Introduction
Bi-graded linear operators (transformations) over n-dimensional vector spaces on a commutative ring . Members of this domain are morphisms . 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
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.
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
"FriCAS 1.3.10 compiled at Wed 10 Jan 02:19:45 CET 2024"
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 09 APR 2024 12:59:48 PM):
; wrote /var/aw/var/LatexWiki/MONAL.NRLIB/MONAL.fasl
; compilation finished in 0:00:00.000
------------------------------------------------------------------------
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 SEC.
compiling exported / : (%,%) -> %
Time: 0 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 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 09 APR 2024 12:59:48 PM):
; wrote /var/aw/var/LatexWiki/PROP.NRLIB/PROP.fasl
; compilation finished in 0:00:00.012
------------------------------------------------------------------------
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
dimension: () -> CardinalNumber
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)$OutputPackage
per coerce [#p,#p,reindex(dat(r).data,p3)]
coerce(p:List None):% == per coerce [0,0,1]
coerce(x:K):% == x*1
-- tensor product
elt(f:%,g:%):% == f * g
elt(f:%,g:NAT):% == f * I^g
elt(f:%,g1:NAT,g2:NAT):% == f * [g1 @ NAT,g2 @ NAT]::List NAT::%
elt(f:%,g1:NAT,g2:NAT,g3:NAT):% == f * [g1 @ NAT,g2 @ NAT,g3 @ NAT]::List NAT::%
apply(f:%,g:%):% == f * g
(f:% * g:%):% == per (rep f * rep g)
leadI(x:Rep):NNI ==
r:=hclf(x,rep(I)^size(x))
size(r)=0 => 0
nthExpon(r,1)
trailI(x:Rep):NNI ==
r:=hcrf(x,rep(I)^size(x))
size(r)=0 => 0
nthExpon(r,1)
-- composition:
-- f/g : A^n -> A^p = f:A^n -> A^m / g:A^m -> A^p
(ff:% / gg:%):% ==
g:=gg; f:=ff
-- partial application from the left
nn:=subtractIfCan(cod ff,dom gg)
if nn case NNI and nn>0 then
-- apply g on f from the left, pass extra f outputs on the right
print(hconcat([message("arity warning: "), _
over(arity(ff)::OutputForm, _
arity(gg)::OutputForm*(arity(I)::OutputForm)^nn::OutputForm) ]))$OutputForm
g:=gg*I^nn
mm:=subtractIfCan(dom gg, cod ff)
-- apply g on f from the left, add extra g inputs on the left
if mm case NNI and mm>0 then
print(hconcat([message("arity warning: "), _
over((arity(I)::OutputForm)^mm::OutputForm*arity(ff)::OutputForm, _
arity(gg)::OutputForm)]))$OutputForm
f:=I^mm*ff
-- parallelize composition f/g = (f1/g1)*(f2/g2)
if cod(f)>0 then
i:Integer:=1
j:Integer:=1
n:NNI:=1
m:NNI:=1
f1 := per coerce nthFactor(rep f,1)
g1 := per coerce nthFactor(rep g,1)
while cod(f1)~=dom(g1) repeat
if cod(f1) < dom(g1) then
if n < nthExpon(rep f,i) then
n:=n+1
else
n:=1
i:=i+1
f1 := f1 * per coerce nthFactor(rep f,i)
else if cod(f1) > dom(g1) then
if m < nthExpon(rep g,j) then
m:=m+1
else
n:=1
j:=j+1
g1 := g1 * per coerce nthFactor(rep g,j)
f2 := per overlap(rep f1, rep f).rm
g2 := per overlap(rep g1,rep g).rm
f := f1
g := g1
else
f2 := per 1
g2 := per 1
-- remove leading and trailing identities
nf := leadI rep f
f := per overlap(rep(I)^nf,rep f).rm
ng := leadI rep g
g := per overlap(rep(I)^ng,rep g).rm
fn := trailI rep f
f := per overlap(rep f,rep(I)^fn).lm
gn := trailI rep g
g := per overlap(rep g,rep(I)^gn).lm
-- parallel factors guarantees that these are just identities
if nf>0 and ng>0 then
return I*(f2/g2)
if fn>0 and gn>0 then
output("Should not happen: trailing [fn,gn] = ",[fn,gn]::OutputForm)$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])$PrimitiveArray(%)::Tuple(%) / t
(f:Tuple % / g:Tuple %):% ==
fs:List % := [select(f,i) for i in 0..#f-1]
gs:List % := [select(g,i) for i in 0..#g-1]
fr:=reduce(elt@(%,%)->%,fs,1)
gr:=reduce(elt@(%,%)->%,gs,1)
fr / gr
(x:K * y:%):% == per coerce [dom y, cod y,x*dat(y).data]
--(x:% * y:K):% == per coerce [dom x,cod x,dat(x).data*y] -- local and exported???
(x:Integer * y:%):% == per coerce [dom y,cod y,x*dat(y).data]
-- display operators using basis
show(x:%):OutputForm ==
dom(x)=0 and cod(x)=0 => return (dat(x).data)::OutputForm
if size()$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/3030360856115257157-25px005.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 SEC.
compiling local per : FreeMonoid L -> %
LOP;per is replaced by x
Time: 0 SEC.
compiling exported 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 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 SEC.
compiling exported basisIn : () -> List %
Time: 0 SEC.
compiling exported ev : PositiveInteger -> %
Time: 0 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 SEC.
compiling exported ravel : % -> List K
Time: 0 SEC.
compiling exported unravel : (Prop %,List K) -> %
Time: 0 SEC.
compiling exported + : (%,%) -> %
Time: 0 SEC.
compiling exported - : (%,%) -> %
Time: 0 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 SEC.
compiling exported coerce : List PositiveInteger -> %
Time: 0.01 SEC.
compiling exported coerce : List None -> %
Time: 0 SEC.
compiling exported coerce : K -> %
Time: 0 SEC.
compiling exported elt : (%,%) -> %
Time: 0 SEC.
compiling exported elt : (%,PositiveInteger) -> %
Time: 0 SEC.
compiling exported elt : (%,PositiveInteger,PositiveInteger) -> %
Time: 0 SEC.
compiling exported elt : (%,PositiveInteger,PositiveInteger,PositiveInteger) -> %
Time: 0 SEC.
compiling exported apply : (%,%) -> %
Time: 0 SEC.
compiling exported * : (%,%) -> %
Time: 0 SEC.
compiling local leadI : FreeMonoid L -> NonNegativeInteger
Time: 0 SEC.
compiling local trailI : FreeMonoid L -> NonNegativeInteger
Time: 0.01 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 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)
;;; *** |LinearOperator| REDEFINED
;;; *** |LinearOperator| 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] prod: data has no value
[6] prod: domain has no value
[7] prod: codomain has no value
[8] dats: gen has no value
[9] dats: exp has no value
[10] retractIfCan: data has no value
[11] retract: data has no value
[12] map: data has no value
[13] ravel: data has no value
[14] +: data has no value
[15] -: data has no value
[16] zero?: data has no value
[17] coerce: data has no value
[18] /: i has no value
[19] /: j has no value
[20] /: data has no value
[21] *: data has no value
[22] show: data has no value
[23] coerce: exp has no value
[24] coerce: gen has no value
Cumulative Statistics for Constructor LinearOperator
Time: 0.40 seconds
finalizing NRLIB LOP
Processing LinearOperator for Browser database:
--->-->LinearOperator(constructor): Not documented!!!!
--->-->LinearOperator((dimension ((CardinalNumber)))): Not documented!!!!
--->-->LinearOperator((arity ((Prop %) %))): Not documented!!!!
--->-->LinearOperator((basisOut ((List %)))): Not documented!!!!
--->-->LinearOperator((basisIn ((List %)))): Not documented!!!!
--->-->LinearOperator((map (% (Mapping K K) %))): Not documented!!!!
--->-->LinearOperator((eval (% %))): Not documented!!!!
--->-->LinearOperator((ravel ((List K) %))): Not documented!!!!
--->-->LinearOperator((unravel (% (Prop %) (List K)))): Not documented!!!!
--------(coerce (% (List (PositiveInteger))))---------
--->-->LinearOperator((coerce (% (List (PositiveInteger))))): Improper first word in comments: identity
"identity for composition and permutations of its products"
--------(coerce (% (List (None))))---------
--->-->LinearOperator((coerce (% (List (None))))): Improper first word in comments: []
"[] = 1"
--->-->LinearOperator((elt (% % %))): Not documented!!!!
--->-->LinearOperator((elt (% % (PositiveInteger)))): Not documented!!!!
--->-->LinearOperator((elt (% % (PositiveInteger) (PositiveInteger)))): Not documented!!!!
--->-->LinearOperator((elt (% % (PositiveInteger) (PositiveInteger) (PositiveInteger)))): Not documented!!!!
--->-->LinearOperator((/ (% (Tuple %) (Tuple %)))): Not documented!!!!
--->-->LinearOperator((/ (% (Tuple %) %))): Not documented!!!!
--------(/ (% % (Tuple %)))---------
--->-->LinearOperator((/ (% % (Tuple %)))): Improper first word in comments: yet
"yet another syntax for product"
--------(ev (% (PositiveInteger)))---------
--->-->LinearOperator((ev (% (PositiveInteger)))): Improper first word in comments:
"(2,{}0)-tensor for evaluation"
--------(co (% (PositiveInteger)))---------
--->-->LinearOperator((co (% (PositiveInteger)))): Improper first word in comments:
"(0,{}2)-tensor for co-evaluation"
--->-->LinearOperator(): Missing Description
; compiling file "/var/aw/var/LatexWiki/LOP.NRLIB/LOP.lsp" (written 09 APR 2024 12:59:48 PM):
; wrote /var/aw/var/LatexWiki/LOP.NRLIB/LOP.fasl
; compilation finished in 0:00:00.388
------------------------------------------------------------------------
LinearOperator is now explicitly exposed in frame initial
LinearOperator will be automatically loaded when needed from
/var/aw/var/LatexWiki/LOP.NRLIB/LOP
Consult the source code above for more details.
Convenient Notation
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-- summation
macro Σ(f,i,b) == reduce(+,[f*b.i for i in 1..#b])
Type: Void
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-- list comprehension
macro Ξ(f,i)==[f for i in 1..retract(dimension()$L)]
Type: Void
Basis
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Q := EXPR INT
Type: Type
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L := LOP(OVAR ['x,'y],Q)
Type: Type
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dim:Integer:=retract dimension()$L
Type: Integer
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Dx:=basisOut()$L
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dx:=basisIn()$L
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matrix Ξ(Ξ( eval(dx.i * Dx.j), i),j)
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matrix Ξ(Ξ( Dx.i / dx.j, i),j)
Tests
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A:L := Σ( Σ( script(a,[[j],[i]]), i,Dx), j,dx)
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-- scalar
3*A
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A/3
1
-
1
arity warning: ------
0 1 1
- (-)
0 1
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-- identity
I:L := [1]
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H:L:=[1,2]
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test( I*I = H )
Type: Boolean
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-- twist
X:L := [2,1]
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test(X/X=H)
Type: Boolean
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-- printing
I*X*X*I
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-- trace
U:L:=ev(1)
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Ω:L:=co(1)
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Ω/U
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test
( I Ω ) /
( U I ) = I
Type: Boolean
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test
( Ω I ) /
( I U ) = I
Type: Boolean
Various special cases of composition
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-- case 1
test( X/X = [1,2] )
Type: Boolean
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test( (I*X)/(I*X) = [1,2,3] )
Type: Boolean
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test( (I*X*I)/(I*X*I) = [1,2,3,4] )
Type: Boolean
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-- case 2
test( (X*I*I)/(X*X) = [1,2,4,3] )
Type: Boolean
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-- case 3
test( (X*X)/(X*I*I) = [1,2,4,3] )
Type: Boolean
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-- case 4
test ( (I*I*X)/(X*X) = [2,1,3,4] )
Type: Boolean
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-- case 5
test( (I*X*I)/(X*X) = [3,1,4,2] )
Type: Boolean
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-- case 6
test( (I*I*X)/(X*I*I)=[2,1,4,3] )
Type: Boolean
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test( (I*X)/(X*I) = [3,1,2] )
Type: Boolean
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test( (I*X*I)/(X*I*I)=[3,1,2,4] )
Type: Boolean
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-- case 7
test( (X*X)/(I*I*X) = [2,1,3,4] )
Type: Boolean
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-- case 8
test( (X*I)/(I*X) = [2,3,1] )
Type: Boolean
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-- case 9
test( (X*X)/(I*X*I) = [2,4,1,3] )
Type: Boolean
Construction
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A1:L := Σ(superscript(a1,[i]),i,dx)
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arity A1
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A2:L := Σ(superscript(a2,[i]),i,dx)
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A:L := A1*I+I*A2
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arity A
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B1:L := Σ(subscript(b1,[i]),i,Dx)
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arity B1
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B2:L := Σ(subscript(b2,[i]),i,Dx)
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B:L := B1*I+I*B2
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arity B
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BB:L := Σ(Σ(subscript(b,[i,j]),i,Dx),j,Dx)
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W:L := Σ(Σ(Σ(script(w,[[k],[i,j]]),k,Dx),i,dx),j,dx)
Composition (evaluation)
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AB2 := A2 / B2; AB2::OutputForm = A2::OutputForm / B2::OutputForm
Type: Equation(OutputForm
?)
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arity(AB2)::OutputForm = arity(A2)::OutputForm / arity(B2)::OutputForm
Type: Equation(OutputForm
?)
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BA1 := B1 / A1; BA1::OutputForm = B1::OutputForm / A1::OutputForm
Type: Equation(OutputForm
?)
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arity(BA1)::OutputForm = arity(B1)::OutputForm / arity(A1)::OutputForm
Type: Equation(OutputForm
?)
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AB1 := A1 / B1; AB1::OutputForm = A1::OutputForm / B1::OutputForm
Type: Equation(OutputForm
?)
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arity(AB1)::OutputForm = arity(A1)::OutputForm / arity(B1)::OutputForm
Type: Equation(OutputForm
?)
Partial Evaluation
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BBA1 := B/A1
1
-
2
arity warning: ------
1 1 1
- (-)
0 1
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BBA2 := B/B1
1
-
2
arity warning: ------
0 1 2
- (-)
1 1
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BBA3 := A1/A
1 2 1
(-) -
1 0
arity warning: ------
2
-
1
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BBA4 := B1/A
1 1 0
(-) -
1 1
arity warning: ------
2
-
1
Powers
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AB3:=(AB1*AB1)*AB1;
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arity(AB3)
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test(AB3=AB1*(AB1*AB1))
Type: Boolean
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test(AB3=AB1^3)
Type: Boolean
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one? (AB3^0)
Type: Boolean
Sums
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A12s := A1 + A2; A12s::OutputForm = A1::OutputForm + A2::OutputForm
Type: Equation(OutputForm
?)
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arity(A12s)::OutputForm = arity(A1)::OutputForm + arity(A2)::OutputForm
Type: Equation(OutputForm
?)
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B12s := B1 + B2; B12s::OutputForm = B1::OutputForm + B2::OutputForm
Type: Equation(OutputForm
?)
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arity(B12s)::OutputForm = arity(B1)::OutputForm + arity(B2)::OutputForm
Type: Equation(OutputForm
?)
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-B12s
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zero? (A12s - A12s)
Type: Boolean
Multiplication
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A3s:=(A1+A1)+A1
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arity(A3s)
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test(A3s=A1+(A1+A1))
Type: Boolean
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test(A3s=A1*3)
Type: Boolean
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zero? (0*A3s)
Type: Boolean
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B3s:=(B1+B1)+B1
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arity(B3s)
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test(B3s=B1+(B1+B1))
Type: Boolean
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test(B3s=B1*3)
Type: Boolean
Product
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AB11:=A1*B1
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arity(AB11)
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BA11:= B1*A1
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arity(BA11)
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AB := A*B
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arity(AB)
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BA := B*A
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arity(BA)
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WB1:=W*B1
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arity(WB1)
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test((A*A)*A=A*(A*A))
Type: Boolean
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test((B*B)*B=B*(B*B))
Type: Boolean
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test((A*B)*A=A*(B*A))
Type: Boolean
Permutations
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-- braid
B3:=(I*X)/(X*I)
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test(B3/B3/B3 = I*I*I)
Type: Boolean
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-- parallel
test((X*X)/(X*X)=H*H)
Type: Boolean
Manipulations
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ravel AB
Type: List(Expression(Integer))
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test(unravel(arity AB,ravel AB)$L=AB)
Type: Boolean
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map(x+->x+1,AB)$L
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[superscript(a1,[i])=i for i in 1..dim]::List EQ Q
Type: List(Equation(Expression(Integer)))
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eval(A,[superscript(a1,[i])=i for i in 1..dim]::List EQ Q)
Examples
Another kind of diagram:
Y = Y
U U
Algebra
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Y:=Σ(Σ(Σ(script(y,[[k],[i,j]]),j,dx),i,dx),k,Dx)
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arity Y
Commutator
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Y - X /
Y
Pairing
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U:=Σ(Σ(script(u,[[],[i,j]]),j,dx),i,dx)
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arity U
3-point function
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YU := Y I
/ U
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YU := Y.I
/ U
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arity YU
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YU:L := (Y,I) / U
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UY:L := (I,Y) / U
Oddities (should work on the right)
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YU := Y [1]
/ U
There are no exposed library operations named Y but there are 2
unexposed operations with that name. Use HyperDoc Browse or issue
)display op Y
to learn more about the available operations.
Cannot find a definition or applicable library operation named Y
with argument type(s)
List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Ok on the left
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UY := [1].Y
/ U
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UY := [1] Y
/ U
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arity UY
Co-algebra
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λ:=Σ(Σ(Σ(script(y,[[i],[j,k]]),i,dx),j,Dx),k,Dx)
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arity λ
Handle
λ
Y
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Φ := λ
/ Y
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arity Φ
Back to the top.
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.
"
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.