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# Edit detail for SandBoxFrobeniusAlgebra revision 3 of 26

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Editor: Bill Page Time: 2011/02/11 20:07:45 GMT-8 Note: reindex

changed:
-An n-dimensional algebra is represented by a tensor $Y=\{ y_{ij}^k \} \ i,j,k =1,2, ... n$ viewed as an operator with two inputs 'i,j' and one output 'k'.
An n-dimensional algebra is represented by a tensor $Y=\{ {y_{ij}}^k \} \ i,j,k =1,2, ... n$ viewed as an operator with two inputs 'i,j' and one output 'k'.

reindex(Yijk,[3,1,2])

reindex(Yijk,[3,1,2])*Ui*Vj


An n-dimensional algebra is represented by a tensor viewed as an operator with two inputs i,j and one output k.

axiom
n:=2
 (1)
Type: PositiveInteger?
axiom
T:=CartesianTensor(1,n,EXPR INT)
 (2)
Type: Domain
axiom
Yijk:=unravel(concat concat
[[[script(y,[[i,j],[k]])
for k in 1..n]
for j in 1..n]
for i in 1..n]
)$T  (3) Type: CartesianTensor?(1,2,Expression(Integer)) axiom reindex(Yijk,[3,1,2])  (4) Type: CartesianTensor?(1,2,Expression(Integer)) axiom Y.[1,1,2]  (5) Type: Symbol axiom Y.[1,2,1]  (6) Type: Symbol axiom Y.[2,1,1]  (7) Type: Symbol Given two vectors U and V axiom Ui:=unravel([script(u,[[],[i]]) for i in 1..n])$T
 (8)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
Vj:=unravel([script(v,[[],[i]]) for i in
1..n])\$T
 (9)
Type: CartesianTensor?(1,2,Expression(Integer))

the tensor Y operates on their tensor product

axiom
UVij:=product(Ui,Vj)
 (10)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
UVij.[1,2]
 (11)
Type: Expression(Integer)
axiom
UVij.[2,1]
 (12)
Type: Expression(Integer)
axiom
YUV:=product(Yijk,UVij)
 (13)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
YUV.[1,1,1,1,2]
 (14)
Type: Expression(Integer)
axiom
YUV.[1,1,1,2,1]
 (15)
Type: Expression(Integer)
axiom
YUV.[1,1,2,1,1]
 (16)
Type: Expression(Integer)
axiom
YUV.[1,2,1,1,1]
 (17)
Type: Expression(Integer)
axiom
YUV.[2,1,1,1,1]
 (18)
Type: Expression(Integer)
axiom
contract(contract(YUV,1,4),1,3)
 (19)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
contract(contract(Yijk,1,UVij,1),1,3)
 (20)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
reindex(Yijk,[3,1,2])*Ui*Vj
 (21)
Type: CartesianTensor?(1,2,Expression(Integer))