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Editor: Bill Page
Time: 2022/09/27 16:08:47 GMT+0
Note:

changed:
-    coerce(x:K):% == 1*x
    coerce(x:K):% == 1$% * x

Introduction

Bi-graded linear operators (transformations) over n-dimensional vector spaces on a commutative ring K. Members of this domain are morphisms K^n \to K^m. Products, permutations and composition (grafting) of morphisms are implemented. Operators are represented internally as tensors.

Operator composition and products can be visualized by directed graphs (read from top to bottom) such as:

      n = 3     inputs
      m = 0     outputs


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External vertices in this graph represent vectors, and tensors. Internal nodes and arcs (edges) represent linear operators. Horizontal juxtaposition (i.e. a horizontal cross-section) represents tensor product. Vertical juxtaposition represents operator composition.

See examples and documentation below

I would like you to make brief comments in the form at the bottom of this web page. For more detailed but related comments click discussion on the top menu.

Regards, Bill Page.

Source Code

We try to start the right way by defining the concept of a monoidal category.

Ref: http://en.wikipedia.org/wiki/PROP_(category_theory)

fricas
(1) -> )set userlevel development
 
fricas
)version
Value = "FriCAS 1.3.8 compiled at Thu Jun 23 16:21:37 UTC 2022"

spad
)abbrev category MONAL Monoidal
Monoidal(R:AbelianSemiGroup):Category == Ring with
    dom: % -> R
      ++ domain
    cod: % -> R
      ++ co-domain
    _/: (%,%) -> %
      ++ vertical composition f/g
    apply:(%,%) -> %
      ++ horizontal product f g = f*g
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2074022068168685528-25px002.spad
      using old system compiler.
   MONAL abbreviates category Monoidal 
------------------------------------------------------------------------
   initializing NRLIB MONAL for Monoidal 
   compiling into NRLIB MONAL 
;;; *** |Monoidal| REDEFINED Time: 0 SEC.
finalizing NRLIB MONAL Processing Monoidal for Browser database: --->-->Monoidal(constructor): Not documented!!!! --------(dom (R %))--------- --------(cod (R %))--------- --------(/ (% % %))--------- --->-->Monoidal((/ (% % %))): Improper first word in comments: vertical "vertical composition \\spad{f/g}" --------(apply (% % %))--------- --->-->Monoidal((apply (% % %))): Improper first word in comments: horizontal "horizontal product \\spad{f} \\spad{g} = \\spad{f*g}" --->-->Monoidal(): Missing Description ; compiling file "/var/aw/var/LatexWiki/MONAL.NRLIB/MONAL.lsp" (written 27 SEP 2022 04:08:43 PM):
; /var/aw/var/LatexWiki/MONAL.NRLIB/MONAL.fasl written ; compilation finished in 0:00:00.005 ------------------------------------------------------------------------ Monoidal is now explicitly exposed in frame initial Monoidal will be automatically loaded when needed from /var/aw/var/LatexWiki/MONAL.NRLIB/MONAL

The initial object in this category is the domain Prop (Products and Permutations). The Prop domain represents everything that is "constant" about all the domains in this category. It can be defined as an endo-functor with only the information available about the category itself.

spad
)abbrev domain PROP Prop
Prop(L:Monoidal NNI): Exports == Implementation where
  NNI ==> NonNegativeInteger
Exports ==> Monoidal NNI with coerce: L -> %
Implementation ==> add Rep ==> Record(domain:NNI, codomain:NNI) -- Rep == L rep(x:%):Rep == x pretend Rep per(x:Rep):% == x pretend %
coerce(f:%):OutputForm == dom(f)::OutputForm / cod(f)::OutputForm
coerce(f:L):% == per [dom f, cod f] -- coerce(f:L):% == per f
dom(x:%):NNI == rep(x).domain -- dom(x:%):NNI == dom rep x cod(x:%):NNI == rep(x).codomain -- cod(x:%):NNI == cod rep x 0:% == per [0,0] -- 0:% == per 0 1:% == per [0,0] -- 1:% == per 1 -- evaluation (f:% / g:%):% == per [dom f, cod g] -- (f:% / g:%):% == per (rep f / rep g) -- product apply(f:%,g:%):% == per [dom f + dom g, cod f + cod g] -- apply(f:%,g:%):% == per apply(rep f,rep g) (f:% * g:%):% == per [dom f + dom g, cod f + cod g] --(f:% * g:%):% == per (rep f * rep g) -- sum (f:% + g:%):% == per [dom f, cod f] --(f:% + g:%):% == per (rep f + rep g)
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3186492205692341477-25px003.spad
      using old system compiler.
   PROP abbreviates domain Prop 
------------------------------------------------------------------------
   initializing NRLIB PROP for Prop 
   compiling into NRLIB PROP 
   processing macro definition Rep ==> Record(domain: NonNegativeInteger,codomain: NonNegativeInteger) 
   compiling local rep : $ -> Record(domain: NonNegativeInteger,codomain: NonNegativeInteger)
      PROP;rep is replaced by x 
Time: 0 SEC.
compiling local per : Record(domain: NonNegativeInteger,codomain: NonNegativeInteger) -> $ PROP;per is replaced by x Time: 0 SEC.
compiling exported coerce : $ -> OutputForm Time: 0 SEC.
compiling exported coerce : L -> $ Time: 0 SEC.
compiling exported dom : $ -> NonNegativeInteger Time: 0 SEC.
compiling exported cod : $ -> NonNegativeInteger Time: 0 SEC.
compiling exported Zero : () -> $ Time: 0 SEC.
compiling exported One : () -> $ Time: 0.01 SEC.
compiling exported / : ($,$) -> $ Time: 0.01 SEC.
compiling exported apply : ($,$) -> $ Time: 0 SEC.
compiling exported * : ($,$) -> $ Time: 0 SEC.
compiling exported + : ($,$) -> $ Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |Prop| REDEFINED
;;; *** |Prop| REDEFINED Time: 0 SEC.
Warnings: [1] dom: domain has no value [2] cod: codomain has no value
Cumulative Statistics for Constructor Prop Time: 0.02 seconds
finalizing NRLIB PROP Processing Prop for Browser database: --->-->Prop(constructor): Not documented!!!! --->-->Prop((coerce (% L))): Not documented!!!! --->-->Prop(): Missing Description ; compiling file "/var/aw/var/LatexWiki/PROP.NRLIB/PROP.lsp" (written 27 SEP 2022 04:08:43 PM):
; /var/aw/var/LatexWiki/PROP.NRLIB/PROP.fasl written ; compilation finished in 0:00:00.027 ------------------------------------------------------------------------ Prop is now explicitly exposed in frame initial Prop will be automatically loaded when needed from /var/aw/var/LatexWiki/PROP.NRLIB/PROP

The LinearOperator? domain is Moniodal over NonNegativeInteger?. The objects of this domain are all tensor powers of a vector space of fixed dimension. The arrows are linear operators that map from one object (tensor power) to another.

Ref: http://en.wikipedia.org/wiki/Category_of_vector_spaces

  • all members of this domain have the same dimension

fricas
)lib CARTEN MONAL PROP
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN Monoidal is already explicitly exposed in frame initial Monoidal will be automatically loaded when needed from /var/aw/var/LatexWiki/MONAL.NRLIB/MONAL Prop is already explicitly exposed in frame initial Prop will be automatically loaded when needed from /var/aw/var/LatexWiki/PROP.NRLIB/PROP

spad
)abbrev domain LOP LinearOperator
LinearOperator(gener:OrderedFinite,K:Field): Exports == Implementation where
  NNI ==> NonNegativeInteger
  NAT ==> PositiveInteger
Exports ==> Join(Ring, FramedModule K, Monoidal NNI, RetractableTo K) with arity: % -> Prop % basisOut: () -> List % basisIn: () -> List % map: (K->K,%) -> % if K has Evalable(K) then Evalable(K) eval: % -> % ravel: % -> List K unravel: (Prop %,List K) -> % coerce:(x:List NAT) -> % ++ identity for composition and permutations of its products coerce:(x:List None) -> % ++ [] = 1 elt: (%,%) -> % elt: (%,NAT) -> % elt: (%,NAT,NAT) -> % elt: (%,NAT,NAT,NAT) -> % _/: (Tuple %,Tuple %) -> % _/: (Tuple %,%) -> % _/: (%,Tuple %) -> % ++ yet another syntax for product ev: NAT -> % ++ (2,0)-tensor for evaluation co: NAT -> % ++ (0,2)-tensor for co-evaluation
Implementation ==> add import List NNI
dim:NNI := size()$gener T := CartesianTensor(1,dim,K) L := Record(domain:NNI, codomain:NNI, data:T) RR := Record(gen:L,exp:NNI) -- FreeMonoid provides unevaluated products Rep ==> FreeMonoid L rep(x:%):Rep == x pretend Rep per(x:Rep):% == x pretend %
dimension():CardinalNumber == coerce dim
-- Prop (arity) dom(f:%):NNI == r:NNI := 0 for y in factors(rep f) repeat r:=r+(y.gen.domain)*(y.exp) return r cod(f:%):NNI == r:NNI := 0 for y in factors(rep f) repeat r:=r+(y.gen.codomain)*(y.exp) return r
prod(f:L,g:L):L == r:T := product(f.data,g.data) -- dom(f) + cod(f) + dom(g) + cod(g) p:List Integer := concat _ [[i for i in 1..(f.domain)], _ [(f.domain)+(f.codomain)+i for i in 1..(g.domain)], _ [(f.domain)+i for i in 1..(f.codomain)], _ [(f.domain)+(g.domain)+(f.codomain)+i for i in 1..(g.codomain)]] -- dom(f) + dom(g) + cod(f) + cod(g) --output("prod p = ",p::OutputForm)$OutputPackage [(f.domain)+(g.domain),(f.codomain)+(g.codomain),reindex(r,p)]
dats(fs:List RR):L == r:L := [0,0,1$T] for y in fs repeat t:L:=y.gen for n in 1..y.exp repeat r:=prod(r,t) return r
dat(f:%):L == dats factors rep f
arity(f:%):Prop % == f::Prop %
eval(f:%):% == per coerce dat(f)
retractIfCan(f:%):Union(K,"failed") == dom(f)=0 and cod(f)=0 => retract(dat(f).data)$T return "failed" retract(f:%):K == dom(f)=0 and cod(f)=0 => retract(dat(f).data)$T error "failed"
-- basis basisOut():List % == [per coerce [0,1,entries(row(1,i)$SquareMatrix(dim,K))::T] for i in 1..dim] basisIn():List % == [per coerce [1,0,entries(row(1,i)$SquareMatrix(dim,K))::T] for i in 1..dim] ev(n:NAT):% == reduce(_+,[ dx^n * dx^n for dx in basisIn()])$List(%) co(n:NAT):% == reduce(_+,[ Dx^n * Dx^n for Dx in basisOut()])$List(%)
-- manipulation map(f:K->K, g:%):% == per coerce [dom g,cod g,unravel(map(f,ravel dat(g).data))$T] if K has Evalable(K) then eval(g:%,f:List Equation K):% == map((x:K):K+->eval(x,f),g) ravel(g:%):List K == ravel dat(g).data unravel(p:Prop %,r:List K):% == dim^(dom(p)+cod(p)) ~= #r => error "failed" per coerce [dom(p),cod(p),unravel(r)$T]
-- sum (f:% + g:%):% == dat(f).data=0 => g dat(g).data=0 => f dom(f) ~= dom(g) or cod(f) ~= cod(g) => error "arity" per coerce [dom f,cod f,dat(f).data+dat(g).data]
(f:% - g:%):% == dat(f).data=0 => g dat(g).data=0 => f dom(f) ~= dom(f) or cod(g) ~= cod(g) => error "arity" per coerce [dom f, cod f,dat(f).data-dat(g).data]
_-(f:%):% == per coerce [dom f, cod f,-dat(f).data]
-- identity for sum (trivial zero map) 0 == per coerce [0,0,0] zero?(f:%):Boolean == dat(f).data = 0 * dat(f).data -- identity for product 1:% == per 1 one?(f:%):Boolean == one? rep f -- identity for composition I := per coerce [1,1,kroneckerDelta()$T] (x:% = y:%):Boolean == rep eval x = rep eval y
-- permutations and identities coerce(p:List NAT):% == r:=I^#p #p = 1 and p.1 = 1 => return r p1:List Integer:=[i for i in 1..#p] p2:List Integer:=[#p+i for i in p] p3:=concat(p1,p2) --output("coerce p3 = ",p3::OutputForm)$OutputPackage per coerce [#p,#p,reindex(dat(r).data,p3)] coerce(p:List None):% == per coerce [0,0,1] coerce(x:K):% == 1$% * x
-- tensor product elt(f:%,g:%):% == f * g elt(f:%,g:NAT):% == f * I^g elt(f:%,g1:NAT,g2:NAT):% == f * [g1 @ NAT,g2 @ NAT]::List NAT::% elt(f:%,g1:NAT,g2:NAT,g3:NAT):% == f * [g1 @ NAT,g2 @ NAT,g3 @ NAT]::List NAT::% apply(f:%,g:%):% == f * g (f:% * g:%):% == per (rep f * rep g)
leadI(x:Rep):NNI == r:=hclf(x,rep(I)^size(x)) size(r)=0 => 0 nthExpon(r,1)
trailI(x:Rep):NNI == r:=hcrf(x,rep(I)^size(x)) size(r)=0 => 0 nthExpon(r,1)
-- composition: -- f/g : A^n -> A^p = f:A^n -> A^m / g:A^m -> A^p (ff:% / gg:%):% == g:=gg; f:=ff -- partial application from the left n:=subtractIfCan(cod ff,dom gg) if n case NNI and n>0 then -- apply g on f from the left, pass extra f outputs on the right print(hconcat([message("arity warning: "), _ over(arity(ff)::OutputForm, _ arity(gg)::OutputForm*(arity(I)::OutputForm)^n::OutputForm) ]))$OutputForm g:=gg*I^n m:=subtractIfCan(dom gg, cod ff) -- apply g on f from the left, add extra g inputs on the left if m case NNI and m>0 then print(hconcat([message("arity warning: "), _ over((arity(I)::OutputForm)^m::OutputForm*arity(ff)::OutputForm, _ arity(gg)::OutputForm)]))$OutputForm f:=I^m*ff
-- parallelize composition f/g = (f1/g1)*(f2/g2) if cod(f)>0 then i:Integer:=1 j:Integer:=1 n:NNI:=1 m:NNI:=1 f1 := per coerce nthFactor(rep f,1) g1 := per coerce nthFactor(rep g,1) while cod(f1)~=dom(g1) repeat if cod(f1) < dom(g1) then if n < nthExpon(rep f,i) then n:=n+1 else n:=1 i:=i+1 f1 := f1 * per coerce nthFactor(rep f,i) else if cod(f1) > dom(g1) then if m < nthExpon(rep g,j) then m:=m+1 else n:=1 j:=j+1 g1 := g1 * per coerce nthFactor(rep g,j) f2 := per overlap(rep f1, rep f).rm g2 := per overlap(rep g1,rep g).rm f := f1 g := g1 else f2 := per 1 g2 := per 1
-- remove leading and trailing identities nf := leadI rep f f := per overlap(rep(I)^nf,rep f).rm ng := leadI rep g g := per overlap(rep(I)^ng,rep g).rm fn := trailI rep f f := per overlap(rep f,rep(I)^fn).lm gn := trailI rep g g := per overlap(rep g,rep(I)^gn).lm
-- parallel factors guarantees that these are just identities if nf>0 and ng>0 then return I*(f2/g2) if fn>0 and gn>0 then output("Should not happen: trailing [fn,gn] = ",[fn,gn]::OutputForm)$OutputPackage return (f/g)*I
-- Exercise for Reader: -- Prove the following contraction and permutation is correct by -- considering all 9 cases for (nf=0 or ng=0) and (fn=0 or gn=0). -- output("leading [nl,nf,ng] = ",[nl,nf,ng]::OutputForm)$OutputPackage -- output("trailing [ln,fn,gn] = ",[ln,fn,gn]::OutputForm)$OutputPackage r:T := contract(cod(f)-ng-gn, dat(f).data,dom(f)+ng+1, dat(g).data,nf+1) p:List Integer:=concat [ _ [dom(f)+gn+i for i in 1..nf], _ [i for i in 1..dom(f)], _ [dom(f)+nf+ng+i for i in 1..fn], _ [dom(f)+i for i in 1..ng], _ [dom(f)+nf+ng+fn+gn+i for i in 1..cod(g)], _ [dom(f)+ng+i for i in 1..gn] ] --print(p::OutputForm)$OutputForm r:=reindex(r,p)
if f2=1 and g2=1 then return per coerce [nf+dom(f)+fn,ng+cod(g)+gn,r] return per coerce [nf+dom(f)+fn,ng+cod(g)+gn,r] * (f2/g2)
-- another notation for composition of products (t:Tuple % / x:%):% == t / construct([x])$PrimitiveArray(%)::Tuple(%) (x:% / t:Tuple %):% == construct([x])$PrimitiveArray(%)::Tuple(%) / t (f:Tuple % / g:Tuple %):% == fs:List % := [select(f,i) for i in 0..#f-1] gs:List % := [select(g,i) for i in 0..#g-1] fr:=reduce(elt@(%,%)->%,fs,1) gr:=reduce(elt@(%,%)->%,gs,1) fr / gr
(x:K * y:%):% == per coerce [dom y, cod y,x*dat(y).data] (x:% * y:K):% == per coerce [dom x,cod x,dat(x).data*y] (x:Integer * y:%):% == per coerce [dom y,cod y,x*dat(y).data]
-- display operators using basis show(x:%):OutputForm == dom(x)=0 and cod(x)=0 => return (dat(x).data)::OutputForm if size()$gener > 0 then gens:List OutputForm:=[index(i::PositiveInteger)$gener::OutputForm for i in 1..dim] else -- default to numeric indices gens:List OutputForm:=[i::OutputForm for i in 1..dim] -- input basis inps:List OutputForm := [] for i in 1..dom(x) repeat empty? inps => inps:=gens inps:=concat [[(inps.k * gens.j) for j in 1..dim] for k in 1..#inps] -- output basis outs:List OutputForm := [] for i in 1..cod(x) repeat empty? outs => outs:=gens outs:=concat [[(outs.k * gens.j) for j in 1..dim] for k in 1..#outs] -- combine input (superscripts) and/or output(subscripts) to form basis symbols bases:List OutputForm if #inps > 0 and #outs > 0 then bases:=concat([[ scripts(message("|"),[i,j]) for i in outs] for j in inps]) else if #inps > 0 then bases:=[super(message("|"),i) for i in inps] else if #outs > 0 then bases:=[sub(message("|"),j) for j in outs] else bases:List OutputForm:= [] -- merge bases with data to form term list terms:=[(k=1 => base;k::OutputForm*base) for base in bases for k in ravel dat(x).data | k~=0] empty? terms => return 0::OutputForm -- combine the terms return reduce(_+,terms)
coerce(x:%):OutputForm == r:OutputForm := empty() for y in factors(rep x) repeat if y.exp = 1 then if size rep x = 1 then r := show per coerce y.gen else r:=r*paren(list show per coerce y.gen) else r:=r*paren(list show per coerce y.gen)^(y.exp::OutputForm) return r
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2301117976732635678-25px005.spad
      using old system compiler.
   LOP abbreviates domain LinearOperator 
------------------------------------------------------------------------
   initializing NRLIB LOP for LinearOperator 
   compiling into NRLIB LOP 
   importing List NonNegativeInteger
   processing macro definition Rep ==> FreeMonoid L 
   compiling local rep : $ -> FreeMonoid L
      LOP;rep is replaced by x 
Time: 0.02 SEC.
compiling local per : FreeMonoid L -> $ LOP;per is replaced by x Time: 0 SEC.
compiling local dimension : () -> CardinalNumber Time: 0 SEC.
compiling exported dom : $ -> NonNegativeInteger Time: 0 SEC.
compiling exported cod : $ -> NonNegativeInteger Time: 0 SEC.
compiling local prod : (L,L) -> L Time: 0.02 SEC.
compiling local dats : List RR -> L Time: 0 SEC.
compiling local dat : $ -> L Time: 0 SEC.
compiling exported arity : $ -> Prop $ Time: 0 SEC.
compiling exported eval : $ -> $ Time: 0 SEC.
compiling exported retractIfCan : $ -> Union(K,failed) Time: 0 SEC.
compiling exported retract : $ -> K Time: 0 SEC.
compiling exported basisOut : () -> List $ Time: 0.01 SEC.
compiling exported basisIn : () -> List $ Time: 0.02 SEC.
compiling exported ev : PositiveInteger -> $ Time: 0.02 SEC.
compiling exported co : PositiveInteger -> $ Time: 0 SEC.
compiling exported map : (K -> K,$) -> $ Time: 0 SEC.
****** Domain: K already in scope augmenting K: (Evalable K) compiling exported eval : ($,List Equation K) -> $ Time: 0.01 SEC.
compiling exported ravel : $ -> List K Time: 0 SEC.
compiling exported unravel : (Prop $,List K) -> $ Time: 0.01 SEC.
compiling exported + : ($,$) -> $ Time: 0.01 SEC.
compiling exported - : ($,$) -> $ Time: 0.01 SEC.
compiling exported - : $ -> $ Time: 0 SEC.
compiling exported Zero : () -> $ Time: 0 SEC.
compiling exported zero? : $ -> Boolean Time: 0 SEC.
compiling exported One : () -> $ Time: 0 SEC.
compiling exported one? : $ -> Boolean Time: 0 SEC.
compiling exported = : ($,$) -> Boolean Time: 0.01 SEC.
compiling exported coerce : List PositiveInteger -> $ Time: 0.02 SEC.
compiling exported coerce : List None -> $ Time: 0.01 SEC.
compiling exported coerce : K -> $ ****** comp fails at level 3 with expression: ****** error in function coerce
(* (|Sel| $ 1) |x|) ****** level 3 ****** $x:= x $m:= $ $f:= ((((|x| # #) (I #) (|dat| #)) ((|dat| #) (|dats| #)) ((|dats| #) (|prod| #)) ((|prod| #) (|dimension| #) (* #) (+ #) ...) ...))
>> Apparent user error: Cannot coerce x of mode K to mode $

Getting Started

Consult the source code above for more details.

Convenient Notation

fricas
-- summation
macro Σ(f,i,b) == reduce(+,[f*b.i for i in 1..#b])
Type: Void
fricas
-- list comprehension
macro Ξ(f,i)==[f for i in 1..retract(dimension()$L)]
Type: Void

Basis

fricas
Q := EXPR INT

\label{eq1}\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ })(1)
Type: Type
fricas
L := LOP(OVAR ['x,'y],Q)
LinearOperator is an unknown constructor and so is unavailable. Did you mean to use -> but type something different instead?

Tests

fricas
A:L := Σ( Σ( script(a,[[j],[i]]), i,Dx), j,dx)
L is not a valid type.

Various special cases of composition

fricas
-- case 1
test( X/X = [1,2] )
There are 3 exposed and 0 unexposed library operations named equation having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op equation to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named equation with argument type(s) Fraction(Polynomial(Integer)) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Construction

fricas
A1:L := Σ(superscript(a1,[i]),i,dx)
L is not a valid type.

Composition (evaluation)


g/f : A^n \to A^{m+p} = f:A^n \to A^m <em> g:A^n \to A^p
 
fricas
AB2 := A2 / B2; AB2::OutputForm = A2::OutputForm / B2::OutputForm

\label{eq2}{\frac{A 2}{B 2}}={\frac{A 2}{B 2}}(2)
Type: Equation(OutputForm?)
fricas
arity(AB2)::OutputForm = arity(A2)::OutputForm / arity(B2)::OutputForm
There are 1 exposed and 0 unexposed library operations named arity having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op arity to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named arity with argument type(s) Fraction(Polynomial(Integer))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Partial Evaluation

fricas
BBA1 := B/A1

\label{eq3}\frac{B}{A 1}(3)
Type: Fraction(Polynomial(Integer))
fricas
BBA2 := B/B1

\label{eq4}\frac{B}{B 1}(4)
Type: Fraction(Polynomial(Integer))
fricas
BBA3 := A1/A

\label{eq5}\frac{A 1}{A}(5)
Type: Fraction(Polynomial(Integer))
fricas
BBA4 := B1/A

\label{eq6}\frac{B 1}{A}(6)
Type: Fraction(Polynomial(Integer))

Powers

fricas
AB3:=(AB1*AB1)*AB1;
Type: Polynomial(Integer)
fricas
arity(AB3)
There are 1 exposed and 0 unexposed library operations named arity having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op arity to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named arity with argument type(s) Polynomial(Integer)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Sums


f+g : A^n \to A^m = f:A^n \to A^m + g:A^n \to A^m
 
fricas
A12s := A1 + A2; A12s::OutputForm = A1::OutputForm + A2::OutputForm

\label{eq7}{A 2 + A 1}={A 1 + A 2}(7)
Type: Equation(OutputForm?)
fricas
arity(A12s)::OutputForm = arity(A1)::OutputForm + arity(A2)::OutputForm
There are 1 exposed and 0 unexposed library operations named arity having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op arity to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named arity with argument type(s) Polynomial(Integer)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Multiplication

fricas
A3s:=(A1+A1)+A1

\label{eq8}3 \  A 1(8)
Type: Polynomial(Integer)
fricas
arity(A3s)
There are 1 exposed and 0 unexposed library operations named arity having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op arity to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named arity with argument type(s) Polynomial(Integer)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
B3s:=(B1+B1)+B1

\label{eq9}3 \  B 1(9)
Type: Polynomial(Integer)
fricas
arity(B3s)
There are 1 exposed and 0 unexposed library operations named arity having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op arity to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named arity with argument type(s) Polynomial(Integer)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Product

fricas
AB11:=A1*B1

\label{eq10}A 1 \  B 1(10)
Type: Polynomial(Integer)
fricas
arity(AB11)
There are 1 exposed and 0 unexposed library operations named arity having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op arity to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named arity with argument type(s) Polynomial(Integer)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Permutations

fricas
-- braid
B3:=(I*X)/(X*I)

\label{eq11}1(11)
Type: Fraction(Polynomial(Integer))
fricas
test(B3/B3/B3 = I*I*I)

\label{eq12} \mbox{\rm false} (12)
Type: Boolean
fricas
-- parallel
test((X*X)/(X*X)=H*H)

\label{eq13} \mbox{\rm false} (13)
Type: Boolean

Manipulations

fricas
ravel AB
There are 1 exposed and 0 unexposed library operations named ravel having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op ravel to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named ravel with argument type(s) Variable(AB)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Examples

Another kind of diagram:

Y = Y U U

Algebra

fricas
Y:=Σ(Σ(Σ(script(y,[[k],[i,j]]),j,dx),i,dx),k,Dx)
There are 2 exposed and 2 unexposed library operations named # having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op # to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named # with argument type(s) Variable(Dx)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Commutator

fricas
Y -   X /
      Y

\label{eq14}\frac{{{Y}^{2}}- X}{Y}(14)
Type: Fraction(Polynomial(Integer))

Pairing

fricas
U:=Σ(Σ(script(u,[[],[i,j]]),j,dx),i,dx)
There are 2 exposed and 2 unexposed library operations named # having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op # to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named # with argument type(s) Variable(dx)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

3-point function

fricas
YU := Y I
  /    U
There are no exposed library operations named Y but there are 2 unexposed operations with that name. Use HyperDoc Browse or issue )display op Y to learn more about the available operations.
Cannot find a definition or applicable library operation named Y with argument type(s) Variable(I)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
YU:L := (Y,I) / U
L is not a valid type.

Oddities (should work on the right)

fricas
YU := Y [1]
  /    U

\label{eq15}\frac{Y_{1}}{U}(15)
Type: Fraction(Polynomial(Integer))
fricas
YU := Y.[1]
  /    U

\label{eq16}\frac{Y_{1}}{U}(16)
Type: Fraction(Polynomial(Integer))

Ok on the left

fricas
UY := [1].Y
  /      U
There are 1 exposed and 1 unexposed library operations named elt having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op elt to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find application of object of type List(NonNegativeInteger) to argument(s) of type(s) Variable(Y)

Co-algebra

fricas
λ:=Σ(Σ(Σ(script(y,[[i],[j,k]]),i,dx),j,Dx),k,Dx)
There are 2 exposed and 2 unexposed library operations named # having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op # to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named # with argument type(s) Variable(Dx)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Handle

λ
Y

fricas
Φ := λ
  /  Y

\label{eq17}\frac{��}{Y}(17)
Type: Fraction(Polynomial(Integer))
fricas
arity Φ
There are 1 exposed and 0 unexposed library operations named arity having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op arity to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named arity with argument type(s) Fraction(Polynomial(Integer))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Back to the top.

Comments

Please leave comments and suggestions.

Thanks

Bill Page

I am trying to understand this page. It might or might not be relevant to the finding of transforms that I am doing. It looks like it would; if I understood it. I hope you have some patience with ignorance! In the process of walking through the above (I presume some load is missing?):

(2) above on my machine gives " L := LOP(OVAR ['x,'y]?,Q) There are no library operations named LOP Use HyperDoc? Browse or issue )what op LOP to learn if there is any operation containing " LOP " in its name.

Cannot find a definition or applicable library operation named LOP with argument type(s) Type Type

Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. "

Re: LOP ? --Bill page, Tue, 30 Dec 2014 00:31:12 +0000 reply
See the SPAD code following:
  )abbrev domain LOP LinearOperator

above. LOP is an abbreviation for a new domain called LinearOperator?.

For a faster reply you might want to email me directly or use one of the FriCAS?/Axiom email lists.