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# Edit detail for Snake Relation revision 9 of 11

 1 2 3 4 5 6 7 8 9 10 11 Editor: Bill Page Time: 2011/05/08 14:16:39 GMT-7 Note: Twist dimension or twist snake?

From BillPage Sun May 8 14:16:39 -0700 2011
From: Bill Page
Date: Sun, 08 May 2011 14:16:39 -0700
Subject: Twist dimension or twist snake?
Message-ID: <20110508141639-0700@axiom-wiki.newsynthesis.org>

TwistedSnakeRelation

Non-degeneracy of the pairing

Ref:

We use the Axiom LinearOperator? library

fricas
)library MONAL PROP LIN
)library cannot find the file LIN. Monoidal is now explicitly exposed in frame initial Monoidal will be automatically loaded when needed from /var/aw/var/LatexWiki/MONAL.NRLIB/MONAL Prop is now explicitly exposed in frame initial Prop will be automatically loaded when needed from /var/aw/var/LatexWiki/PROP.NRLIB/PROP

and convenient notation

fricas
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
fricas
macro Ξ(f,i,n)==[f for i in n]
Type: Void
fricas
macro sb == subscript
Type: Void
fricas
macro sp == superscript
Type: Void

Let 𝐋 be the domain of 2-dimensional linear operators

fricas
dim:=2
 (1)
Type: PositiveInteger?
fricas
macro ℒ == List
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
𝐋 := LinearOperator(dim, OVAR [], ℚ)
There are no library operations named LinearOperator Use HyperDoc Browse or issue )what op LinearOperator to learn if there is any operation containing " LinearOperator " in its name.
Cannot find a definition or applicable library operation named LinearOperator with argument type(s) PositiveInteger Type Type
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. 𝐞:ℒ 𝐋 := basisVectors() There are no library operations named basisVectors Use HyperDoc Browse or issue )what op basisVectors to learn if there is any operation containing " basisVectors " in its name. Cannot find a no-argument definition or library operation named basisVectors . 𝐝:ℒ 𝐋 := basisForms() There are no library operations named basisForms Use HyperDoc Browse or issue )what op basisForms to learn if there is any operation containing " basisForms " in its name. Cannot find a no-argument definition or library operation named basisForms . I:𝐋:=[1] -- identity for composition  (2) Type: List(PositiveInteger?) fricas X:𝐋:=[2,1] -- twist  (3) Type: List(PositiveInteger?) ## Pairing A scalar product (pairing) is represented by fricas U:=Σ(Σ(sp('u,[i,j])*𝐝.i*𝐝.j, i,1..dim), j,1..dim) >> System error: The index 3 is too large. In general we do not require that it be symmetric. ## Co-pairing Solve the "snake relation" as a system of linear equations. fricas Ω:𝐋:=Σ(Σ(sb('u,[i,j])*𝐞.i*𝐞.j, i,1..dim), j,1..dim) >> System error: The index 3 is too large. This is equivalent to a matrix inverse (transposed!) fricas Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim) >> System error: The index 3 is too large. Check that the snake relation holds fricas test ( I Ω ) / ( U I ) = I There are no exposed library operations named I but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op I to learn more about the available operation. Cannot find a definition or applicable library operation named I with argument type(s) Variable(Ω) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
test ( Ω I ) / ( I U ) = I
There are no library operations named Ω Use HyperDoc Browse or issue )what op Ω to learn if there is any operation containing " Ω " in its name.
Cannot find a definition or applicable library operation named Ω with argument type(s) List(PositiveInteger)

## Symmetric Pairing

Repeat the calculation, assuming that U is symmetric.

fricas
sym:=groebner ravel(U-X/U)
There are 13 exposed and 12 unexposed library operations named / having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op / to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named / with argument type(s) List(PositiveInteger) Variable(U)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. vars:=map(x+->kernels(x).1,sym)::List Symbol >> Error detected within library code: index out of range fricas Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim) >> System error: The index 3 is too large. Check that the snake relation holds fricas test ( I Ω ) / ( U I ) = I There are no exposed library operations named I but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op I to learn more about the available operation. Cannot find a definition or applicable library operation named I with argument type(s) Variable(Ω) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
test ( Ω I ) / ( I U ) = I
There are no library operations named Ω Use HyperDoc Browse or issue )what op Ω to learn if there is any operation containing " Ω " in its name.
Cannot find a definition or applicable library operation named Ω with argument type(s) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. These quantities no longer depends on ! fricas d:= Ω / U  (5) Type: Fraction(Polynomial(Integer)) fricas d':= Ω / X / U There are 13 exposed and 12 unexposed library operations named / having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op / to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named / with argument type(s) Variable(Ω) List(PositiveInteger) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Twist dimension or twist snake? --Bill Page, Sun, 08 May 2011 14:16:39 -0700 reply
TwistedSnakeRelation?