displaystyle {{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \ 2 \ 1}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \ 3 \ 4}}}-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \ 4 \ 3}}}+{{p_{1}}\ {|_{\ }^{2 \ 5 \ 6}}}-{{p_{1}}\ {|_{\ }^{2 \ 6 \ 5}}}- \ \ \displaystyle {{p_{1}}\ {|_{\ }^{2 \ 7 \ 8}}}+{{p_{1}}\ {|_{\ }^{2 \ 8 \ 7}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \ 1 \ 3}}}-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \ 2 \ 4}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \ 3 \ 1}}}+ \ \ \displaystyle {{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \ 4 \ 2}}}+{{p_{1}}\ {|_{\ }^{3 \ 5 \ 7}}}+{{p_{1}}\ {|_{\ }^{3 \ 6 \ 8}}}-{{p_{1}}\ {|_{\ }^{3 \ 7 \ 5}}}-{{p_{1}}\ {|_{\ }^{3 \ 8 \ 6}}}+ \ \ \displaystyle {{{p_{1}}\over{q_{2}}}\ {|_{\ }^{4 \ 1 \ 4}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{4 \ 2 \ 3}}}-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{4 \ 3 \ 2}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{4 \ 4 \ 1}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{4 \ 5 \ 8}}}-{{p_{1}}\ {|_{\ }^{4 \ 6 \ 7}}}+{{p_{1}}\ {|_{\ }^{4 \ 7 \ 6}}}-{{p_{1}}\ {|_{\ }^{4 \ 8 \ 5}}}+{{p_{1}}\ {|_{\ }^{5 \ 1 \ 5}}}- \ \ \displaystyle {{p_{1}}\ {|_{\ }^{5 \ 2 \ 6}}}-{{p_{1}}\ {|_{\ }^{5 \ 3 \ 7}}}-{{p_{1}}\ {|_{\ }^{5 \ 4 \ 8}}}+{{p_{1}}\ {|_{\ }^{5 \ 5 \ 1}}}+{{p_{1}}\ {|_{\ }^{5 \ 6 \ 2}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{5 \ 7 \ 3}}}+{{p_{1}}\ {|_{\ }^{5 \ 8 \ 4}}}+{{p_{1}}\ {|_{\ }^{6 \ 1 \ 6}}}+{{p_{1}}\ {|_{\ }^{6 \ 2 \ 5}}}-{{p_{1}}\ {|_{\ }^{6 \ 3 \ 8}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{6 \ 4 \ 7}}}-{{p_{1}}\ {|_{\ }^{6 \ 5 \ 2}}}+{{p_{1}}\ {|_{\ }^{6 \ 6 \ 1}}}-{{p_{1}}\ {|_{\ }^{6 \ 7 \ 4}}}+{{p_{1}}\ {|_{\ }^{6 \ 8 \ 3}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{7 \ 1 \ 7}}}+{{p_{1}}\ {|_{\ }^{7 \ 2 \ 8}}}+{{p_{1}}\ {|_{\ }^{7 \ 3 \ 5}}}-{{p_{1}}\ {|_{\ }^{7 \ 4 \ 6}}}-{{p_{1}}\ {|_{\ }^{7 \ 5 \ 3}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{7 \ 6 \ 4}}}+{{p_{1}}\ {|_{\ }^{7 \ 7 \ 1}}}-{{p_{1}}\ {|_{\ }^{7 \ 8 \ 2}}}+{{p_{1}}\ {|_{\ }^{8 \ 1 \ 8}}}-{{p_{1}}\ {|_{\ }^{8 \ 2 \ 7}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{8 \ 3 \ 6}}}+{{p_{1}}\ {|_{\ }^{8 \ 4 \ 5}}}-{{p_{1}}\ {|_{\ }^{8 \ 5 \ 4}}}-{{p_{1}}\ {|_{\ }^{8 \ 6 \ 3}}}+{{p_{1}}\ {|_{\ }^{8 \ 7 \ 2}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{8 \ 8 \ 1}}}$

(30)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

λ:=(Ω,I,Ω)/(I,W,I)

$\label{eq31}\begin{array}{@{}l} \displaystyle -{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 1}^{1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 2}^{1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 3}^{1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{1}}}+{{1 \over{p_{1}}}\ {|_{5 \ 5}^{1}}}+ \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{6 \ 6}^{1}}}+{{1 \over{p_{1}}}\ {|_{7 \ 7}^{1}}}+{{1 \over{p_{1}}}\ {|_{8 \ 8}^{1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{2}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{2}}}-{{1 \over{p_{1}}}\ {|_{5 \ 6}^{2}}}+{{1 \over{p_{1}}}\ {|_{6 \ 5}^{2}}}+{{1 \over{p_{1}}}\ {|_{7 \ 8}^{2}}}- \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{8 \ 7}^{2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{3}}}- \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{5 \ 7}^{3}}}-{{1 \over{p_{1}}}\ {|_{6 \ 8}^{3}}}+{{1 \over{p_{1}}}\ {|_{7 \ 5}^{3}}}+{{1 \over{p_{1}}}\ {|_{8 \ 6}^{3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{4}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{4}}}-{{1 \over{p_{1}}}\ {|_{5 \ 8}^{4}}}+{{1 \over{p_{1}}}\ {|_{6 \ 7}^{4}}}- \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{7 \ 6}^{4}}}+{{1 \over{p_{1}}}\ {|_{8 \ 5}^{4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 5}^{5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 6}^{5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 7}^{5}}}+ \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 1}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 2}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 3}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{5}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{6}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{7}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{7}}}+ \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{8}}}+ \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{8}}}$

(31)

Type: LinearOperator?(8,OrderedVariableList?([]),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}}}\ {|_{1 \ 1}^{1 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 2}^{1 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 3}^{1 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{1 \ 1}}}+
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 5}^{1 \ 1}}}+{{1 \over{p_{1}}}\ {|_{6 \ 6}^{1 \ 1}}}+{{1 \over{p_{1}}}\ {|_{7 \ 7}^{1 \ 1}}}+{{1 \over{p_{1}}}\ {|_{8 \ 8}^{1 \ 1}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{1 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{1 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{1 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{1 \ 2}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 6}^{1 \ 2}}}+{{1 \over{p_{1}}}\ {|_{6 \ 5}^{1 \ 2}}}+{{1 \over{p_{1}}}\ {|_{7 \ 8}^{1 \ 2}}}-{{1 \over{p_{1}}}\ {|_{8 \ 7}^{1 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{1 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{1 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{1 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{1 \ 3}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 7}^{1 \ 3}}}-{{1 \over{p_{1}}}\ {|_{6 \ 8}^{1 \ 3}}}+{{1 \over{p_{1}}}\ {|_{7 \ 5}^{1 \ 3}}}+{{1 \over{p_{1}}}\ {|_{8 \ 6}^{1 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{1 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{1 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{1 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{1 \ 4}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 8}^{1 \ 4}}}+{{1 \over{p_{1}}}\ {|_{6 \ 7}^{1 \ 4}}}-{{1 \over{p_{1}}}\ {|_{7 \ 6}^{1 \ 4}}}+{{1 \over{p_{1}}}\ {|_{8 \ 5}^{1 \ 4}}}-
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{1 \ 5}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{1 \ 5}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{1 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{1 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{1 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{1 \ 6}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{1 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{1 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{1 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{1 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{1 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{1 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{1 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{1 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{1 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{1 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{1 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{1 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{1 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{1 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{1 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{1 \ 8}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{1 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{1 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{1 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{1 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{2 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{2 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{2 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{2 \ 1}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 6}^{2 \ 1}}}+{{1 \over{p_{1}}}\ {|_{6 \ 5}^{2 \ 1}}}+{{1 \over{p_{1}}}\ {|_{7 \ 8}^{2 \ 1}}}-{{1 \over{p_{1}}}\ {|_{8 \ 7}^{2 \ 1}}}+
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{2 \ 2}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 5}^{2 \ 2}}}-{{1 \over{p_{1}}}\ {|_{6 \ 6}^{2 \ 2}}}-{{1 \over{p_{1}}}\ {|_{7 \ 7}^{2 \ 2}}}-{{1 \over{p_{1}}}\ {|_{8 \ 8}^{2 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{2 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{2 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{2 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{2 \ 3}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 8}^{2 \ 3}}}+{{1 \over{p_{1}}}\ {|_{6 \ 7}^{2 \ 3}}}-{{1 \over{p_{1}}}\ {|_{7 \ 6}^{2 \ 3}}}+{{1 \over{p_{1}}}\ {|_{8 \ 5}^{2 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{2 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{2 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{2 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{2 \ 4}}}+
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 7}^{2 \ 4}}}+{{1 \over{p_{1}}}\ {|_{6 \ 8}^{2 \ 4}}}-{{1 \over{p_{1}}}\ {|_{7 \ 5}^{2 \ 4}}}-{{1 \over{p_{1}}}\ {|_{8 \ 6}^{2 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{2 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{2 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{2 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{2 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{2 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{2 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{2 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{2 \ 5}}}+
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{2 \ 6}}}+
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{2 \ 6}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{2 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{2 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{2 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{2 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{2 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{2 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{2 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{2 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{2 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{2 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{2 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{2 \ 8}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{2 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{2 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{2 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{2 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{3 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{3 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{3 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{3 \ 1}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 7}^{3 \ 1}}}-{{1 \over{p_{1}}}\ {|_{6 \ 8}^{3 \ 1}}}+{{1 \over{p_{1}}}\ {|_{7 \ 5}^{3 \ 1}}}+{{1 \over{p_{1}}}\ {|_{8 \ 6}^{3 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{3 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{3 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{3 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{3 \ 2}}}+
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 8}^{3 \ 2}}}-{{1 \over{p_{1}}}\ {|_{6 \ 7}^{3 \ 2}}}+{{1 \over{p_{1}}}\ {|_{7 \ 6}^{3 \ 2}}}-{{1 \over{p_{1}}}\ {|_{8 \ 5}^{3 \ 2}}}+
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{3 \ 3}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 5}^{3 \ 3}}}-{{1 \over{p_{1}}}\ {|_{6 \ 6}^{3 \ 3}}}-{{1 \over{p_{1}}}\ {|_{7 \ 7}^{3 \ 3}}}-{{1 \over{p_{1}}}\ {|_{8 \ 8}^{3 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{3 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{3 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{3 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{3 \ 4}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 6}^{3 \ 4}}}+{{1 \over{p_{1}}}\ {|_{6 \ 5}^{3 \ 4}}}+{{1 \over{p_{1}}}\ {|_{7 \ 8}^{3 \ 4}}}-{{1 \over{p_{1}}}\ {|_{8 \ 7}^{3 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{3 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{3 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{3 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{3 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{3 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{3 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{3 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{3 \ 5}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{3 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{3 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{3 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{3 \ 6}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{3 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{3 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{3 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{3 \ 6}}}+
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{3 \ 7}}}+
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{3 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{3 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{3 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{3 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{3 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{3 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{3 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{3 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{3 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{4 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{4 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{4 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{4 \ 1}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 8}^{4 \ 1}}}+{{1 \over{p_{1}}}\ {|_{6 \ 7}^{4 \ 1}}}-{{1 \over{p_{1}}}\ {|_{7 \ 6}^{4 \ 1}}}+{{1 \over{p_{1}}}\ {|_{8 \ 5}^{4 \ 1}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{4 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{4 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{4 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{4 \ 2}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 7}^{4 \ 2}}}-{{1 \over{p_{1}}}\ {|_{6 \ 8}^{4 \ 2}}}+{{1 \over{p_{1}}}\ {|_{7 \ 5}^{4 \ 2}}}+{{1 \over{p_{1}}}\ {|_{8 \ 6}^{4 \ 2}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{4 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{4 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{4 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{4 \ 3}}}+
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 6}^{4 \ 3}}}-{{1 \over{p_{1}}}\ {|_{6 \ 5}^{4 \ 3}}}-{{1 \over{p_{1}}}\ {|_{7 \ 8}^{4 \ 3}}}+{{1 \over{p_{1}}}\ {|_{8 \ 7}^{4 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 1}^{4 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 2}^{4 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 3}^{4 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{4 \ 4}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 5}^{4 \ 4}}}-{{1 \over{p_{1}}}\ {|_{6 \ 6}^{4 \ 4}}}-{{1 \over{p_{1}}}\ {|_{7 \ 7}^{4 \ 4}}}-{{1 \over{p_{1}}}\ {|_{8 \ 8}^{4 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{4 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{4 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{4 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{4 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{4 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{4 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{4 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{4 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{4 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{4 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{4 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{4 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{4 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{4 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{4 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{4 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{4 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{4 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{4 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{4 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{4 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{4 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{4 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{4 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 5}^{4 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 6}^{4 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 7}^{4 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{4 \ 8}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 1}^{4 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 2}^{4 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 3}^{4 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{4 \ 8}}}-
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{5 \ 1}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{5 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{5 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{5 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{5 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{5 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{5 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{5 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{5 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{5 \ 2}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{5 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{5 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{5 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{5 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{5 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{5 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{5 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{5 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{5 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{5 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{5 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{5 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{5 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{5 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{5 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{5 \ 4}}}+
\
\
\displaystyle
{{{{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}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 4}^{5 \ 5}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 8}^{5 \ 5}}}-
\
\
\displaystyle
{{{{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 \ 4}^{5 \ 6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 3}^{5 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 6}^{5 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 5}^{5 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 8}^{5 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 7}^{5 \ 6}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 3}^{5 \ 7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 4}^{5 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 1}^{5 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 2}^{5 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 7}^{5 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 8}^{5 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 5}^{5 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 6}^{5 \ 7}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 4}^{5 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 3}^{5 \ 8}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 2}^{5 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 1}^{5 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 8}^{5 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 7}^{5 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 6}^{5 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 5}^{5 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{6 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{6 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{6 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{6 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{6 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{6 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{6 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{6 \ 1}}}-
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{6 \ 2}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{6 \ 2}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{6 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{6 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{6 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{6 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{6 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{6 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{6 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{6 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{6 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{6 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{6 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{6 \ 4}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{6 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{6 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{6 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{6 \ 4}}}+
\
\
\displaystyle
{{{{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 \ 4}^{6 \ 5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 3}^{6 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 6}^{6 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 5}^{6 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 8}^{6 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 7}^{6 \ 5}}}+
\
\
\displaystyle
{{{{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}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 4}^{6 \ 6}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 8}^{6 \ 6}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 4}^{6 \ 7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 3}^{6 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 2}^{6 \ 7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 1}^{6 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 8}^{6 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 7}^{6 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 6}^{6 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 5}^{6 \ 7}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 3}^{6 \ 8}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 4}^{6 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 1}^{6 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 2}^{6 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 7}^{6 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 8}^{6 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 5}^{6 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 6}^{6 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{7 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{7 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{7 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{7 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{7 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{7 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{7 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{7 \ 1}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{7 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{7 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{7 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{7 \ 2}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{7 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{7 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{7 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{7 \ 2}}}-
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{7 \ 3}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{7 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{7 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{7 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{7 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{7 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{7 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{7 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{7 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{7 \ 4}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 3}^{7 \ 5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 4}^{7 \ 5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 1}^{7 \ 5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 2}^{7 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 7}^{7 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 8}^{7 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 5}^{7 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 6}^{7 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 4}^{7 \ 6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 3}^{7 \ 6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 2}^{7 \ 6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 1}^{7 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 8}^{7 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 7}^{7 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 6}^{7 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 5}^{7 \ 6}}}+
\
\
\displaystyle
{{{{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}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 4}^{7 \ 7}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 8}^{7 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 2}^{7 \ 8}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 1}^{7 \ 8}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 4}^{7 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 3}^{7 \ 8}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 6}^{7 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 5}^{7 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 8}^{7 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 7}^{7 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{8 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{8 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{8 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{8 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{8 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{8 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{8 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{8 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{8 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{8 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{8 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{8 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{8 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{8 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{8 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{8 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{8 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{8 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{8 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{8 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{8 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{8 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{8 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{8 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 5}^{8 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 6}^{8 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 7}^{8 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{8 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 1}^{8 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 2}^{8 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 3}^{8 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{8 \ 4}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 4}^{8 \ 5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 3}^{8 \ 5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 2}^{8 \ 5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 1}^{8 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 8}^{8 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 7}^{8 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 6}^{8 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 5}^{8 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 3}^{8 \ 6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 4}^{8 \ 6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 1}^{8 \ 6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 2}^{8 \ 6}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 7}^{8 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 8}^{8 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 5}^{8 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 6}^{8 \ 6}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 2}^{8 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 1}^{8 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 4}^{8 \ 7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 3}^{8 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 6}^{8 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 5}^{8 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 8}^{8 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 7}^{8 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 1}^{8 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 2}^{8 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 3}^{8 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 4}^{8 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 5}^{8 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 6}^{8 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 7}^{8 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 8}^{8 \ 8}}}

(34)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

Χr := (λ,I)/(I,Y)

\label{eq35}\begin{array}{@{}l}
\displaystyle
-{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{1 \ 1}}}+
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 5}^{1 \ 1}}}+{{1 \over{p_{1}}}\ {|_{6 \ 6}^{1 \ 1}}}+{{1 \over{p_{1}}}\ {|_{7 \ 7}^{1 \ 1}}}+{{1 \over{p_{1}}}\ {|_{8 \ 8}^{1 \ 1}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{1 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{1 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{1 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{1 \ 2}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 6}^{1 \ 2}}}+{{1 \over{p_{1}}}\ {|_{6 \ 5}^{1 \ 2}}}+{{1 \over{p_{1}}}\ {|_{7 \ 8}^{1 \ 2}}}-{{1 \over{p_{1}}}\ {|_{8 \ 7}^{1 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{1 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{1 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{1 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{1 \ 3}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 7}^{1 \ 3}}}-{{1 \over{p_{1}}}\ {|_{6 \ 8}^{1 \ 3}}}+{{1 \over{p_{1}}}\ {|_{7 \ 5}^{1 \ 3}}}+{{1 \over{p_{1}}}\ {|_{8 \ 6}^{1 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{1 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{1 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{1 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{1 \ 4}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 8}^{1 \ 4}}}+{{1 \over{p_{1}}}\ {|_{6 \ 7}^{1 \ 4}}}-{{1 \over{p_{1}}}\ {|_{7 \ 6}^{1 \ 4}}}+{{1 \over{p_{1}}}\ {|_{8 \ 5}^{1 \ 4}}}-
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{1 \ 5}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{1 \ 5}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{1 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{1 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{1 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{1 \ 6}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{1 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{1 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{1 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{1 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{1 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{1 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{1 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{1 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{1 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{1 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{1 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{1 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{1 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{1 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{1 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{1 \ 8}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{1 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{1 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{1 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{1 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{2 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{2 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{2 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{2 \ 1}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 6}^{2 \ 1}}}+{{1 \over{p_{1}}}\ {|_{6 \ 5}^{2 \ 1}}}+{{1 \over{p_{1}}}\ {|_{7 \ 8}^{2 \ 1}}}-{{1 \over{p_{1}}}\ {|_{8 \ 7}^{2 \ 1}}}+
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{2 \ 2}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 5}^{2 \ 2}}}-{{1 \over{p_{1}}}\ {|_{6 \ 6}^{2 \ 2}}}-{{1 \over{p_{1}}}\ {|_{7 \ 7}^{2 \ 2}}}-{{1 \over{p_{1}}}\ {|_{8 \ 8}^{2 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{2 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{2 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{2 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{2 \ 3}}}+
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 8}^{2 \ 3}}}-{{1 \over{p_{1}}}\ {|_{6 \ 7}^{2 \ 3}}}+{{1 \over{p_{1}}}\ {|_{7 \ 6}^{2 \ 3}}}-{{1 \over{p_{1}}}\ {|_{8 \ 5}^{2 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{2 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{2 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{2 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{2 \ 4}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 7}^{2 \ 4}}}-{{1 \over{p_{1}}}\ {|_{6 \ 8}^{2 \ 4}}}+{{1 \over{p_{1}}}\ {|_{7 \ 5}^{2 \ 4}}}+{{1 \over{p_{1}}}\ {|_{8 \ 6}^{2 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{2 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{2 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{2 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{2 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{2 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{2 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{2 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{2 \ 5}}}+
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{2 \ 6}}}+
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{2 \ 6}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{2 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{2 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{2 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{2 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{2 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{2 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{2 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{2 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{2 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{2 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{2 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{2 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{2 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{2 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{2 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{2 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{3 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{3 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{3 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{3 \ 1}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 7}^{3 \ 1}}}-{{1 \over{p_{1}}}\ {|_{6 \ 8}^{3 \ 1}}}+{{1 \over{p_{1}}}\ {|_{7 \ 5}^{3 \ 1}}}+{{1 \over{p_{1}}}\ {|_{8 \ 6}^{3 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{3 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{3 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{3 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{3 \ 2}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 8}^{3 \ 2}}}+{{1 \over{p_{1}}}\ {|_{6 \ 7}^{3 \ 2}}}-{{1 \over{p_{1}}}\ {|_{7 \ 6}^{3 \ 2}}}+{{1 \over{p_{1}}}\ {|_{8 \ 5}^{3 \ 2}}}+
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{3 \ 3}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 5}^{3 \ 3}}}-{{1 \over{p_{1}}}\ {|_{6 \ 6}^{3 \ 3}}}-{{1 \over{p_{1}}}\ {|_{7 \ 7}^{3 \ 3}}}-{{1 \over{p_{1}}}\ {|_{8 \ 8}^{3 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{3 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{3 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{3 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{3 \ 4}}}+
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 6}^{3 \ 4}}}-{{1 \over{p_{1}}}\ {|_{6 \ 5}^{3 \ 4}}}-{{1 \over{p_{1}}}\ {|_{7 \ 8}^{3 \ 4}}}+{{1 \over{p_{1}}}\ {|_{8 \ 7}^{3 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{3 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{3 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{3 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{3 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{3 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{3 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{3 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{3 \ 5}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{3 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{3 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{3 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{3 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{3 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{3 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{3 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{3 \ 6}}}+
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{3 \ 7}}}+
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{3 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{3 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{3 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{3 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{3 \ 8}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{3 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{3 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{3 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{3 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{4 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{4 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{4 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{4 \ 1}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 8}^{4 \ 1}}}+{{1 \over{p_{1}}}\ {|_{6 \ 7}^{4 \ 1}}}-{{1 \over{p_{1}}}\ {|_{7 \ 6}^{4 \ 1}}}+{{1 \over{p_{1}}}\ {|_{8 \ 5}^{4 \ 1}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{4 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{4 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{4 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{4 \ 2}}}+
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 7}^{4 \ 2}}}+{{1 \over{p_{1}}}\ {|_{6 \ 8}^{4 \ 2}}}-{{1 \over{p_{1}}}\ {|_{7 \ 5}^{4 \ 2}}}-{{1 \over{p_{1}}}\ {|_{8 \ 6}^{4 \ 2}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{4 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{4 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{4 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{4 \ 3}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 6}^{4 \ 3}}}+{{1 \over{p_{1}}}\ {|_{6 \ 5}^{4 \ 3}}}+{{1 \over{p_{1}}}\ {|_{7 \ 8}^{4 \ 3}}}-{{1 \over{p_{1}}}\ {|_{8 \ 7}^{4 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 1}^{4 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 2}^{4 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 3}^{4 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{4 \ 4}}}-
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ 5}^{4 \ 4}}}-{{1 \over{p_{1}}}\ {|_{6 \ 6}^{4 \ 4}}}-{{1 \over{p_{1}}}\ {|_{7 \ 7}^{4 \ 4}}}-{{1 \over{p_{1}}}\ {|_{8 \ 8}^{4 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{4 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{4 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{4 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{4 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{4 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{4 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{4 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{4 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{4 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{4 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{4 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{4 \ 6}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{4 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{4 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{4 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{4 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{4 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{4 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{4 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{4 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{4 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{4 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{4 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{4 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 5}^{4 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 6}^{4 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 7}^{4 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{4 \ 8}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 1}^{4 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 2}^{4 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 3}^{4 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{4 \ 8}}}-
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{5 \ 1}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{5 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{5 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{5 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{5 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{5 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{5 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{5 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{5 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{5 \ 2}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{5 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{5 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{5 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{5 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{5 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{5 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{5 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{5 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{5 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{5 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{5 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{5 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{5 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{5 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{5 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{5 \ 4}}}+
\
\
\displaystyle
{{{{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}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 4}^{5 \ 5}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 8}^{5 \ 5}}}-
\
\
\displaystyle
{{{{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 \ 4}^{5 \ 6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 3}^{5 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 6}^{5 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 5}^{5 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 8}^{5 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 7}^{5 \ 6}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 3}^{5 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 4}^{5 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 1}^{5 \ 7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 2}^{5 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 7}^{5 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 8}^{5 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 5}^{5 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 6}^{5 \ 7}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 4}^{5 \ 8}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 3}^{5 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 2}^{5 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 1}^{5 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 8}^{5 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 7}^{5 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 6}^{5 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 5}^{5 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{6 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{6 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{6 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{6 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{6 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{6 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{6 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{6 \ 1}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{6 \ 2}}}-
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{6 \ 2}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{6 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{6 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{6 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{6 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{6 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{6 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{6 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{6 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{6 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{6 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{6 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{6 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{6 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{6 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{6 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{6 \ 4}}}+
\
\
\displaystyle
{{{{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 \ 4}^{6 \ 5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 3}^{6 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 6}^{6 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 5}^{6 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 8}^{6 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 7}^{6 \ 5}}}+
\
\
\displaystyle
{{{{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}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 4}^{6 \ 6}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 8}^{6 \ 6}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 4}^{6 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 3}^{6 \ 7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 2}^{6 \ 7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 1}^{6 \ 7}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 8}^{6 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 7}^{6 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 6}^{6 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 5}^{6 \ 7}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 3}^{6 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 4}^{6 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 1}^{6 \ 8}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 2}^{6 \ 8}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 7}^{6 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 8}^{6 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 5}^{6 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 6}^{6 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{7 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{7 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{7 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{7 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{7 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{7 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{7 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{7 \ 1}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{7 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{7 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{7 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{7 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{7 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{7 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{7 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{7 \ 2}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{7 \ 3}}}-
\
\
\displaystyle
{{{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}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{7 \ 3}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{7 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{7 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{7 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{7 \ 4}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{7 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{7 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{7 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{7 \ 4}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 3}^{7 \ 5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 4}^{7 \ 5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 1}^{7 \ 5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 2}^{7 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 7}^{7 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 8}^{7 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 5}^{7 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 6}^{7 \ 5}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 4}^{7 \ 6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 3}^{7 \ 6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 2}^{7 \ 6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 1}^{7 \ 6}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 8}^{7 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 7}^{7 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 6}^{7 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 5}^{7 \ 6}}}+
\
\
\displaystyle
{{{{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}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 4}^{7 \ 7}}}-
\
\
\displaystyle
{{{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}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 8}^{7 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 2}^{7 \ 8}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 1}^{7 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 4}^{7 \ 8}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 3}^{7 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 6}^{7 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 5}^{7 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 8}^{7 \ 8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 7}^{7 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{8 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{8 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{8 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{8 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{8 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{8 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{8 \ 1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{8 \ 1}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{8 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{8 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{8 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{8 \ 2}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{8 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{8 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{8 \ 2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{8 \ 2}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{8 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{8 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{8 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{8 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{8 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{8 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{8 \ 3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{8 \ 3}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 5}^{8 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 6}^{8 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 7}^{8 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{8 \ 4}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 1}^{8 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 2}^{8 \ 4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 3}^{8 \ 4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{8 \ 4}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 4}^{8 \ 5}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 3}^{8 \ 5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 2}^{8 \ 5}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 1}^{8 \ 5}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 8}^{8 \ 5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 7}^{8 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 6}^{8 \ 5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 5}^{8 \ 5}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 3}^{8 \ 6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 4}^{8 \ 6}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 1}^{8 \ 6}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 2}^{8 \ 6}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 7}^{8 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 8}^{8 \ 6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 5}^{8 \ 6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 6}^{8 \ 6}}}-
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 2}^{8 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 1}^{8 \ 7}}}+{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 4}^{8 \ 7}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 3}^{8 \ 7}}}+
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 6}^{8 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 5}^{8 \ 7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 8}^{8 \ 7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 7}^{8 \ 7}}}+
\
\
\displaystyle
{{{{q_{2}}^2}\over{p_{1}}}\ {|_{1 \ 1}^{8 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{2 \ 2}^{8 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{3 \ 3}^{8 \ 8}}}-{{{{q_{2}}^2}\over{p_{1}}}\ {|_{4 \ 4}^{8 \ 8}}}-
\
\
\displaystyle
{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 5}^{8 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 6}^{8 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 7}^{8 \ 8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 8}^{8 \ 8}}}

(35)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

test(Χr = Χ )

$\label{eq36} \mbox{\rm false}$

(36)

Type: Boolean

axiom

Χl := (I,λ)/(Y,I);

Type: LinearOperator?(8,OrderedVariableList?([]),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

$\label{eq39}|_{1}$

(39)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

test
         i     /
         λ     =    Ω

$\label{eq40} \mbox{\rm true}$

(40)

Type: Boolean

Handle

axiom

$\label{eq41}\begin{array}{@{}l} \displaystyle -{{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{1}^{1}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{2}^{2}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{3}^{3}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{4}^{4}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{5}^{5}}}+ \ \ \displaystyle {{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{6}^{6}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{7}^{7}}}+{{{4 \ {q_{2}}}\over{p_{1}}}\ {|_{8}^{8}}}$

(41)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

Definition 5

Co-unit

  i 
  U

axiom

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

$\label{eq42}-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1}}}$

(42)

Type: LinearOperator?(8,OrderedVariableList?([]),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)

$\label{eq45}-{|_{\ }^{1 \ 1}}+{|_{\ }^{2 \ 2}}+{|_{\ }^{3 \ 3}}+{|_{\ }^{4 \ 4}}+{|_{\ }^{5 \ 5}}+{|_{\ }^{6 \ 6}}+{|_{\ }^{7 \ 7}}+{|_{\ }^{8 \ 8}}$

(45)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

Ω0:𝐋  :=eval(Ω,ex1)$𝐋

$\label{eq46}-{|_{1 \ 1}}+{|_{2 \ 2}}+{|_{3 \ 3}}+{|_{4 \ 4}}+{|_{5 \ 5}}+{|_{6 \ 6}}+{|_{7 \ 7}}+{|_{8 \ 8}}$

(46)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

λ0:𝐋  :=eval(λ,ex1)$𝐋

$\label{eq47}\begin{array}{@{}l} \displaystyle -{|_{1 \ 1}^{1}}+{|_{2 \ 2}^{1}}+{|_{3 \ 3}^{1}}+{|_{4 \ 4}^{1}}+{|_{5 \ 5}^{1}}+{|_{6 \ 6}^{1}}+{|_{7 \ 7}^{1}}+ \ \ \displaystyle {|_{8 \ 8}^{1}}-{|_{1 \ 2}^{2}}-{|_{2 \ 1}^{2}}-{|_{3 \ 4}^{2}}+{|_{4 \ 3}^{2}}-{|_{5 \ 6}^{2}}+{|_{6 \ 5}^{2}}+{|_{7 \ 8}^{2}}- \ \ \displaystyle {|_{8 \ 7}^{2}}-{|_{1 \ 3}^{3}}+{|_{2 \ 4}^{3}}-{|_{3 \ 1}^{3}}-{|_{4 \ 2}^{3}}-{|_{5 \ 7}^{3}}-{|_{6 \ 8}^{3}}+{|_{7 \ 5}^{3}}+ \ \ \displaystyle {|_{8 \ 6}^{3}}-{|_{1 \ 4}^{4}}-{|_{2 \ 3}^{4}}+{|_{3 \ 2}^{4}}-{|_{4 \ 1}^{4}}-{|_{5 \ 8}^{4}}+{|_{6 \ 7}^{4}}-{|_{7 \ 6}^{4}}+ \ \ \displaystyle {|_{8 \ 5}^{4}}-{|_{1 \ 5}^{5}}+{|_{2 \ 6}^{5}}+{|_{3 \ 7}^{5}}+{|_{4 \ 8}^{5}}-{|_{5 \ 1}^{5}}-{|_{6 \ 2}^{5}}-{|_{7 \ 3}^{5}}- \ \ \displaystyle {|_{8 \ 4}^{5}}-{|_{1 \ 6}^{6}}-{|_{2 \ 5}^{6}}+{|_{3 \ 8}^{6}}-{|_{4 \ 7}^{6}}+{|_{5 \ 2}^{6}}-{|_{6 \ 1}^{6}}+{|_{7 \ 4}^{6}}- \ \ \displaystyle {|_{8 \ 3}^{6}}-{|_{1 \ 7}^{7}}-{|_{2 \ 8}^{7}}-{|_{3 \ 5}^{7}}+{|_{4 \ 6}^{7}}+{|_{5 \ 3}^{7}}-{|_{6 \ 4}^{7}}-{|_{7 \ 1}^{7}}+ \ \ \displaystyle {|_{8 \ 2}^{7}}-{|_{1 \ 8}^{8}}+{|_{2 \ 7}^{8}}-{|_{3 \ 6}^{8}}-{|_{4 \ 5}^{8}}+{|_{5 \ 4}^{8}}+{|_{6 \ 3}^{8}}-{|_{7 \ 2}^{8}}- \ \ \displaystyle {|_{8 \ 1}^{8}}$

(47)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

H0:𝐋 :=eval(H,ex1)$𝐋

$\label{eq48}-{8 \ {|_{1}^{1}}}+{4 \ {|_{2}^{2}}}+{4 \ {|_{3}^{3}}}+{4 \ {|_{4}^{4}}}+{4 \ {|_{5}^{5}}}+{4 \ {|_{6}^{6}}}+{4 \ {|_{7}^{7}}}+{4 \ {|_{8}^{8}}}$

(48)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))