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## Introduction

Bi-graded two-color linear operators (transformations) over a finite n-dimensional vector space, including products and duals on a field . Members of this domain are morphisms where and (dual), etc. Products, permutations and composition (grafting) of morphisms are implemented. Operators are represented internally as tensors.

Operator composition and products can be visualized by oriented directed graphs. The graphs have a top and a bottom and arrows are directed either downwards or upwards.

External vertices in this graph represent vectors, tensors and their duals. Internal nodes and arcs (edges) represent linear operators. Edges may be one of two different "colors" (labelled + and - in the code that follows) or marked with arrows directed either downwards (+) or upwards (-). Horizontal juxtaposition (i.e. a horizontal cross-section) represents tensor product. Vertical juxtaposition of edges of the same color represents operator composition.

See examples and documentation below

I would be glad if 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 by defining the concept of a compact closed category.

Ref:

fricas
(1) -> <spad>
fricas
)abbrev domain ARITY Arity
Arity():  Exports == Implementation where
COLOR ==> OrderedVariableList ['_+,'_-]
LIST2 ==> ListFunctions2
NNI ==> NonNegativeInteger
RS ==> Record(gen:COLOR,exp:NNI)
Exports ==> Join(Monoid, RetractableTo COLOR) with
index: PositiveInteger -> %
dn: () -> %
++ default
up: () -> %
++ dual
0: %
++ use 0 for identity
"+": (%,%) -> %
++ composition
size: % -> NonNegativeInteger
len: % -> Integer
color: (%,NNI) -> %
overlap: (%,%) -> Record(lm:%, mm:%, rm:%)
Rep ==> FreeMonoid COLOR
rep(x:%):Rep == x pretend Rep
per(x:Rep):% == x pretend %
index(x:PositiveInteger):% == per(index(x)$COLOR ::Rep) up():% == index(2) dn():% == index(1) 0:% == per 1 (f:% + g:%):% == per(rep f * rep g) coerce(x:%):OutputForm == (rep(x)=1 => message "0";rep(x)::OutputForm) -- size of FreeMonoid is just number of factors -- len is total length len(f:%):Integer == reduce(_+,map(x+->x.exp,factors rep f)$LIST2(RS,NNI),0)$List(NNI) pretend Integer -- i'th element color(f:%,i:NNI):% == j:NNI:=0 for x in factors rep f repeat j:=j+x.exp if j>=i then return per coerce x.gen error "index error"</spad> fricas Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7355925127427701597-25px001.spad using old system compiler. ARITY abbreviates domain Arity ------------------------------------------------------------------------ initializing NRLIB ARITY for Arity compiling into NRLIB ARITY processing macro definition Rep ==> FreeMonoid OrderedVariableList construct(QUOTE +,QUOTE -) compiling local rep : % -> FreeMonoid OrderedVariableList construct(QUOTE +,QUOTE -) ARITY;rep is replaced by x Time: 0 SEC. compiling local per : FreeMonoid OrderedVariableList construct(QUOTE +,QUOTE -) -> % ARITY;per is replaced by x Time: 0 SEC. compiling exported index : PositiveInteger -> % Time: 0 SEC. compiling exported up : () -> % Time: 0 SEC. compiling exported dn : () -> % Time: 0 SEC. compiling exported Zero : () -> % Time: 0 SEC. compiling exported + : (%,%) -> % Time: 0 SEC. compiling exported coerce : % -> OutputForm Time: 0 SEC. compiling exported len : % -> Integer Time: 0 SEC. compiling exported color : (%,NonNegativeInteger) -> % Time: 0 SEC. (time taken in buildFunctor: 0) ;;; *** |Arity| REDEFINED ;;; *** |Arity| REDEFINED Time: 0 SEC. Warnings: [1] len: exp has no value [2] len: pretend(Integer) -- should replace by @ [3] color: exp has no value [4] color: gen has no value Cumulative Statistics for Constructor Arity Time: 0.02 seconds --------------non extending category---------------------- .. Arity of cat (|Join| (|Monoid|) (|RetractableTo| (|OrderedVariableList| (|construct| '+ '-))) (CATEGORY |domain| (SIGNATURE |index| (% (|PositiveInteger|))) (SIGNATURE |dn| (%)) (SIGNATURE |up| (%)) (SIGNATURE (|Zero|) (%) |constant|) (SIGNATURE + (% % %)) (SIGNATURE |size| ((|NonNegativeInteger|) %)) (SIGNATURE |len| ((|Integer|) %)) (SIGNATURE |color| (% % (|NonNegativeInteger|))) (SIGNATURE |overlap| ((|Record| (|:| |lm| %) (|:| |mm| %) (|:| |rm| %)) % %)))) has no (|RetractableTo| (|OrderedVariableList| (|construct| + -))) finalizing NRLIB ARITY Processing Arity for Browser database: --->-->Arity(constructor): Not documented!!!! --->-->Arity((index (% (PositiveInteger)))): Not documented!!!! --------(dn (%))--------- --------(up (%))--------- --------((Zero) (%) constant)--------- --->-->Arity(((Zero) (%) constant)): Improper first word in comments: use "use 0 for identity" --------(+ (% % %))--------- --->-->Arity((size ((NonNegativeInteger) %))): Not documented!!!! --->-->Arity((len ((Integer) %))): Not documented!!!! --->-->Arity((color (% % (NonNegativeInteger)))): Not documented!!!! --->-->Arity((overlap ((Record (: lm %) (: mm %) (: rm %)) % %))): Not documented!!!! --->-->Arity(): Missing Description ; compiling file "/var/aw/var/LatexWiki/ARITY.NRLIB/ARITY.lsp" (written 09 APR 2024 01:02:33 PM): ; wrote /var/aw/var/LatexWiki/ARITY.NRLIB/ARITY.fasl ; compilation finished in 0:00:00.012 ------------------------------------------------------------------------ Arity is now explicitly exposed in frame initial Arity will be automatically loaded when needed from /var/aw/var/LatexWiki/ARITY.NRLIB/ARITY spad )abbrev category CMONAL ClosedMonoidal ClosedMonoidal():Category == Ring with dom: % -> Arity ++ domain cod: % -> Arity ++ co-domain _/: (%,%) -> % ++ vertical composition f/g apply: (%,%) -> % ++ horizontal product f g = f*g "^": (%,Arity) -> % ++ colored power dagger: % -> % ++ reverse arrows: ++ cod f = dom dagger f ++ dom f = cod dagger f spad  Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7854049118560040482-25px002.spad using old system compiler. CMONAL abbreviates category ClosedMonoidal ------------------------------------------------------------------------ initializing NRLIB CMONAL for ClosedMonoidal compiling into NRLIB CMONAL ;;; *** |ClosedMonoidal| REDEFINED Time: 0 SEC. finalizing NRLIB CMONAL Processing ClosedMonoidal for Browser database: --->-->ClosedMonoidal(constructor): Not documented!!!! --------(dom ((Arity) %))--------- --------(cod ((Arity) %))--------- --------(/ (% % %))--------- --->-->ClosedMonoidal((/ (% % %))): Improper first word in comments: vertical "vertical composition \\spad{f/g}" --------(apply (% % %))--------- --->-->ClosedMonoidal((apply (% % %))): Improper first word in comments: horizontal "horizontal product \\spad{f} \\spad{g} = \\spad{f*g}" --------(^ (% % (Arity)))--------- --------(dagger (% %))--------- --->-->ClosedMonoidal(): Missing Description ; compiling file "/var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL.lsp" (written 09 APR 2024 01:02:33 PM): ; wrote /var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL.fasl ; compilation finished in 0:00:00.000 ------------------------------------------------------------------------ ClosedMonoidal is now explicitly exposed in frame initial ClosedMonoidal will be automatically loaded when needed from /var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL The initial object in this category is the domain ClosedProp (Products and Permutations). The ClosedProp domain represents everything that is "constant" about all the domains in this category. It can be defined as an endofunctor with only the information available about the category itself. spad )abbrev domain CPROP ClosedProp ClosedProp(L:ClosedMonoidal): Exports == Implementation where Exports ==> ClosedMonoidal with coerce: L -> % Implementation ==> add Rep ==> Record(domain:Arity, codomain:Arity) 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:%):Arity == rep(x).domain -- dom(x:%):NNI == dom rep x cod(x:%):Arity == rep(x).codomain -- cod(x:%):NNI == cod rep x --0:% == per [1,1] -- 0:% == per 0 1:% == per [1,1] -- 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) dagger(f:%):% == per [cod f, dom f] (f:% ^ p:Arity):% == r:% := 1 for i in 1..len(p) repeat if color(p,i)=dn() then r:=r*f else r:=r*dagger(f) return r spad  Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8579968413172847989-25px003.spad using old system compiler. CPROP abbreviates domain ClosedProp ------------------------------------------------------------------------ initializing NRLIB CPROP for ClosedProp compiling into NRLIB CPROP processing macro definition Rep ==> Record(domain: Arity,codomain: Arity) compiling local rep : % -> Record(domain: Arity,codomain: Arity) CPROP;rep is replaced by x Time: 0 SEC. compiling local per : Record(domain: Arity,codomain: Arity) -> % CPROP;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 : % -> Arity Time: 0 SEC. compiling exported cod : % -> Arity Time: 0 SEC. compiling exported One : () -> % Time: 0 SEC. compiling exported / : (%,%) -> % Time: 0 SEC. compiling exported apply : (%,%) -> % Time: 0 SEC. compiling exported * : (%,%) -> % Time: 0 SEC. compiling exported dagger : % -> % Time: 0 SEC. compiling exported ^ : (%,Arity) -> % Time: 0 SEC. (time taken in buildFunctor: 0) ;;; *** |ClosedProp| REDEFINED ;;; *** |ClosedProp| REDEFINED Time: 0 SEC. Warnings: [1] dom: domain has no value [2] cod: codomain has no value Cumulative Statistics for Constructor ClosedProp Time: 0 seconds finalizing NRLIB CPROP Processing ClosedProp for Browser database: --->-->ClosedProp(constructor): Not documented!!!! --->-->ClosedProp((coerce (% L))): Not documented!!!! --->-->ClosedProp(): Missing Description ; compiling file "/var/aw/var/LatexWiki/CPROP.NRLIB/CPROP.lsp" (written 09 APR 2024 01:02:33 PM): ; wrote /var/aw/var/LatexWiki/CPROP.NRLIB/CPROP.fasl ; compilation finished in 0:00:00.016 ------------------------------------------------------------------------ ClosedProp is now explicitly exposed in frame initial ClosedProp will be automatically loaded when needed from /var/aw/var/LatexWiki/CPROP.NRLIB/CPROP The ClosedLinearOperator domain is ClosedMonoidal over FreeMonoid OrderedVariables? ['+,'-], i.e. strings of symbols. 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 powers of vector space and it's dual) to another such power. • all members of this domain have the same dimension Faster tensors fricas )lib CARTEN CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN spad )abbrev domain CLOP ClosedLinearOperator ClosedLinearOperator(gener:OrderedFinite,K:Field): Exports == Implementation where NNI ==> NonNegativeInteger Exports ==> Join(Ring, FramedModule K, ClosedMonoidal, RetractableTo K) with dimension: () -> CardinalNumber arity: % -> ClosedProp % basisOut: () -> List % basisIn: () -> List % map: (K->K,%) -> % if K has Evalable(K) then Evalable(K) eval: % -> % ravel: % -> List K unravel: (ClosedProp %,List K) -> % coerce:(x:List Integer) -> % ++ identity for composition and permutations of its products coerce:(x:List None) -> % ++ [] = 1 elt: (%,%) -> % elt: (%,Integer) -> % elt: (%,Integer,Integer) -> % elt: (%,Integer,Integer,Integer) -> % _/: (Tuple %,Tuple %) -> % _/: (Tuple %,%) -> % _/: (%,Tuple %) -> % ++ yet another syntax for product ev: NNI -> % ++ (2,0)-tensor for evaluation co: NNI -> % ++ (0,2)-tensor for co-evaluation Implementation ==> add LIST2 ==> ListFunctions2 dim:NNI := size()$gener
dimension():CardinalNumber == coerce dim
T := CartesianTensor(1,dim,K)
L := Record(domain:Arity, codomain:Arity, 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:%):Arity ==
reduce(_*,map(x+->(x.gen.domain)^(x.exp),factors rep f)$LIST2(RR,Arity),1)$List(Arity)
cod(f:%):Arity ==
reduce(_*,map(x+->(x.gen.codomain)^(x.exp),factors rep f)$LIST2(RR,Arity),1)$List(Arity)
dagger(f:%):% ==
r:=1
for f1 in factors rep f repeat
p:List Integer := concat [ _
[len(f1.gen.domain)+i for i in 1..len(f1.gen.codomain)], _
[i for i in 1..len(f1.gen.domain)]]
r:= per(coerce [f1.gen.codomain, f1.gen.domain, reindex(f1.gen.data,p) ])^f1.exp * r
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..len(f.domain)],              _
[len(f.domain)+len(f.codomain)+i for i in 1..len(g.domain)], _
[len(f.domain)+i for i in 1..len(f.codomain)],        _
[len(f.domain)+len(g.domain)+len(f.codomain)+i for i in 1..len(g.codomain)]]
-- dom(f) + dom(g) + cod(f) + cod(g)
--output("prod p = ",p::OutputForm)$OutputPackage [(f.domain)*(g.domain),(f.codomain)*(g.codomain),reindex(r,p)] dats(fs:List RR):L == r:L := [1,1,1$T]  -- scalar 1 as tensor
for y in fs repeat
t:L:=y.gen
for n in 1..y.exp repeat
r:=prod(r,t)
return r
dat(f:Rep):L == dats factors f
arity(f:%):ClosedProp % == f::ClosedProp %
eval(f:%):% == per coerce dat(rep f)
retractIfCan(f:%):Union(K,"failed") ==
dom(f)=1 and cod(f)=1 => retract(dat(rep f).data)$T return "failed" retract(f:%):K == dom(f)=1 and cod(f)=1 => retract(dat(rep f).data)$T
error "failed"
-- basis
basisOut():List % == [per coerce [1,dn(),entries(row(1,i)$SquareMatrix(dim,K))::T] for i in 1..dim] basisIn():List % == [per coerce [dn(),1,entries(row(1,i)$SquareMatrix(dim,K))::T] for i in 1..dim]
ev(n:NNI):% == reduce(_+,[ dx^n * dx^n for dx in basisIn()])$List(%) co(n:NNI):% == reduce(_+,[ Dx^n * Dx^n for Dx in basisOut()])$List(%)
-- manipulation
map(f:K->K, g:%):% == per coerce [dom g,cod g,unravel(map(f,ravel dat(rep g).data))$T] if K has Evalable(K) then eval(g:%,f:List Equation K):% == map((x:K):K+->eval(x,f),g) ravel(g:%):List K == ravel dat(rep g).data unravel(p:ClosedProp %,r:List K):% == dim^(len(dom(p)*cod(p)) pretend NNI) ~= #r => error "failed" per coerce [dom(p),cod(p),unravel(r)$T]
-- sum
(f:% + g:%):% ==
dat(rep f).data=0 => g
dat(rep g).data=0 => f
dom(f) ~= dom(g) or cod(f) ~= cod(g) => error "arity"
per coerce [dom f,cod f,dat(rep f).data+dat(rep g).data]
(f:% - g:%):% ==
dat(rep f).data=0 => g
dat(rep g).data=0 => f
dom(f) ~= dom(f) or cod(g) ~= cod(g) => error "arity"
per coerce [dom f, cod f,dat(rep f).data-dat(rep g).data]
_-(f:%):% == per coerce [dom f, cod f,-dat(rep f).data]
(x:% = y:%):Boolean ==
if rep x = rep y then true
else rep eval x = rep eval y
-- identity for sum (trivial zero map)
0 == per coerce [1,1,0]
zero?(f:%):Boolean == dat(rep f).data = 0 * dat(rep f).data
-- identity for product
1:% == per 1
one?(f:%):Boolean == one? rep f
-- identity for composition
I := per coerce [dn(),dn(),kroneckerDelta()$T] -- permutations and identities coerce(p:List Integer):% == r:=I^#p --#p = 1 and p.1 = 1 => return r p3:List Integer:=concat [ _ [i for i in 1..#p], _ [#p+abs(i) for i in p]] d:Arity:=dn()^(#p) c:Arity := reduce(_*,map((x:Integer):Arity+->(x>0=>dn();up()),p)$LIST2(Integer,Arity))$List(Arity) r:=per coerce [d, c, reindex(dat(rep r).data,p3)] return r coerce(p:List None):% == per coerce [1,1,1] -- product elt(f:%,g:%):% == f * g elt(f:%,g:Integer):% == f * [g @ Integer]::List Integer::% elt(f:%,g1:Integer,g2:Integer):% == f * [g1 @ Integer,g2 @ Integer]::List Integer::% elt(f:%,g1:Integer,g2:Integer,g3:Integer):% == f * [g1 @ Integer,g2 @ Integer,g3 @ Integer]::List Integer::% apply(f:%,g:%):% == f * g -- just free monoid product (f:% * g:%):% == r := per (rep f * rep g) -- evaluate scalars if dom(f)=1 and cod(f)=1 then return eval(r) if dom(g)=1 and cod(g)=1 then return eval(r) return r coerce(x:K):% == x*1 -- tensor powers (f:% ^ p:Arity):% == r:% := 1 for i in 1..len(p) repeat if color(p,i)=dn() then r:=r*f else r:=r*dagger(f) return r -- returns arity of leading identities leadI(x:Rep):Arity == xx := x s:Arity:=1 repeat r:=overlap(rep(I),xx) if r.rm=1 and r.mm=rep(I) then s := s * dn() xx:= r.lm else r:=overlap(rep(dagger I),xx) if r.rm=1 and r.mm=rep(I) then s := s * up() xx:= r.lm else break return s -- returns arity of trailing identities trailI(x:Rep):Arity == xx := x s:Arity:=1 repeat r:=overlap(xx,rep(I)) if r.lm=1 and r.mm=rep(I) then s := dn() * s xx:= r.rm else r:=overlap(xx,rep(dagger I)) if r.lm=1 and r.mm=rep(I) then s := up() * s xx:= r.rm else break return s -- tensor composition: compose(lnf:Integer,f:Rep,lfn:Integer, lng:Integer,g:Rep,lgn:Integer):T == ldf := len dom(per f) lcf := len cod(per f) ldg := len dom(per g) lcg := len cod(per g) --output("ldf, lcf, ldg, lcg = ", [ldf,lcf,ldg,lcg]::List Integer::OutputForm)$OutputPackage
r:T := contract(lcf-lng-lgn, dat(f).data,ldf+lng+1, dat(g).data,lnf+1)
p:List Integer:=concat [ _
[ldf+lgn+i for i in 1..lnf], _
[i for i in 1..ldf], _
[ldf+lnf+lng+i for i in 1..lfn], _
[ldf+i for i in 1..lng], _
[ldf+lnf+lng+lfn+lgn+i for i in 1..lcg], _
[ldf+lng+i for i in 1..lgn] ]
--print(p::OutputForm)$OutputForm return reindex(r,p) parallelize(f:Rep,g:Rep):Record(f1:Rep,g1:Rep,f2:Rep,g2:Rep) == -- parallelize composition f/g = (f1/g1)*(f2/g2) r:Record(f1:Rep,g1:Rep,f2:Rep,g2:Rep):=[1,1,1,1] if cod(per f)~=1 then i:Integer:=1 j:Integer:=1 n:NNI:=1 m:NNI:=1 r.f1 := nthFactor(f,1)::Rep r.g1 := nthFactor(g,1)::Rep while cod(per r.f1) ~= dom(per r.g1) repeat if len(cod per r.f1) < len(dom per r.g1) then if n < nthExpon(f,i) then n:=n+1 else n:=1 i:=i+1 r.f1 := r.f1 * nthFactor(f,i)::Rep else if len(cod per r.f1) > len(dom per r.g1) then if m < nthExpon(g,j) then m:=m+1 else n:=1 j:=j+1 r.g1 := r.g1 * nthFactor(g,j)::Rep r.f2 := overlap(r.f1, f).rm r.g2 := overlap(r.g1, g).rm --print(r::OutputForm)$OutputForm
return r
-- f/g : A^n -> A^p = f:A^n -> A^m / g:A^m -> A^p
(ff:% / gg:%):% ==
-- scalars
if dom(ff)=1 and cod(ff)=1 and dom(gg)=1 and cod(gg)=1 then return ff*gg
fg:=overlap(cod ff,dom gg)
if fg.rm~=1 or fg.lm~=1 then
-- pass extra f inputs on the left
-- pass extra g outputs on the right
print(hconcat([message("arity warning: "), _
over(arity(ff)::OutputForm * arity(I^(fg.rm))::OutputForm, _
arity(I^(fg.lm))::OutputForm * arity(gg)::OutputForm) ]))$OutputForm r:=parallelize(rep(ff*I^(fg.rm)),rep(I^(fg.lm)*gg)) -- remove leading and trailing identities nf := leadI r.f1 r.f1 := overlap(rep(I^nf),r.f1).rm ng := leadI r.g1 r.g1 := overlap(rep(I^ng),r.g1).rm --output("nf,ng = ",[nf,ng]::List Arity::OutputForm)$OutputPackage
fn := trailI r.f1
f := overlap(r.f1,rep(I^fn)).lm
gn := trailI r.g1
g := overlap(r.g1,rep(I^gn)).lm
--output("fn,gn = ",[fn,gn]::List Arity::OutputForm)$OutputPackage -- parallel factors guarantees that these are just identities if nf~=1 and nf=ng then return I^nf*(per(r.f2)/per(r.g2)) if fn~=1 and gn~=1 then return (per(f)/per(g))*I^fn return per([nf*dom(per f)*fn, ng*cod(per g)*gn, _ compose(len nf,f,len fn, len ng,g,len gn)]::Rep) * _ (per(r.f2)/per(r.g2)) -- another notation for composition of products (t:Tuple % / x:%):% == t / construct([x])$PrimitiveArray(%)::Tuple(%)
(x:% / t:Tuple %):% == construct([x])$PrimitiveArray(%)::Tuple(%) / t (f:Tuple % / g:Tuple %):% == fs:List % := [select(f,i) for i in 0..#f-1] gs:List % := [select(g,i) for i in 0..#g-1] fr:=reduce(elt@(%,%)->%,fs,1) gr:=reduce(elt@(%,%)->%,gs,1) fr / gr (x:K * y:%):% == per coerce [dom y, cod y,x*dat(rep y).data] --(x:% * y:K):% == per coerce [dom x,cod x,dat(rep x).data*y] (x:Integer * y:%):% == per coerce [dom y,cod y,x*dat(rep y).data] -- display operators using basis show(x:%):OutputForm == dom(x)=1 and cod(x)=1 => return (dat(rep x).data)::OutputForm gens:List OutputForm:=[index(i::PositiveInteger)$gener::OutputForm for i in 1..dim]
-- input basis
inps:List OutputForm := list empty()
dx:=dom(x)
for i in 1..len(dx) repeat
--empty? inps => inps:=gens
inps:=concat [[(inps.k * (color(dx,i)=dn()=>gens.j;super(gens.j,message "*"))) _
for j in 1..dim] for k in 1..#inps]
-- output basis
outs:List OutputForm := list empty()
cx:=cod(x)
for i in 1..len(cx) repeat
--empty? outs => outs:=gens
outs:=concat [[(outs.k * (color(cx,i)=dn()=>gens.j;super(gens.j,message "*"))) _
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(x) | 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
   Compiling FriCAS source code from file
using old system compiler.
CLOP abbreviates domain ClosedLinearOperator
------------------------------------------------------------------------
initializing NRLIB CLOP for ClosedLinearOperator
compiling into NRLIB CLOP
processing macro definition LIST2 ==> ListFunctions2
compiling exported dimension : () -> CardinalNumber
Time: 0 SEC.
processing macro definition Rep ==> FreeMonoid L
compiling local rep : % -> FreeMonoid L
CLOP;rep is replaced by x
Time: 0 SEC.
compiling local per : FreeMonoid L -> %
CLOP;per is replaced by x
Time: 0 SEC.
compiling exported dimension : () -> CardinalNumber
Time: 0 SEC.
compiling exported dom : % -> Arity
Time: 0 SEC.
compiling exported cod : % -> Arity
Time: 0 SEC.
compiling exported dagger : % -> %
Time: 0.12 SEC.
compiling local prod : (L,L) -> L
Time: 0 SEC.
compiling local dats : List RR -> L
Time: 0 SEC.
compiling local dat : FreeMonoid L -> L
Time: 0 SEC.
compiling exported arity : % -> ClosedProp %
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 SEC.
compiling exported basisIn : () -> List %
Time: 0 SEC.
compiling exported ev : NonNegativeInteger -> %
Time: 0 SEC.
compiling exported co : NonNegativeInteger -> %
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 SEC.
compiling exported ravel : % -> List K
Time: 0 SEC.
compiling exported unravel : (ClosedProp %,List K) -> %
Time: 0.02 SEC.
compiling exported + : (%,%) -> %
Time: 0 SEC.
compiling exported - : (%,%) -> %
Time: 0 SEC.
compiling exported - : % -> %
Time: 0 SEC.
compiling exported = : (%,%) -> Boolean
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 coerce : List Integer -> %
Time: 0 SEC.
compiling exported coerce : List None -> %
Time: 0 SEC.
compiling exported elt : (%,%) -> %
Time: 0 SEC.
compiling exported elt : (%,Integer) -> %
Time: 0 SEC.
compiling exported elt : (%,Integer,Integer) -> %
Time: 0 SEC.
compiling exported elt : (%,Integer,Integer,Integer) -> %
Time: 0 SEC.
compiling exported apply : (%,%) -> %
Time: 0 SEC.
compiling exported * : (%,%) -> %
Time: 0 SEC.
compiling exported coerce : K -> %
Time: 0 SEC.
compiling exported ^ : (%,Arity) -> %
Time: 0 SEC.
compiling local leadI : FreeMonoid L -> Arity
Time: 0 SEC.
compiling local trailI : FreeMonoid L -> Arity
Time: 0 SEC.
compiling local compose : (Integer,FreeMonoid L,Integer,Integer,FreeMonoid L,Integer) -> T$Time: 0.02 SEC. compiling local parallelize : (FreeMonoid L,FreeMonoid L) -> Record(f1: FreeMonoid L,g1: FreeMonoid L,f2: FreeMonoid L,g2: FreeMonoid L) Time: 0 SEC. compiling exported / : (%,%) -> % Time: 0.26 SEC. compiling exported / : (Tuple %,%) -> % Time: 0 SEC. compiling exported / : (%,Tuple %) -> % Time: 0 SEC. compiling exported / : (Tuple %,Tuple %) -> % Time: 0 SEC. compiling exported * : (K,%) -> % Time: 0 SEC. compiling exported * : (Integer,%) -> % Time: 0 SEC. compiling local show : % -> OutputForm Time: 0.01 SEC. compiling exported coerce : % -> OutputForm Time: 0.01 SEC. ****** Domain: K already in scope augmenting K: (Evalable K) ****** Domain: K already in scope augmenting K: (Finite) ****** Domain: K already in scope augmenting K: (Hashable) (time taken in buildFunctor: 0) ;;; *** |ClosedLinearOperator| REDEFINED ;;; *** |ClosedLinearOperator| REDEFINED Time: 0 SEC. Warnings: [1] dom: gen has no value [2] dom: exp has no value [3] cod: gen has no value [4] cod: exp has no value [5] dagger: gen has no value [6] dagger: codomain has no value [7] dagger: exp has no value [8] prod: data has no value [9] prod: domain has no value [10] prod: codomain has no value [11] dats: gen has no value [12] dats: exp has no value [13] retractIfCan: data has no value [14] retract: data has no value [15] map: data has no value [16] ravel: data has no value [17] +: data has no value [18] -: data has no value [19] zero?: data has no value [20] coerce: data has no value [21] leadI: rm has no value [22] leadI: mm has no value [23] leadI: lm has no value [24] trailI: lm has no value [25] trailI: mm has no value [26] trailI: rm has no value [27] compose: data has no value [28] parallelize: f1 has no value [29] parallelize: g1 has no value [30] parallelize: i has no value [31] parallelize: j has no value [32] parallelize: f2 has no value [33] parallelize: g2 has no value [34] /: rm has no value [35] /: lm has no value [36] /: f1 has no value [37] /: g1 has no value [38] /: f2 has no value [39] /: g2 has no value [40] *: data has no value [41] show: data has no value [42] coerce: exp has no value [43] coerce: gen has no value Cumulative Statistics for Constructor ClosedLinearOperator Time: 0.55 seconds finalizing NRLIB CLOP Processing ClosedLinearOperator for Browser database: --->-->ClosedLinearOperator(constructor): Not documented!!!! --->-->ClosedLinearOperator((dimension ((CardinalNumber)))): Not documented!!!! --->-->ClosedLinearOperator((arity ((ClosedProp %) %))): Not documented!!!! --->-->ClosedLinearOperator((basisOut ((List %)))): Not documented!!!! --->-->ClosedLinearOperator((basisIn ((List %)))): Not documented!!!! --->-->ClosedLinearOperator((map (% (Mapping K K) %))): Not documented!!!! --->-->ClosedLinearOperator((eval (% %))): Not documented!!!! --->-->ClosedLinearOperator((ravel ((List K) %))): Not documented!!!! --->-->ClosedLinearOperator((unravel (% (ClosedProp %) (List K)))): Not documented!!!! --------(coerce (% (List (Integer))))--------- --->-->ClosedLinearOperator((coerce (% (List (Integer))))): Improper first word in comments: identity "identity for composition and permutations of its products" --------(coerce (% (List (None))))--------- --->-->ClosedLinearOperator((coerce (% (List (None))))): Improper first word in comments: [] "[] = 1" --->-->ClosedLinearOperator((elt (% % %))): Not documented!!!! --->-->ClosedLinearOperator((elt (% % (Integer)))): Not documented!!!! --->-->ClosedLinearOperator((elt (% % (Integer) (Integer)))): Not documented!!!! --->-->ClosedLinearOperator((elt (% % (Integer) (Integer) (Integer)))): Not documented!!!! --->-->ClosedLinearOperator((/ (% (Tuple %) (Tuple %)))): Not documented!!!! --->-->ClosedLinearOperator((/ (% (Tuple %) %))): Not documented!!!! --------(/ (% % (Tuple %)))--------- --->-->ClosedLinearOperator((/ (% % (Tuple %)))): Improper first word in comments: yet "yet another syntax for product" --------(ev (% (NonNegativeInteger)))--------- --->-->ClosedLinearOperator((ev (% (NonNegativeInteger)))): Improper first word in comments: "(2,{}0)-tensor for evaluation" --------(co (% (NonNegativeInteger)))--------- --->-->ClosedLinearOperator((co (% (NonNegativeInteger)))): Improper first word in comments: "(0,{}2)-tensor for co-evaluation" --->-->ClosedLinearOperator(): Missing Description ; compiling file "/var/aw/var/LatexWiki/CLOP.NRLIB/CLOP.lsp" (written 09 APR 2024 01:02:34 PM): ; wrote /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP.fasl ; compilation finished in 0:00:00.428 ------------------------------------------------------------------------ ClosedLinearOperator is now explicitly exposed in frame initial ClosedLinearOperator will be automatically loaded when needed from /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP ## 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

Example

fricas
Q := EXPR INT
 (1)
Type: Type
fricas
L := CLOP(OVAR ['x,'y],Q)
 (2)
Type: Type
fricas
)show L
ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)) is a domain constructor.
Abbreviation for ClosedLinearOperator is CLOP
This constructor is exposed in this frame.
48 Names for 69 Operations in this Domain.
------------------------------- Operations --------------------------------
?*? : (Integer, %) -> %               ?*? : (PositiveInteger, %) -> %
?*? : (%, %) -> %                     ?+? : (%, %) -> %
?-? : (%, %) -> %                     -? : % -> %
?/? : (Tuple(%), Tuple(%)) -> %       ?/? : (Tuple(%), %) -> %
?/? : (%, Tuple(%)) -> %              ?/? : (%, %) -> %
?=? : (%, %) -> Boolean               1 : () -> %
0 : () -> %                           ?^? : (%, Arity) -> %
?^? : (%, PositiveInteger) -> %       annihilate? : (%, %) -> Boolean
antiCommutator : (%, %) -> %          apply : (%, %) -> %
arity : % -> ClosedProp(%)            associator : (%, %, %) -> %
basis : () -> Vector(%)               basisIn : () -> List(%)
basisOut : () -> List(%)              co : NonNegativeInteger -> %
cod : % -> Arity                      coerce : % -> OutputForm
coerce : Expression(Integer) -> %     coerce : Integer -> %
coerce : List(Integer) -> %           coerce : List(None) -> %
commutator : (%, %) -> %              dagger : % -> %
dimension : () -> CardinalNumber      dom : % -> Arity
elt : (%, Integer, Integer) -> %      elt : (%, Integer) -> %
elt : (%, %) -> %                     ev : NonNegativeInteger -> %
eval : % -> %                         latex : % -> String
one? : % -> Boolean                   opposite? : (%, %) -> Boolean
rank : () -> PositiveInteger          recip : % -> Union(%,"failed")
sample : () -> %                      zero? : % -> Boolean
?~=? : (%, %) -> Boolean
?*? : (Expression(Integer), %) -> %
?*? : (NonNegativeInteger, %) -> %
?^? : (%, NonNegativeInteger) -> %
characteristic : () -> NonNegativeInteger
convert : % -> Vector(Expression(Integer))
convert : Vector(Expression(Integer)) -> %
coordinates : Vector(%) -> Matrix(Expression(Integer))
coordinates : % -> Vector(Expression(Integer))
elt : (%, Integer, Integer, Integer) -> %
leftPower : (%, NonNegativeInteger) -> %
leftPower : (%, PositiveInteger) -> %
leftRecip : % -> Union(%,"failed")
map : ((Expression(Integer) -> Expression(Integer)), %) -> %
ravel : % -> List(Expression(Integer))
represents : Vector(Expression(Integer)) -> %
retract : % -> Expression(Integer)
retractIfCan : % -> Union(Expression(Integer),"failed")
rightPower : (%, NonNegativeInteger) -> %
rightPower : (%, PositiveInteger) -> %
rightRecip : % -> Union(%,"failed")
subtractIfCan : (%, %) -> Union(%,"failed")
unravel : (ClosedProp(%), List(Expression(Integer))) -> %

Basis

fricas
dim:Integer:=retract dimension()$L  (3) Type: Integer fricas Dx:=basisOut()$L
 (4)
Type: List(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))
fricas
dx:=basisIn()\$L
 (5)
Type: List(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))
fricas
matrix Ξ(Ξ( eval(dx.i * Dx.j), i),j)
 (6)
Type: Matrix(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))
fricas
matrix Ξ(Ξ( Dx.i / dx.j, i),j)
 (7)
Type: Matrix(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))

(1,1)-tensor

fricas
A:L := Σ( Σ( script(a,[[j],[i]]), i,Dx), j,dx)
 (8)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
a:=arity(A)
 (9)
Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))
fricas
p:=dom(A)
 (10)
Type: Arity

scalar

fricas
s:= 3::L
 (11)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
arity s
 (12)
Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))
fricas
2*s
 (13)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
s*2
 (14)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))

Powers

fricas
A^p
 (15)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
a^p
 (16)
Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))

fricas
s/A
0 +
- -
0 +
arity warning: ---
0 +
- -
0 +
 (17)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
A/s
+ 0
- -
+ 0
arity warning: ---
+ 0
- -
+ 0
 (18)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
3*A
 (19)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
A/3
+ 0
- -
+ 0
arity warning: ---
+ 0
- -
+ 0
 (20)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))

identities

fricas
I:L := [1]
 (21)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
I*I
 (22)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
arity(I*I)
 (23)
Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))
fricas
I/I
 (24)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
H:L:=[1,2]
 (25)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
arity(H)
 (26)
Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))
fricas
eval(I*I)
 (27)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
arity eval(I*I)
 (28)
Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer)))
fricas
test( I*I = H )
 (29)
Type: Boolean
fricas
-- twist
X:L := [2,1]
 (30)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
test(X/X=H)
 (31)
Type: Boolean
fricas
-- printing
I*X*X*I
 (32)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
-- trace
U:L:=ev(1)
 (33)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
Ω:L:=co(1)
 (34)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
Ω/U
 (35)
Type: ClosedLinearOperator(OrderedVariableList([x,y]),Expression(Integer))
fricas
test
( I Ω  ) /
(  U I ) = I
 (36)
Type: Boolean
fricas
test
(  Ω I ) /
( I U  ) = I
 (37)
Type: Boolean

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