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SandBoxTensorProduct3

Edit detail for SandBoxTensorProduct3 revision 1 of 2

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Editor: pagani Time: 2016/12/09 20:43:31 GMT+0
Note:

changed:
-
\begin{axiom}
R ==> EXPR INT

e:=[subscript('e, [k]) for k in 1..3]
g:=[subscript('g, [k]) for k in 1..4]
h:=[superscript('h, [k]) for k in 1..4]

B1:=OrderedVariableList e
B2:=OrderedVariableList g 
B3:=OrderedVariableList h 

M1:=FreeModule(R,B1)
M2:=FreeModule(R,B2)
M3:=FreeModule(R,B3)

M12:=TensorProduct(R,B1,B2,M1,M2)
M23:=TensorProduct(R,B2,B3,M2,M3)
M123:=TensorProduct(R,Product(B1,B2),B3,M12,M3)

v1:=x*(e.1)::M1 - y*(e.3)::M1
v2:=q*(g.2)::M2 + r*(g.4)::M2
v3:=3*(h.1)::M3 - z*(h.3)::M3

t12:=tensor(v1,v2)$M12
t23:=tensor(v2,v3)$M23

t123:=tensor(t12,v3)$M123

N:=TensorPower(3,R,B2,M2)
tt3:=tensor([(g.1)::M2,(g.2)::M2,(g.4)::M2])$N

-- 

construct(e.1,g.1)$Product(B1,B2)::M12
BB12:=concat [[construct((e.i)::B1,(g.j)::B2)$Product(B1,B2)::M12 for j in 1..4] for i in 1..3]

typeOf(%) has OrderedSet
\end{axiom}

Tensor product of three or more different spaces:
$$ U\otimes V \otimes W$$
where $$U=[e_1,e_2,_e3], V=[g_1,\ldots,g_4]$$ and 
$$W=[h^1,\ldots,h^4]$$. The problem can be seen in
the output of equation $(18)$. 

In order to get the output correct how should B12 be set?

fricas

R ==> EXPR INT

Type: Void

fricas

e:=[subscript('e, [k]) for k in 1..3]

$\label{eq1}\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]$

(1)

Type: List(Symbol)

fricas

g:=[subscript('g, [k]) for k in 1..4]

$\label{eq2}\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]$

(2)

Type: List(Symbol)

fricas

h:=[superscript('h, [k]) for k in 1..4]

$\label{eq3}\left[{h^{1}}, \:{h^{2}}, \:{h^{3}}, \:{h^{4}}\right]$

(3)

Type: List(Symbol)

fricas

B1:=OrderedVariableList e

$\label{eq4}\hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ])$

(4)

Type: Type

fricas

B2:=OrderedVariableList g

$\label{eq5}\hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ])$

(5)

Type: Type

fricas

B3:=OrderedVariableList h

$\label{eq6}\hbox{\axiomType{OrderedVariableList}\ } ([ h [ ; 1 ] , h [ ; 2 ] , h [ ; 3 ] , h [ ; 4 ] ])$

(6)

Type: Type

fricas

M1:=FreeModule(R,B1)

$\label{eq7}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ]))$

(7)

Type: Type

fricas

M2:=FreeModule(R,B2)

$\label{eq8}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ]))$

(8)

Type: Type

fricas

M3:=FreeModule(R,B3)

$\label{eq9}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ h [ ; 1 ] , h [ ; 2 ] , h [ ; 3 ] , h [ ; 4 ] ]))$

(9)

Type: Type

fricas

M12:=TensorProduct(R,B1,B2,M1,M2)

$\label{eq10}\hbox{\axiomType{TensorProduct}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ]) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ]) , \hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ])) , \hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ])))$

(10)

Type: Type

fricas

M23:=TensorProduct(R,B2,B3,M2,M3)

$\label{eq11}\hbox{\axiomType{TensorProduct}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ]) , \hbox{\axiomType{OrderedVariableList}\ } ([ h [ ; 1 ] , h [ ; 2 ] , h [ ; 3 ] , h [ ; 4 ] ]) , \hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ])) , \hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ h [ ; 1 ] , h [ ; 2 ] , h [ ; 3 ] , h [ ; 4 ] ])))$

(11)

Type: Type

fricas

M123:=TensorProduct(R,Product(B1,B2),B3,M12,M3)

$\label{eq12}\hbox{\axiomType{TensorProduct}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{Product}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ]) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ])) , \hbox{\axiomType{OrderedVariableList}\ } ([ h [ ; 1 ] , h [ ; 2 ] , h [ ; 3 ] , h [ ; 4 ] ]) , \hbox{\axiomType{TensorProduct}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ]) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ]) , \hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ])) , \hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ]))) , \hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ h [ ; 1 ] , h [ ; 2 ] , h [ ; 3 ] , h [ ; 4 ] ])))$

(12)

Type: Type

fricas

v1:=x*(e.1)::M1 - y*(e.3)::M1

$\label{eq13}{x \ {e_{1}}}-{y \ {e_{3}}}$

(13)

Type: FreeModule?(Expression(Integer),OrderedVariableList?([e[1],e[2],e[3]]))

fricas

v2:=q*(g.2)::M2 + r*(g.4)::M2

$\label{eq14}{q \ {g_{2}}}+{r \ {g_{4}}}$

(14)

Type: FreeModule?(Expression(Integer),OrderedVariableList?([g[1],g[2],g[3],g[4]]))

fricas

v3:=3*(h.1)::M3 - z*(h.3)::M3

$\label{eq15}{3 \ {h^{1}}}-{z \ {h^{3}}}$

(15)

Type: FreeModule?(Expression(Integer),OrderedVariableList?([h[;1],h[;2],h[;3],h[;4]]))

fricas

t12:=tensor(v1,v2)$M12

$\label{eq16}\begin{array}{@{}l} \displaystyle {q \ x \ {{e_{1}}\otimes{g_{2}}}}+{r \ x \ {{e_{1}}\otimes{g_{4}}}}-{q \ y \ {{e_{3}}\otimes{g_{2}}}}- \ \ \displaystyle {r \ y \ {{e_{3}}\otimes{g_{4}}}}$

(16)

Type: TensorProduct?(Expression(Integer),OrderedVariableList?([e[1],e[2],e[3]]),OrderedVariableList?([g[1],g[2],g[3],g[4]]),FreeModule?(Expression(Integer),OrderedVariableList?([e[1],e[2],e[3]])),FreeModule?(Expression(Integer),OrderedVariableList?([g[1],g[2],g[3],g[4]])))

fricas

t23:=tensor(v2,v3)$M23

$\label{eq17}\begin{array}{@{}l} \displaystyle {3 \ q \ {{g_{2}}\otimes{h^{1}}}}-{q \ z \ {{g_{2}}\otimes{h^{3}}}}+{3 \ r \ {{g_{4}}\otimes{h^{1}}}}- \ \ \displaystyle {r \ z \ {{g_{4}}\otimes{h^{3}}}}$

(17)

Type: TensorProduct?(Expression(Integer),OrderedVariableList?([g[1],g[2],g[3],g[4]]),OrderedVariableList?([h[;1],h[;2],h[;3],h[;4]]),FreeModule?(Expression(Integer),OrderedVariableList?([g[1],g[2],g[3],g[4]])),FreeModule?(Expression(Integer),OrderedVariableList?([h[;1],h[;2],h[;3],h[;4]])))

fricas

t123:=tensor(t12,v3)$M123

$\label{eq18}\begin{array}{@{}l} \displaystyle {3 \ q \ x \ {{\left[{e_{1}}, \:{g_{2}}\right]}\otimes{h^{1}}}}-{q \ x \ z \ {{\left[{e_{1}}, \:{g_{2}}\right]}\otimes{h^{3}}}}+ \ \ \displaystyle {3 \ r \ x \ {{\left[{e_{1}}, \:{g_{4}}\right]}\otimes{h^{1}}}}-{r \ x \ z \ {{\left[{e_{1}}, \:{g_{4}}\right]}\otimes{h^{3}}}}- \ \ \displaystyle {3 \ q \ y \ {{\left[{e_{3}}, \:{g_{2}}\right]}\otimes{h^{1}}}}+{q \ y \ z \ {{\left[{e_{3}}, \:{g_{2}}\right]}\otimes{h^{3}}}}- \ \ \displaystyle {3 \ r \ y \ {{\left[{e_{3}}, \:{g_{4}}\right]}\otimes{h^{1}}}}+{r \ y \ z \ {{\left[{e_{3}}, \:{g_{4}}\right]}\otimes{h^{3}}}}$

(18)

Type: TensorProduct?(Expression(Integer),Product(OrderedVariableList?([e[1],e[2],e[3]]),OrderedVariableList?([g[1],g[2],g[3],g[4]])),OrderedVariableList?([h[;1],h[;2],h[;3],h[;4]]),TensorProduct?(Expression(Integer),OrderedVariableList?([e[1],e[2],e[3]]),OrderedVariableList?([g[1],g[2],g[3],g[4]]),FreeModule?(Expression(Integer),OrderedVariableList?([e[1],e[2],e[3]])),FreeModule?(Expression(Integer),OrderedVariableList?([g[1],g[2],g[3],g[4]]))),FreeModule?(Expression(Integer),OrderedVariableList?([h[;1],h[;2],h[;3],h[;4]])))

fricas

N:=TensorPower(3,R,B2,M2)

$\label{eq19}\hbox{\axiomType{TensorPower}\ } (3, \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ]) , \hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ g [ 1 ] , g [ 2 ] , g [ 3 ] , g [ 4 ] ])))$

(19)

Type: Type

fricas

tt3:=tensor([(g.1)::M2,(g.2)::M2,(g.4)::M2])$N

$\label{eq20}{{g_{1}}\otimes{g_{2}}}\otimes{g_{4}}$

(20)

Type: TensorPower?(3,Expression(Integer),OrderedVariableList?([g[1],g[2],g[3],g[4]]),FreeModule?(Expression(Integer),OrderedVariableList?([g[1],g[2],g[3],g[4]])))

fricas

-- 

construct(e.1,g.1)$Product(B1,B2)::M12

$\label{eq21}{e_{1}}\otimes{g_{1}}$

(21)

fricas

BB12:=concat [[construct((e.i)::B1,(g.j)::B2)$Product(B1,B2)::M12 for j in 1..4] for i in 1..3]

$\label{eq22}\begin{array}{@{}l} \displaystyle \left[{{e_{1}}\otimes{g_{1}}}, \:{{e_{1}}\otimes{g_{2}}}, \:{{e_{1}}\otimes{g_{3}}}, \:{{e_{1}}\otimes{g_{4}}}, \: \right. \ \ \displaystyle \left.{{e_{2}}\otimes{g_{1}}}, \:{{e_{2}}\otimes{g_{2}}}, \:{{e_{2}}\otimes{g_{3}}}, \:{{e_{2}}\otimes{g_{4}}}, \: \right. \ \ \displaystyle \left.{{e_{3}}\otimes{g_{1}}}, \:{{e_{3}}\otimes{g_{2}}}, \:{{e_{3}}\otimes{g_{3}}}, \:{{e_{3}}\otimes{g_{4}}}\right]$

(22)

Type: List(TensorProduct?(Expression(Integer),OrderedVariableList?([e[1],e[2],e[3]]),OrderedVariableList?([g[1],g[2],g[3],g[4]]),FreeModule?(Expression(Integer),OrderedVariableList?([e[1],e[2],e[3]])),FreeModule?(Expression(Integer),OrderedVariableList?([g[1],g[2],g[3],g[4]]))))

fricas

typeOf(%) has OrderedSet

$\label{eq23} \mbox{\rm false}$

(23)

Type: Boolean

Tensor product of three or more different spaces:

$U\otimes V \otimes W$

where

$U=[e_1,e_2,_e3], V=[g_1,\ldots,g_4]$

and

$W=[h^1,\ldots,h^4]$

. The problem can be seen in the output of equation

In order to get the output correct how should B12 be set?