MathAction SandBoxTensorProduct3

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SandBoxTensorProduct ← ↑ → SandBoxTestTexOutput

SandBoxTensorProduct3
last edited 8 years ago by pagani

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(1) -> R ==> EXPR INT
Type: Void
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e:=[subscript('e, [k]) for k in 1..3]
\label{eq1}\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right] (1)
Type: List(Symbol)
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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)
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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)
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B1:=OrderedVariableList e
\label{eq4}\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]}\right) (4)
Type: Type
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B2:=OrderedVariableList g
\label{eq5}\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right) (5)
Type: Type
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B3:=OrderedVariableList h
\label{eq6}\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{h^{1}}, \:{h^{2}}, \:{h^{3}}, \:{h^{4}}\right]}\right) (6)
Type: Type
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M1:=FreeModule(R,B1)
\label{eq7}\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]}\right)}}\right) (7)
Type: Type
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M2:=FreeModule(R,B2)
\label{eq8}\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}}\right) (8)
Type: Type
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M3:=FreeModule(R,B3)
\label{eq9}\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{h^{1}}, \:{h^{2}}, \:{h^{3}}, \:{h^{4}}\right]}\right)}}\right) (9)
Type: Type
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M12:=TensorProduct(R,B1,B2,M1,M2)
\label{eq10}\hbox{\axiomType{TensorProduct}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}, \:{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]}\right)}}\right)}, \:{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}}\right)}}\right) (10)
Type: Type
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M23:=TensorProduct(R,B2,B3,M2,M3)
\label{eq11}\hbox{\axiomType{TensorProduct}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{h^{1}}, \:{h^{2}}, \:{h^{3}}, \:{h^{4}}\right]}\right)}, \:{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}}\right)}, \:{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{h^{1}}, \:{h^{2}}, \:{h^{3}}, \:{h^{4}}\right]}\right)}}\right)}}\right) (11)
Type: Type
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M123:=TensorProduct(R,Product(B1,B2),B3,M12,M3)
\label{eq12}\hbox{\axiomType{TensorProduct}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{Product}\ } \left({{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{h^{1}}, \:{h^{2}}, \:{h^{3}}, \:{h^{4}}\right]}\right)}, \:{\hbox{\axiomType{TensorProduct}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}, \:{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]}\right)}}\right)}, \:{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}}\right)}}\right)}, \:{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{h^{1}}, \:{h^{2}}, \:{h^{3}}, \:{h^{4}}\right]}\right)}}\right)}}\right) (12)
Type: Type
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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]]))
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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]]))
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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]]))
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t12:=tensor(v1,v2)$M12
\label{eq16}\begin{array}{@{}l} \displaystyle {q \ x \ {{e_{1}}\otimes{g_{2}}}}+{r \ x \ {{e_{1}}\otimes{g_{4}}}}-{q \ y \ {{e_{3}}\otimes{g_{2}}}}- \ \ \displaystyle {r \ y \ {{e_{3}}\otimes{g_{4}}}} (16)
Type: TensorProduct?(Expression(Integer),OrderedVariableList([e[1],e[2],e[3]]),OrderedVariableList([g[1],g[2],g[3],g[4]]),FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3]])),FreeModule(Expression(Integer),OrderedVariableList([g[1],g[2],g[3],g[4]])))
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t23:=tensor(v2,v3)$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]])))
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t123:=tensor(t12,v3)$M123
\label{eq18}\begin{array}{@{}l} \displaystyle {3 \ q \ x \ {{\left[{e_{1}}, \:{g_{2}}\right]}\otimes{h^{1}}}}-{q \ x \ z \ {{\left[{e_{1}}, \:{g_{2}}\right]}\otimes{h^{3}}}}+ \ \ \displaystyle {3 \ r \ x \ {{\left[{e_{1}}, \:{g_{4}}\right]}\otimes{h^{1}}}}-{r \ x \ z \ {{\left[{e_{1}}, \:{g_{4}}\right]}\otimes{h^{3}}}}- \ \ \displaystyle {3 \ q \ y \ {{\left[{e_{3}}, \:{g_{2}}\right]}\otimes{h^{1}}}}+{q \ y \ z \ {{\left[{e_{3}}, \:{g_{2}}\right]}\otimes{h^{3}}}}- \ \ \displaystyle {3 \ r \ y \ {{\left[{e_{3}}, \:{g_{4}}\right]}\otimes{h^{1}}}}+{r \ y \ z \ {{\left[{e_{3}}, \:{g_{4}}\right]}\otimes{h^{3}}}} (18)
Type: TensorProduct?(Expression(Integer),Product(OrderedVariableList([e[1],e[2],e[3]]),OrderedVariableList([g[1],g[2],g[3],g[4]])),OrderedVariableList([h[;1],h[;2],h[;3],h[;4]]),TensorProduct?(Expression(Integer),OrderedVariableList([e[1],e[2],e[3]]),OrderedVariableList([g[1],g[2],g[3],g[4]]),FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3]])),FreeModule(Expression(Integer),OrderedVariableList([g[1],g[2],g[3],g[4]]))),FreeModule(Expression(Integer),OrderedVariableList([h[;1],h[;2],h[;3],h[;4]])))
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N:=TensorPower(3,R,B2,M2)
\label{eq19}\hbox{\axiomType{TensorPower}\ } \left({3, \:{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}, \:{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{g_{1}}, \:{g_{2}}, \:{g_{3}}, \:{g_{4}}\right]}\right)}}\right)}}\right) (19)
Type: Type
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tt3:=tensor([(g.1)::M2,(g.2)::M2,(g.4)::M2])$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]])))
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-- 

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

 Cannot convert the value from type Product(OrderedVariableList([e[1]
 ,e[2],e[3]]),OrderedVariableList([g[1],g[2],g[3],g[4]])) to 
 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]]))) .

Tensor product of three or more different spaces:

U\otimes V \otimes W
where
U=[e_1,e_2,e_3], 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?




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