|
spad )abbrev domain TENSALG TensorAlgebra
++ Author: Kurt Pagani
++ Date Created: Thu Jan 30 23:04:02 CET 2020
++ License: BSD
++ References:
++ https://en.wikipedia.org/wiki/Tensor_algebra
++ Description:
++ Quote Ref:
++ -- Non-commutative polynomials
++ -- Quotients
++ Because of the generality of the tensor algebra, many other algebras
++ of interest can be constructed by starting with the tensor algebra and
++ then imposing certain relations on the generators, i.e. by constructing
++ certain quotient algebras of T(V). Examples of this are the
++ exterior algebra, the symmetric algebra, Clifford algebras,
++ the Weyl algebra and universal enveloping algebras.
++
TensorAlgebra(M,R,B) : Exports == Implementation where
M:FreeModuleCategory(R, B)
R:Ring
B:OrderedSet
OF ==> OutputForm
NNI ==> NonNegativeInteger
FMB ==> FreeMonoid B
CTOF ==> CoercibleTo OutputForm
FMCRB ==> FreeModuleCategory(R,FMB)
XFABR ==> XFreeAlgebra(B,R)
XDPBR ==> XDistributedPolynomial(B,R)
TERM ==> Record(k:FMB,c:R)
Exports == Join(FMCRB, XFABR) with
coerce : B -> %
convert : FMB -> OutputForm
Implementation == XDPBR add
Rep := XDPBR
convert(x:FMB):OutputForm ==
x=1$FMB => empty()$OF
length(x)$FMB = 1 => x::OF
length(x)$FMB = 2 => tensor(first(x)::OF,rest(x)::OF)$OF
tensor(first(x)::OF, convert(rest x))
coerce(x:%):OutputForm ==
zero? x => empty()$OF
x=1$% => outputForm(1)$OF
c:R:=leadingCoefficient(x)
if c=1 then cof:=empty()$OF else cof:=c::OF
kof:OF:=cof * convert(leadingSupport(x))
zero? reductum(x) => kof
kof + reductum(x)::OF
)abbrev domain OOBJ OrderedObjectList
++ Description:
++ This domain implements ordered objects such that the type
++ has OrderedSet. Adapted from OrderedVariableList.
OrderedObjectList(T:Comparable, ObjectList : List T):
Join(OrderedFinite, ConvertibleTo T) with
object : T -> Union(%,"failed")
++ object(s) returns a member of the object set or failed
== add
ObjectList := removeDuplicates ObjectList
Rep := PositiveInteger
s1, s2 : %
convert(s1) : T == ObjectList.((s1::Rep)::PositiveInteger)
coerce(s1) : OutputForm == (convert(s1)@T)::OutputForm
index i == i::%
lookup j == j :: Rep
size () == #ObjectList
object(exp : T) ==
for i in 1.. for exp2 in ObjectList repeat
if exp = exp2 then return i::PositiveInteger::%
"failed"
s1 < s2 == s2 <$Rep s1
s1 = s2 == s1 =$Rep s2
latex(x : %) : String == latex(convert(x)@T)
hashUpdate!(hs, s) == update!(hs, SXHASH(s)$Lisp)$HashState
-- B:=OrderedObjectList(BOP, map(operator,[a,b,c]))
-- index(1)$B
-- enumerate()$B
-- size()$B
-- object(operator a)$B
spad Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3277064617359698758-25px001.spad
using old system compiler.
TENSALG abbreviates domain TensorAlgebra
------------------------------------------------------------------------
initializing NRLIB TENSALG for TensorAlgebra
compiling into NRLIB TENSALG
compiling exported convert : FreeMonoid B -> OutputForm
Time: 0.01 SEC.
compiling exported coerce : $ -> OutputForm
Time: 0.01 SEC.
****** Domain: R already in scope
augmenting R: (OrderedAbelianMonoidSup)
****** Domain: R already in scope
augmenting R: (CommutativeRing)
(time taken in buildFunctor: 10)
;;; *** |TensorAlgebra| REDEFINED
;;; *** |TensorAlgebra| REDEFINED
Time: 0.01 SEC.
Cumulative Statistics for Constructor TensorAlgebra
Time: 0.03 seconds
--------------non extending category----------------------
.. TensorAlgebra(#1,#2,#3) of cat
(|Join| (|FreeModuleCategory| |#2| (|FreeMonoid| |#3|))
(|XFreeAlgebra| |#3| |#2|)
(CATEGORY |domain| (SIGNATURE |coerce| ($ |#3|))
(SIGNATURE |convert| ((|OutputForm|) (|FreeMonoid| |#3|))))) has no
(|XPolynomialsCat| |#3| |#2|) finalizing NRLIB TENSALG
Processing TensorAlgebra for Browser database:
--------constructor---------
--->-->TensorAlgebra((coerce (% B))): Not documented!!!!
--->-->TensorAlgebra((convert ((OutputForm) (FreeMonoid B)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/TENSALG.NRLIB/TENSALG.lsp" (written 03 NOV 2020 07:33:59 PM):
; /var/aw/var/LatexWiki/TENSALG.NRLIB/TENSALG.fasl written
; compilation finished in 0:00:00.030
------------------------------------------------------------------------
TensorAlgebra is now explicitly exposed in frame initial
TensorAlgebra will be automatically loaded when needed from
/var/aw/var/LatexWiki/TENSALG.NRLIB/TENSALG
OOBJ abbreviates domain OrderedObjectList
------------------------------------------------------------------------
initializing NRLIB OOBJ for OrderedObjectList
compiling into NRLIB OOBJ
****** Domain: T$ already in scope
Local variable Rep type redefined: (Join (OrderedAbelianSemiGroup) (OrderedMonoid) (CommutativeStar) (ConvertibleTo (InputForm)) (CATEGORY domain (SIGNATURE gcd ($ $ $)) (SIGNATURE qcoerce ($ (Integer))))) to (Join (FreeModuleCategory R (FreeMonoid B)) (XPolynomialsCat B R))
compiling exported convert : $ -> T$
Time: 0.02 SEC.
compiling exported coerce : $ -> OutputForm
Time: 0 SEC.
compiling exported index : PositiveInteger -> $
OOBJ;index;Pi$;3 is replaced by i
Time: 0 SEC.
compiling exported lookup : $ -> PositiveInteger
OOBJ;lookup;$Pi;4 is replaced by j
Time: 0 SEC.
compiling exported size : () -> NonNegativeInteger
Time: 0 SEC.
compiling exported object : T$ -> Union($,failed)
Time: 0 SEC.
compiling exported < : ($,$) -> Boolean
OOBJ;<;2$B;7 is replaced by <s2s1
Time: 0 SEC.
compiling exported = : ($,$) -> Boolean
OOBJ;=;2$B;8 is replaced by EQL
Time: 0 SEC.
compiling exported latex : $ -> String
Time: 0 SEC.
compiling exported hashUpdate! : (HashState,$) -> HashState
Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |OrderedObjectList| REDEFINED
;;; *** |OrderedObjectList| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor OrderedObjectList
Time: 0.02 seconds
finalizing NRLIB OOBJ
Processing OrderedObjectList for Browser database:
--------constructor---------
--------(object ((Union % failed) T$))---------
; compiling file "/var/aw/var/LatexWiki/OOBJ.NRLIB/OOBJ.lsp" (written 03 NOV 2020 07:33:59 PM):
; /var/aw/var/LatexWiki/OOBJ.NRLIB/OOBJ.fasl written
; compilation finished in 0:00:00.032
------------------------------------------------------------------------
OrderedObjectList is now explicitly exposed in frame initial
OrderedObjectList will be automatically loaded when needed from
/var/aw/var/LatexWiki/OOBJ.NRLIB/OOBJfricas n:=3
fricas B1:=OrderedVariableList [e[i] for i in 1..n]
Type: Type
fricas R1:=Expression Integer
Type: Type
fricas M1:=FreeModule(R1, B1)
Type: Type
fricas TA1:=TensorAlgebra(M1,R1,B1)
Type: Type
fricas e:=[a::TA1 for a in enumerate()$B1]
fricas T1:=x*e.1+y*e.2-z*e.3
fricas T11:=T1*T1
fricas ---
B2:=OrderedObjectList(BOP, map(operator,[a,b,c]))
Type: Type
fricas R2:=DeRhamComplex(Integer,[x,y,z])
Type: Type
fricas M2:=FreeModule(R2, B2)
Type: Type
fricas TA2:=TensorAlgebra(M2,R2,B2)
Type: Type
fricas g:=[a::TA2 for a in enumerate()$B2]
Type: List(TensorAlgebra ?( FreeModule(DeRhamComplex ?(Integer, [x, y, z]), OrderedObjectList ?(BasicOperator ?, [a, b, c])), DeRhamComplex ?(Integer, [x, y, z]), OrderedObjectList ?(BasicOperator ?, [a, b, c])))
fricas h:=[generator(i)$R2 for i in 1..3]
Type: List(DeRhamComplex ?(Integer, [x, y, z]))
fricas T2:=h.1*g.1+h.2*g.2-h.3*g.3
Type: TensorAlgebra ?( FreeModule(DeRhamComplex ?(Integer, [x, y, z]), OrderedObjectList ?(BasicOperator ?, [a, b, c])), DeRhamComplex ?(Integer, [x, y, z]), OrderedObjectList ?(BasicOperator ?, [a, b, c]))
fricas T3:=g.2 * T2 - T2 * g.1
Type: TensorAlgebra ?( FreeModule(DeRhamComplex ?(Integer, [x, y, z]), OrderedObjectList ?(BasicOperator ?, [a, b, c])), DeRhamComplex ?(Integer, [x, y, z]), OrderedObjectList ?(BasicOperator ?, [a, b, c]))
fricas T4:=h.2 * T3
Type: TensorAlgebra ?( FreeModule(DeRhamComplex ?(Integer, [x, y, z]), OrderedObjectList ?(BasicOperator ?, [a, b, c])), DeRhamComplex ?(Integer, [x, y, z]), OrderedObjectList ?(BasicOperator ?, [a, b, c]))
fricas )show TENSALG
TensorAlgebra(M: FreeModuleCategory(R,B),R: Ring,B: OrderedSet) is a domain constructor
Abbreviation for TensorAlgebra is TENSALG
This constructor is exposed in this frame.
------------------------------- Operations --------------------------------
?*? : (%, %) -> % ?*? : (Integer, %) -> %
?*? : (B, %) -> % ?*? : (FreeMonoid(B), R) -> %
?*? : (R, FreeMonoid(B)) -> % ?*? : (%, R) -> %
?*? : (R, %) -> % ?*? : (PositiveInteger, %) -> %
?+? : (%, %) -> % ?-? : (%, %) -> %
-? : % -> % ?=? : (%, %) -> Boolean
1 : () -> % 0 : () -> %
?^? : (%, PositiveInteger) -> % annihilate? : (%, %) -> Boolean
antiCommutator : (%, %) -> % associator : (%, %, %) -> %
coef : (%, FreeMonoid(B)) -> R coef : (%, %) -> R
coefficients : % -> List(R) coerce : Integer -> %
coerce : R -> % coerce : FreeMonoid(B) -> %
coerce : B -> % coerce : % -> OutputForm
commutator : (%, %) -> % constant : % -> R
constant? : % -> Boolean hash : % -> SingleInteger
latex : % -> String lquo : (%, B) -> %
lquo : (%, FreeMonoid(B)) -> % lquo : (%, %) -> %
map : ((R -> R), %) -> % mindeg : % -> FreeMonoid(B)
mirror : % -> % monom : (FreeMonoid(B), R) -> %
monomial? : % -> Boolean monomials : % -> List(%)
one? : % -> Boolean opposite? : (%, %) -> Boolean
quasiRegular : % -> % quasiRegular? : % -> Boolean
recip : % -> Union(%,"failed") retract : % -> FreeMonoid(B)
rquo : (%, B) -> % rquo : (%, FreeMonoid(B)) -> %
rquo : (%, %) -> % sample : () -> %
varList : % -> List(B) zero? : % -> Boolean
?~=? : (%, %) -> Boolean
?*? : (NonNegativeInteger, %) -> %
?<? : (%, %) -> Boolean if R has OAMON and FreeMonoid(B) has ORDSET or R has OAMONS and FreeMonoid(B) has ORDSET
?<=? : (%, %) -> Boolean if R has OAMON and FreeMonoid(B) has ORDSET or R has OAMONS and FreeMonoid(B) has ORDSET
?>? : (%, %) -> Boolean if R has OAMON and FreeMonoid(B) has ORDSET or R has OAMONS and FreeMonoid(B) has ORDSET
?>=? : (%, %) -> Boolean if R has OAMON and FreeMonoid(B) has ORDSET or R has OAMONS and FreeMonoid(B) has ORDSET
?^? : (%, NonNegativeInteger) -> %
characteristic : () -> NonNegativeInteger
coefficient : (%, FreeMonoid(B)) -> R
construct : List(Record(k: FreeMonoid(B),c: R)) -> %
constructOrdered : List(Record(k: FreeMonoid(B),c: R)) -> % if FreeMonoid(B) has COMPAR
convert : FreeMonoid(B) -> OutputForm
hashUpdate! : (HashState, %) -> HashState
leadingCoefficient : % -> R if FreeMonoid(B) has COMPAR
leadingMonomial : % -> % if FreeMonoid(B) has COMPAR
leadingSupport : % -> FreeMonoid(B) if FreeMonoid(B) has COMPAR
leadingTerm : % -> Record(k: FreeMonoid(B),c: R) if FreeMonoid(B) has COMPAR
leftPower : (%, PositiveInteger) -> %
leftPower : (%, NonNegativeInteger) -> %
leftRecip : % -> Union(%,"failed")
linearExtend : ((FreeMonoid(B) -> R), %) -> R if R has COMRING
listOfTerms : % -> List(Record(k: FreeMonoid(B),c: R))
max : (%, %) -> % if R has OAMON and FreeMonoid(B) has ORDSET or R has OAMONS and FreeMonoid(B) has ORDSET
min : (%, %) -> % if R has OAMON and FreeMonoid(B) has ORDSET or R has OAMONS and FreeMonoid(B) has ORDSET
mindegTerm : % -> Record(k: FreeMonoid(B),c: R)
monomial : (R, FreeMonoid(B)) -> %
numberOfMonomials : % -> NonNegativeInteger
reductum : % -> % if FreeMonoid(B) has COMPAR
retractIfCan : % -> Union(FreeMonoid(B),"failed")
rightPower : (%, PositiveInteger) -> %
rightPower : (%, NonNegativeInteger) -> %
rightRecip : % -> Union(%,"failed")
sh : (%, %) -> % if R has COMRING
sh : (%, NonNegativeInteger) -> % if R has COMRING
smaller? : (%, %) -> Boolean if R has COMPAR and FreeMonoid(B) has COMPAR or R has OAMON and FreeMonoid(B) has ORDSET or R has OAMONS and FreeMonoid(B) has ORDSET
subtractIfCan : (%, %) -> Union(%,"failed")
sup : (%, %) -> % if R has OAMONS and FreeMonoid(B) has ORDSET
support : % -> List(FreeMonoid(B))
|
![\label{eq2}\hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ])](images/2198763490073114834-16.0px.png "\label{eq2}\hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ])")
![\label{eq4}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ]))](images/5129712195218335516-16.0px.png "\label{eq4}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ]))")
![\label{eq5}\hbox{\axiomType{TensorAlgebra}\ } (\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ])) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ]))](images/3101873648314391718-16.0px.png "\label{eq5}\hbox{\axiomType{TensorAlgebra}\ } (\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ])) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] ]))")
![\label{eq6}\left[{\ {e_{1}}}, \:{\ {e_{2}}}, \:{\ {e_{3}}}\right]](images/2500432007490566947-16.0px.png "\label{eq6}\left[{\ {e_{1}}}, \:{\ {e_{2}}}, \:{\ {e_{3}}}\right]")
![\label{eq8}\begin{array}{@{}l}
\displaystyle
{{{x}^{2}}\ {{e_{1}}\otimes{e_{1}}}}+{x \ y \ {{e_{1}}\otimes{e_{2}}}}-{x \ z \ {{e_{1}}\otimes{e_{3}}}}+
\
\
\displaystyle
{x \ y \ {{e_{2}}\otimes{e_{1}}}}+{{{y}^{2}}\ {{e_{2}}\otimes{e_{2}}}}-{y \ z \ {{e_{2}}\otimes{e_{3}}}}-
\
\
\displaystyle
{x \ z \ {{e_{3}}\otimes{e_{1}}}}-{y \ z \ {{e_{3}}\otimes{e_{2}}}}+{{{z}^{2}}\ {{e_{3}}\otimes{e_{3}}}}](images/7361074830312898928-16.0px.png "\label{eq8}\begin{array}{@{}l}
\displaystyle
{{{x}^{2}}\ {{e_{1}}\otimes{e_{1}}}}+{x \ y \ {{e_{1}}\otimes{e_{2}}}}-{x \ z \ {{e_{1}}\otimes{e_{3}}}}+
\
\
\displaystyle
{x \ y \ {{e_{2}}\otimes{e_{1}}}}+{{{y}^{2}}\ {{e_{2}}\otimes{e_{2}}}}-{y \ z \ {{e_{2}}\otimes{e_{3}}}}-
\
\
\displaystyle
{x \ z \ {{e_{3}}\otimes{e_{1}}}}-{y \ z \ {{e_{3}}\otimes{e_{2}}}}+{{{z}^{2}}\ {{e_{3}}\otimes{e_{3}}}}")
![\label{eq9}\hbox{\axiomType{OrderedObjectList}\ } (\hbox{\axiomType{BasicOperator}\ } , [ a , b , c ])](images/8018644099321842291-16.0px.png "\label{eq9}\hbox{\axiomType{OrderedObjectList}\ } (\hbox{\axiomType{BasicOperator}\ } , [ a , b , c ])")
![\label{eq10}\hbox{\axiomType{DeRhamComplex}\ } (\hbox{\axiomType{Integer}\ } , [ x , y , z ])](images/3430935700335925816-16.0px.png "\label{eq10}\hbox{\axiomType{DeRhamComplex}\ } (\hbox{\axiomType{Integer}\ } , [ x , y , z ])")
![\label{eq11}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{DeRhamComplex}\ } (\hbox{\axiomType{Integer}\ } , [ x , y , z ]) , \hbox{\axiomType{OrderedObjectList}\ } (\hbox{\axiomType{BasicOperator}\ } , [ a , b , c ]))](images/4556370371390928014-16.0px.png "\label{eq11}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{DeRhamComplex}\ } (\hbox{\axiomType{Integer}\ } , [ x , y , z ]) , \hbox{\axiomType{OrderedObjectList}\ } (\hbox{\axiomType{BasicOperator}\ } , [ a , b , c ]))")
![\label{eq12}\hbox{\axiomType{TensorAlgebra}\ } (\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{DeRhamComplex}\ } (\hbox{\axiomType{Integer}\ } , [ x , y , z ]) , \hbox{\axiomType{OrderedObjectList}\ } (\hbox{\axiomType{BasicOperator}\ } , [ a , b , c ])) , \hbox{\axiomType{DeRhamComplex}\ } (\hbox{\axiomType{Integer}\ } , [ x , y , z ]) , \hbox{\axiomType{OrderedObjectList}\ } (\hbox{\axiomType{BasicOperator}\ } , [ a , b , c ]))](images/6613057779332652815-16.0px.png "\label{eq12}\hbox{\axiomType{TensorAlgebra}\ } (\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{DeRhamComplex}\ } (\hbox{\axiomType{Integer}\ } , [ x , y , z ]) , \hbox{\axiomType{OrderedObjectList}\ } (\hbox{\axiomType{BasicOperator}\ } , [ a , b , c ])) , \hbox{\axiomType{DeRhamComplex}\ } (\hbox{\axiomType{Integer}\ } , [ x , y , z ]) , \hbox{\axiomType{OrderedObjectList}\ } (\hbox{\axiomType{BasicOperator}\ } , [ a , b , c ]))")
![\label{eq13}\left[{\ a}, \:{\ b}, \:{\ c}\right]](images/782915223677107193-16.0px.png "\label{eq13}\left[{\ a}, \:{\ b}, \:{\ c}\right]")
![\label{eq14}\left[ dx , \: dy , \: dz \right]](images/4524090445907504999-16.0px.png "\label{eq14}\left[ dx , \: dy , \: dz \right]")
![\label{eq16}\begin{array}{@{}l}
\displaystyle
-{dx \ {a \otimes a}}+{{\left(- dy + dx \right)}\ {b \otimes a}}+{dy \ {b \otimes b}}-
\
\
\displaystyle
{dz \ {b \otimes c}}+{dz \ {c \otimes a}}](images/1190886619837943122-16.0px.png "\label{eq16}\begin{array}{@{}l}
\displaystyle
-{dx \ {a \otimes a}}+{{\left(- dy + dx \right)}\ {b \otimes a}}+{dy \ {b \otimes b}}-
\
\
\displaystyle
{dz \ {b \otimes c}}+{dz \ {c \otimes a}}")
![\label{eq17}\begin{array}{@{}l}
\displaystyle
{dx \ dy \ {a \otimes a}}-{dx \ dy \ {b \otimes a}}-{dy \ dz \ {b \otimes c}}+
\
\
\displaystyle
{dy \ dz \ {c \otimes a}}](images/3127845189858916110-16.0px.png "\label{eq17}\begin{array}{@{}l}
\displaystyle
{dx \ dy \ {a \otimes a}}-{dx \ dy \ {b \otimes a}}-{dy \ dz \ {b \otimes c}}+
\
\
\displaystyle
{dy \ dz \ {c \otimes a}}")