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fricas
(1) -> <spad>
fricas
)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()$OF else cof:=c::OF kof:OF:=cof * convert(leadingSupport(x)) zero? reductum(x) => kof kof + reductum(x)::OF
fricas
)abbrev domain OOBJ OrderedObjectList
++ Description:
++   This domain implements ordered objects such that the type
++   has OrderedSet. Adapted from OrderedVariableList.
OrderedObjectList(T:Comparable, ObjectList : List T):
  Join(OrderedFinite, ConvertibleTo T) with
         object : T -> Union(%,"failed")
           ++ object(s) returns a member of the object set or failed
    == add
       ObjectList := removeDuplicates ObjectList
       Rep := PositiveInteger
       s1, s2 : %
       convert(s1) : T == ObjectList.((s1::Rep)::PositiveInteger)
       coerce(s1) : OutputForm == (convert(s1)@T)::OutputForm
       index i   == i::%
       lookup j  == j :: Rep
       size ()   == #ObjectList
       object(exp : T) ==
            for i in 1.. for exp2 in ObjectList repeat
                if exp = exp2 then return i::PositiveInteger::%
            "failed"
       s1 < s2 == s2 <$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>
fricas
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: 0.02 SEC.
compiling exported coerce : % -> OutputForm Time: 0 SEC.
****** Domain: R already in scope augmenting R: (Comparable) ****** Domain: R already in scope augmenting R: (CommutativeRing) (time taken in buildFunctor: 1196)
;;; *** |TensorAlgebra| REDEFINED
;;; *** |TensorAlgebra| REDEFINED Time: 0 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 10 DEC 2024 01:20:24 AM):
; wrote /var/aw/var/LatexWiki/TENSALG.NRLIB/TENSALG.fasl ; compilation finished in 0:00:00.020 ------------------------------------------------------------------------ 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)) (Hashable) (CATEGORY domain (SIGNATURE gcd (% % %)) (SIGNATURE qcoerce (% (Integer))))) to (Join (XPolynomialsCat B R) (FreeModuleCategory R (FreeMonoid B))) compiling exported convert : % -> T$ Time: 0.01 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: 413)
;;; *** |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 10 DEC 2024 01:20:24 AM):
; wrote /var/aw/var/LatexWiki/OOBJ.NRLIB/OOBJ.fasl ; compilation finished in 0:00:00.012 ------------------------------------------------------------------------ OrderedObjectList is now explicitly exposed in frame initial OrderedObjectList will be automatically loaded when needed from /var/aw/var/LatexWiki/OOBJ.NRLIB/OOBJ

fricas
n:=3

\label{eq1}3(1)
Type: PositiveInteger?
fricas
B1:=OrderedVariableList [e[i] for i in 1..n]

\label{eq2}\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]}\right)(2)
Type: Type
fricas
R1:=Expression Integer

\label{eq3}\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)(3)
Type: Type
fricas
M1:=FreeModule(R1, B1)

\label{eq4}\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)(4)
Type: Type
fricas
TA1:=TensorAlgebra(M1,R1,B1)

\label{eq5}\hbox{\axiomType{TensorAlgebra}\ } \left({{\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{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}\right]}\right)}}\right)(5)
Type: Type
fricas
e:=[a::TA1 for a in enumerate()$B1]

\label{eq6}\left[{\ {e_{1}}}, \:{\ {e_{2}}}, \:{\ {e_{3}}}\right](6)
Type: List(TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3]])))
fricas
T1:=x*e.1+y*e.2-z*e.3

\label{eq7}{x \ {e_{1}}}+{y \ {e_{2}}}-{z \ {e_{3}}}(7)
Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3]]))
fricas
T11:=T1*T1

\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}}}}
(8)
Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3]]))
fricas
---
B2:=OrderedObjectList(BOP, map(operator,[a,b,c]))

\label{eq9}\hbox{\axiomType{OrderedObjectList}\ } \left({\hbox{\axiomType{BasicOperator}\ } , \:{\left[ a , \: b , \: c \right]}}\right)(9)
Type: Type
fricas
R2:=DeRhamComplex(Integer,[x,y,z])

\label{eq10}\hbox{\axiomType{DeRhamComplex}\ } \left({\hbox{\axiomType{Integer}\ } , \:{\left[ x , \: y , \: z \right]}}\right)(10)
Type: Type
fricas
M2:=FreeModule(R2, B2)

\label{eq11}\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{DeRhamComplex}\ } \left({\hbox{\axiomType{Integer}\ } , \:{\left[ x , \: y , \: z \right]}}\right)}, \:{\hbox{\axiomType{OrderedObjectList}\ } \left({\hbox{\axiomType{BasicOperator}\ } , \:{\left[ a , \: b , \: c \right]}}\right)}}\right)(11)
Type: Type
fricas
TA2:=TensorAlgebra(M2,R2,B2)

\label{eq12}\hbox{\axiomType{TensorAlgebra}\ } \left({{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{DeRhamComplex}\ } \left({\hbox{\axiomType{Integer}\ } , \:{\left[ x , \: y , \: z \right]}}\right)}, \:{\hbox{\axiomType{OrderedObjectList}\ } \left({\hbox{\axiomType{BasicOperator}\ } , \:{\left[ a , \: b , \: c \right]}}\right)}}\right)}, \:{\hbox{\axiomType{DeRhamComplex}\ } \left({\hbox{\axiomType{Integer}\ } , \:{\left[ x , \: y , \: z \right]}}\right)}, \:{\hbox{\axiomType{OrderedObjectList}\ } \left({\hbox{\axiomType{BasicOperator}\ } , \:{\left[ a , \: b , \: c \right]}}\right)}}\right)(12)
Type: Type
fricas
g:=[a::TA2 for a in enumerate()$B2]

\label{eq13}\left[{\  a}, \:{\  b}, \:{\  c}\right](13)
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]

\label{eq14}\left[ dx , \: dy , \: dz \right](14)
Type: List(DeRhamComplex?(Integer,[x,y,z]))
fricas
T2:=h.1*g.1+h.2*g.2-h.3*g.3

\label{eq15}{dx \  a}+{dy \  b}-{dz \  c}(15)
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

\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}}
(16)
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

\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}}
(17)
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, %) -> % ?*? : (%, 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 latex : % -> String lquo : (%, B) -> % lquo : (%, FreeMonoid(B)) -> % lquo : (%, %) -> % map : ((R -> R), %) -> % mindeg : % -> FreeMonoid(B) mirror : % -> % monomial? : % -> Boolean monomials : % -> List(%) one? : % -> Boolean opposite? : (%, %) -> Boolean quasiRegular : % -> % quasiRegular? : % -> Boolean recip : % -> Union(%,"failed") retract : % -> R retract : % -> FreeMonoid(B) rquo : (%, B) -> % rquo : (%, FreeMonoid(B)) -> % rquo : (%, %) -> % sample : () -> % varList : % -> List(B) zero? : % -> Boolean ?~=? : (%, %) -> Boolean ?*? : (NonNegativeInteger, %) -> % ?^? : (%, 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 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)) mindegTerm : % -> Record(k: FreeMonoid(B),c: R) monomial : (R, FreeMonoid(B)) -> % numberOfMonomials : % -> NonNegativeInteger plenaryPower : (%, PositiveInteger) -> % if R has COMRING reductum : % -> % if FreeMonoid(B) has COMPAR retractIfCan : % -> Union(R,"failed") 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 subtractIfCan : (%, %) -> Union(%,"failed") support : % -> List(FreeMonoid(B))




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