Edit detail for LinearOperator revision 45 of 63

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.

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)

)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

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/2074022068168685528-25px001.spad using 
      old system compiler.
   MONAL abbreviates category Monoidal 
------------------------------------------------------------------------
   initializing NRLIB MONAL for Monoidal 
   compiling into NRLIB MONAL 

;;;     ***       |Monoidal| REDEFINED
Time: 0.02 SEC.

   finalizing NRLIB MONAL 
   Processing Monoidal for Browser database:
--------(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(constructor): Not documented!!!!
--->-->Monoidal(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL.lsp" (written 26 APR 2011 10:27:23 PM):

; /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL.fasl written
; compilation finished in 0:00:00.047
------------------------------------------------------------------------
   Monoidal is now explicitly exposed in frame initial 
   Monoidal will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

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.

)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)

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/3714416860228184972-25px002.spad using 
      old system compiler.
   PROP abbreviates domain Prop 
------------------------------------------------------------------------
   initializing NRLIB PROP for Prop 
   compiling into NRLIB PROP 
   compiling local rep : $ -> Record(domain: NonNegativeInteger,codomain: NonNegativeInteger)
      PROP;rep is replaced by x 
Time: 0.04 SEC.

   compiling local per : Record(domain: NonNegativeInteger,codomain: NonNegativeInteger) -> $
      PROP;per is replaced by x 
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.01 SEC.

   compiling exported coerce : L -> $
Time: 0 SEC.

   compiling exported dom : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling exported cod : $ -> NonNegativeInteger
Time: 0.01 SEC.

   compiling exported Zero : () -> $
Time: 0 SEC.

   compiling exported One : () -> $
Time: 0 SEC.

   compiling exported / : ($,$) -> $
Time: 0.01 SEC.

   compiling exported apply : ($,$) -> $
Time: 0 SEC.

   compiling exported * : ($,$) -> $
Time: 0.01 SEC.

   compiling exported + : ($,$) -> $
Time: 0.01 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |Prop| REDEFINED

;;;     ***       |Prop| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor Prop
      Time: 0.10 seconds

   finalizing NRLIB PROP 
   Processing Prop for Browser database:
--->-->Prop((coerce (% L))): Not documented!!!!
--->-->Prop(constructor): Not documented!!!!
--->-->Prop(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/PROP.NRLIB/PROP.lsp" (written 26 APR 2011 10:27:23 PM):

; /var/zope2/var/LatexWiki/PROP.NRLIB/PROP.fasl written
; compilation finished in 0:00:00.203
------------------------------------------------------------------------
   Prop is now explicitly exposed in frame initial 
   Prop will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/PROP.NRLIB/PROP

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

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
• it might be useful (but much more complicated) to allow source and target dimensions to be different

)abbrev domain LIN LinearOperator
)lib CARTEN
LinearOperator(dim:NNI,gen:OrderedFinite,K:CommutativeRing): Exports == Implementation where
  NNI ==> NonNegativeInteger
  NAT ==> PositiveInteger
  T ==> CartesianTensor(1,dim,K)

  Exports ==> Join(Ring, BiModule(K,K), Monoidal NNI, RetractableTo K) with
    inp: List K -> %
      ++ incoming vector
    inp: List % -> %
    out: List K -> %
      ++ output vector
    out: List % -> %
    arity: % -> Prop %
    basisVectors: () -> List %
    basisForms: () -> List %
    tensor: % -> T
    map: (K->K,%) -> %
    if K has Evalable(K) then Evalable(K)
    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) -> %
    ev: NAT -> %
      ++ (2,0)-tensor for evaluation
    co: NAT -> %
      ++ (0,2)-tensor for co-evaluation

  Implementation ==> add
    import List NNI
    import NAT
    Rep == Record(domain:NNI, codomain:NNI, data:T)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    -- Prop (arity)
    dom(f:%):NNI == rep(f).domain
    cod(f:%):NNI == rep(f).codomain
    dat(f:%):T == rep(f).data
    arity(f:%):Prop % == f::Prop %

    retractIfCan(f:%):Union(K,"failed") ==
      dom(f)=0 and cod(f)=0 => retract(dat f)$T
      return "failed"
    retract(f:%):K ==
      dom(f)=0 and cod(f)=0 => retract(dat f)$T
      error "failed"

    -- basis
    basisVectors():List % == [per [0,1,entries(row(1,i)$SquareMatrix(dim,K))::T] for i in 1..dim]
    basisForms():List % == [per [1,0,entries(row(1,i)$SquareMatrix(dim,K))::T] for i in 1..dim]
    ev(n:NAT):% ==
      dx:= basisForms()
      reduce(_+,[ (dx.i)^n * (dx.i)^n for i in 1..dim])
    co(n:NAT):% == 
      Dx:= basisVectors()
      reduce(_+,[ (Dx.i)^n * (Dx.i)^n for i in 1..dim])

    -- manipulation
    map(f:K->K, g:%):% == per [dom g,cod g,unravel(map(f,ravel dat g))$T]
    if K has Evalable(K) then
      eval(g:%,f:List Equation K):% == map((x:K):K+->eval(x,f),g)
    ravel(g:%):List K == ravel dat g
    unravel(p:Prop %,r:List K):% ==
      dim^(dom(p)+cod(p)) ~= #r => error "failed"
      per [dom(p),cod(p),unravel(r)$T]
    tensor(x:%):T == dat(x)

    -- sum
    (f:% + g:%):% ==
      dat(f)=0 => g
      dat(g)=0 => f
      dom(f) ~= dom(g) or cod(f) ~= cod(g) => error "arity"
      per [dom f,cod f,dat(f)+dat(g)]

    (f:% - g:%):% ==
      dat(f)=0 => g
      dat(g)=0 => f
      dom(f) ~= dom(f) or cod(g) ~= cod(g) => error "arity"
      per [dom f, cod f,dat(f)-dat(g)]

    _-(f:%):% == per [dom f, cod f,-dat(f)]

    -- repeated sum
    (p:NNI * f:%):% ==
      p=1 => f
      q:=subtractIfCan(p,1)
      q case NNI => q*f + f
      -- zero map (non-trivial)
      per [dom f,cod f,0*dat(f)]

    -- identity for sum (trivial zero map)
    0 == per [0,0,0]
    zero?(f:%):Boolean == dat f = 0 * dat f
    -- identity for product
    1:% == per [0,0,1]
    one?(f:%):Boolean == dat f = 1$T
    -- identity for composition
    I == per([1,1,kroneckerDelta()$T])
    -- inherited from Ring
    --(x:% = y:%):Boolean ==
    --  dom(x) ~= dom(y) or cod(x) ~= cod(y) => error "arity"
    --  dat(x) = dat(y)
    (x:% = y:%):Boolean == zero? (x - 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)
      per [#p,#p,reindex(dat r,p3)]
    coerce(p:List None):% == per [0,0,1]
    coerce(x:K):% == 1*x

    -- 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:%):% ==
      r:T := product(dat f,dat g)
      -- dom(f) + cod(f) + dom(g) + cod(g)
      p:List Integer := concat                _
        [[i for i in 1..dom(f)],              _
        [dom(f)+cod(f)+i for i in 1..dom(g)], _
        [dom(f)+i for i in 1..cod(f)],        _
        [dom(f)+dom(g)+cod(f)+i for i in 1..cod(g)]]
        -- dom(f) + dom(g) + cod(f) + cod(g)
      per [dom(f)+dom(g),cod(f)+cod(g),reindex(r,p)]

    -- repeated product
    (f:% ^ p:NNI):% ==
      p=1 => f
      q:=subtractIfCan(p,1)
      q case NNI => f^q * f
      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
      -- dom(g) ~= cod(f) => error "arity"
      f1:Integer:=dom(f)+1
      g1:Integer:=dom(f)+cod(f)+1
      if cod(f)=0 then
        r:T := product(dat f, dat g)
      else
        -- product and first contraction
        r:T := contract(dat f,f1, dat g,1)
        -- more contractions
        for i in 1..cod(f)-1 repeat
          r := contract(r,f1,g1-i)
      per [dom(f),cod(g),r]

    (x:K * y:%):% == per [dom y, cod y,x*dat(y)]
    (x:% * y:K):% == per [dom x,cod x,dat(x)*y]
    (x:Integer * y:%):% == per [dom y,cod y,x*dat(y)]

    -- constructors
    inp(x:List K):% == per [1,0,entries(x)::T]
    inp(x:List %):% ==
      #removeDuplicates([dom(y) for y in x]) ~= 1 or
        #removeDuplicates([cod(y) for y in x]) ~= 1 => error "arity"
      per [(dom(first x)+1),cod(first x),[dat(y) for y in x]::T]$Rep
    out(x:List K):% == per [0,1,entries(x)::T]
    out(x:List %):% ==
      #removeDuplicates([dom(y) for y in x])~=1 or
        #removeDuplicates([cod(y) for y in x])~=1 => error "arity"
      per [dom(first x),(cod(first x)+1),[dat(y) for y in x]::T]$Rep

    -- display operators using basis
    coerce(x:%):OutputForm ==
      dom(x)=0 and cod(x)=0 => return dat(x)::OutputForm
      if size()$gen > 0 then
        gens:List OutputForm:=[index(i::PositiveInteger)$gen::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:= []
      -- 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) | k~=0]
      empty? terms => return 0::OutputForm
      -- combine the terms
      return reduce(_+,terms)

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/5462585282849818384-25px003.spad using 
      old system compiler.
   LIN abbreviates domain LinearOperator 
   CartesianTensor is now explicitly exposed in frame initial 
   CartesianTensor will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CARTEN.NRLIB/CARTEN
------------------------------------------------------------------------
   initializing NRLIB LIN for LinearOperator 
   compiling into NRLIB LIN 
   importing List NonNegativeInteger
   importing PositiveInteger
   compiling local rep : $ -> Record(domain: NonNegativeInteger,codomain: NonNegativeInteger,data: CartesianTensor(One,dim,K))
      LIN;rep is replaced by x 
Time: 0.06 SEC.

   compiling local per : Record(domain: NonNegativeInteger,codomain: NonNegativeInteger,data: CartesianTensor(One,dim,K)) -> $
      LIN;per is replaced by x 
Time: 0.01 SEC.

   compiling exported dom : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling exported cod : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling local dat : $ -> CartesianTensor(One,dim,K)
Time: 0 SEC.

   compiling exported arity : $ -> Prop $
Time: 0.01 SEC.

   compiling exported retractIfCan : $ -> Union(K,failed)
Time: 0.01 SEC.

   compiling exported retract : $ -> K
Time: 0 SEC.

   compiling exported basisVectors : () -> List $
Time: 0.04 SEC.

   compiling exported basisForms : () -> List $
Time: 0.03 SEC.

   compiling exported ev : PositiveInteger -> $
Time: 0.32 SEC.

   compiling exported co : PositiveInteger -> $
Time: 0.07 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: 0.01 SEC.

   compiling exported unravel : (Prop $,List K) -> $
Time: 0.02 SEC.

   compiling exported tensor : $ -> CartesianTensor(One,dim,K)
Time: 0 SEC.

   compiling exported + : ($,$) -> $
Time: 0.01 SEC.

   compiling exported - : ($,$) -> $
Time: 0.01 SEC.

   compiling exported - : $ -> $
Time: 0 SEC.

   compiling exported * : (NonNegativeInteger,$) -> $
Time: 0.01 SEC.

   compiling exported Zero : () -> $
Time: 0 SEC.

   compiling exported zero? : $ -> Boolean
Time: 0 SEC.

   compiling exported One : () -> $
Time: 0.01 SEC.

   compiling exported one? : $ -> Boolean
Time: 0 SEC.

   compiling exported = : ($,$) -> Boolean
Time: 0.01 SEC.

   compiling exported coerce : List PositiveInteger -> $
Time: 0.03 SEC.

   compiling exported coerce : List None -> $
Time: 0.01 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: 0.02 SEC.

   compiling exported apply : ($,$) -> $
Time: 0 SEC.

   compiling exported * : ($,$) -> $
Time: 0.09 SEC.

   compiling exported ^ : ($,NonNegativeInteger) -> $
Time: 0.01 SEC.

   compiling exported / : ($,$) -> $
Time: 0.06 SEC.

   compiling exported * : (K,$) -> $
Time: 0 SEC.

   compiling exported * : ($,K) -> $
Time: 0 SEC.

   compiling exported * : (Integer,$) -> $
Time: 0 SEC.

   compiling exported inp : List K -> $
Time: 0 SEC.

   compiling exported inp : List $ -> $
Time: 0.02 SEC.

   compiling exported out : List K -> $
Time: 0.01 SEC.

   compiling exported out : List $ -> $
Time: 0.02 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.06 SEC.

****** Domain: K already in scope
augmenting K: (Evalable K)
(time taken in buildFunctor:  0)

;;;     ***       |LinearOperator| REDEFINED

;;;     ***       |LinearOperator| REDEFINED
Time: 0.02 SEC.

   Cumulative Statistics for Constructor LinearOperator
      Time: 0.99 seconds

   finalizing NRLIB LIN 
   Processing LinearOperator for Browser database:
--------(inp (% (List K)))---------
--->-->LinearOperator((inp (% (List %)))): Not documented!!!!
--------(out (% (List K)))---------
--->-->LinearOperator((out (% (List %)))): Not documented!!!!
--->-->LinearOperator((arity ((Prop %) %))): Not documented!!!!
--->-->LinearOperator((basisVectors ((List %)))): Not documented!!!!
--->-->LinearOperator((basisForms ((List %)))): Not documented!!!!
--->-->LinearOperator((tensor (T$ %))): Not documented!!!!
--->-->LinearOperator((map (% (Mapping K K) %))): Not documented!!!!
--->-->LinearOperator((ravel ((List K) %))): Not documented!!!!
--->-->LinearOperator((unravel (% (Prop %) (List K)))): Not documented!!!!
--------(coerce (% (List NAT)))---------
--->-->LinearOperator((coerce (% (List NAT)))): 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 (% % NAT))): Not documented!!!!
--->-->LinearOperator((elt (% % NAT NAT))): Not documented!!!!
--->-->LinearOperator((elt (% % NAT NAT NAT))): Not documented!!!!
--------(ev (% NAT))---------
--->-->LinearOperator((ev (% NAT))): Improper first word in comments: 
"(2,{}0)-tensor for evaluation"
--------(co (% NAT))---------
--->-->LinearOperator((co (% NAT))): Improper first word in comments: 
"(0,{}2)-tensor for co-evaluation"
--->-->LinearOperator(constructor): Not documented!!!!
--->-->LinearOperator(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/LIN.NRLIB/LIN.lsp" (written 26 APR 2011 10:27:25 PM):

; /var/zope2/var/LatexWiki/LIN.NRLIB/LIN.fasl written
; compilation finished in 0:00:01.816
------------------------------------------------------------------------
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

Getting Started

Consult the source code above for more details.

Conveniences

-- summation
macro Σ(x,i)==reduce(+,[x for i in 1..dim])

-- list comprehension
macro Ξ(f,i)==[f for i in 1..dim]

Basis

dim:=2

(1)

Q:=EXPR INT

(2)

L:=LinearOperator(dim,OVAR [x,y],Q)

(3)

Dx:=basisVectors()$L

(4)

dx:=basisForms()$L

(5)

matrix Ξ(Ξ( dx.i * Dx.j, i),j)

(6)

matrix Ξ(Ξ( Dx.i / dx.j, i),j)

(7)

Ω:=co(1)$L

(8)

U:=ev(1)$L

(9)

Ω / U

(10)

I:L:=[1]

(11)

test
  ( I Ω  ) /
  (  U I ) = I

(12)

test
  (  Ω I ) /
  ( I U  ) = I

(13)

Construction

A1:L := Σ(superscript(a1,[i])*dx.i,i)

(14)

arity A1

(15)

A2:L := Σ(superscript(a2,[i])*dx.i,i)

(16)

A:L := inp[A1,A2]

(17)

arity A

(18)

B1:L := Σ(subscript(b1,[i])*Dx.i,i)

(19)

arity B1

(20)

B2:L := Σ(subscript(b2,[i])*Dx.i,i)

(21)

B:L := out[B1,B2]

(22)

arity B

(23)

BB:L := Σ(Σ(subscript(b,[i,j])*Dx.i*Dx.j,i),j)

(24)

W:L := Σ(Σ(Σ(script(w,[[k],[i,j]])*(Dx.k*dx.i*dx.j),k),i),j)

(25)

Composition (evaluation)

AB2 := A2 / B2; AB2::OutputForm = A2::OutputForm / B2::OutputForm

(26)

arity(AB2)::OutputForm = arity(A2)::OutputForm / arity(B2)::OutputForm

(27)

BA1 := B1 / A1; BA1::OutputForm = B1::OutputForm / A1::OutputForm

(28)

arity(BA1)::OutputForm = arity(B1)::OutputForm / arity(A1)::OutputForm

(29)

AB1 := A1 / B1; AB1::OutputForm = A1::OutputForm / B1::OutputForm

(30)

arity(AB1)::OutputForm = arity(A1)::OutputForm / arity(B1)::OutputForm

(31)

Partial Evaluation

BBA1 := B/A1

                     0
                     -
                     2
   arity warning: ------
                  1  1 1
                  - (-)
                  0  1

(32)

BBA2 := B/B1

                     0
                     -
                     2
   arity warning: ------
                  0  1 2
                  - (-)
                  1  1

(33)

BBA3 := A1/A

                   1 2 1
                  (-)  -
                   1   0
   arity warning: ------
                     2
                     -
                     0

(34)

BBA4 := B1/A

                   1 1 0
                  (-)  -
                   1   1
   arity warning: ------
                     2
                     -
                     0

(35)

Powers

AB3:=(AB1*AB1)*AB1;

arity(AB3)

(36)

test(AB3=AB1*(AB1*AB1))

(37)

test(AB3=AB1^3)

(38)

one? (AB3^0)

(39)

Sums

A12s := A1 + A2; A12s::OutputForm = A1::OutputForm + A2::OutputForm

(40)

arity(A12s)::OutputForm = arity(A1)::OutputForm + arity(A2)::OutputForm

(41)

B12s := B1 + B2; B12s::OutputForm = B1::OutputForm + B2::OutputForm

(42)

arity(B12s)::OutputForm = arity(B1)::OutputForm + arity(B2)::OutputForm

(43)

-B12s

(44)

zero? (A12s - A12s)

(45)

Multiplication

A3s:=(A1+A1)+A1

(46)

arity(A3s)

(47)

test(A3s=A1+(A1+A1))

(48)

test(A3s=A1*3)

(49)

zero? (0*A3s)

(50)

B3s:=(B1+B1)+B1

(51)

arity(B3s)

(52)

test(B3s=B1+(B1+B1))

(53)

test(B3s=B1*3)

(54)

Product

AB11:=A1*B1

(55)

arity(AB11)

(56)

BA11:= B1*A1

(57)

arity(BA11)

(58)

AB := A*B

(59)

arity(AB)

(60)

BA := B*A

(61)

arity(BA)

(62)

WB1:=W*B1

(63)

arity(WB1)

(64)

X2:L:=inp[inp([script(x,[[],[i,j]]) for j in 1..2])$L for i in 1..dim]

(65)

XB21:=X2*B1

(66)

arity(XB21)

(67)

Y2:L:=out[out([script(y,[[i,j]]) for j in 1..2])$L for i in 1..dim]

(68)

XY22:=X2*Y2

(69)

arity(XY22)

(70)

YX22:=Y2*X2

(71)

arity(XY22)

(72)

test((A*A)*A=A*(A*A))

(73)

test((B*B)*B=B*(B*B))

(74)

test((A*B)*A=A*(B*A))

(75)

Permutations and Identities

H:L:=[1,2]

(76)

X:L:=[2,1]

(77)

test(X/X=H)

(78)

Manipulations

tensor AB

(79)

ravel AB

(80)

test(unravel(arity AB,ravel AB)$L=AB)

(81)

map(x+->x+1,AB)$L

(82)

[superscript(a1,[i])=i for i in 1..dim]::List EQ Q

(83)

eval(A,[superscript(a1,[i])=i for i in 1..dim]::List EQ Q)

(84)

Examples

Another kind of diagram:

     Y  =  Y 
    U       U

Algebra

Y:=Σ(Σ(Σ(script(y,[[k],[i,j]])*Dx.k*dx.i*dx.j,j),i),k)

(85)

arity Y

(86)

tensor Y

(87)

Commutator Algebra

Y - [2,1]
  /   Y

(88)

Pairing

U:=inp[inp([script(u,[[],[i,j]]) for j in 1..2])$L for i in 1..2]

(89)

arity U

(90)

3-point function

YU := Y I
  /    U

(91)

YU := Y.I
  /    U

(92)

arity YU

(93)

Oddities (should work on the right)

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.
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

UY := [1].Y
  /      U

(94)

UY := [1] Y
  /      U

(95)

arity UY

(96)

Co-algebra

λ:=inp[out[out([script(y,[[i],[j,k]]) for k in 1..2])$L for j in 1..2] for i in 1..2]

(97)

arity λ

(98)

tensor λ

(99)

Handle

λ
Y

Φ := λ
  /  Y

(100)

arity Φ

(101)

Back to the top.

Comments

Please leave comments and suggestions.

Thanks

Bill Page

  Rep == Record(Dom:VectorSpace, Cod:VectorSpace, t:T)

But VectorSpace? is a category which would make Dom and Cod domains. The domains that currently satisfy VectorSpace? in Axiom are rather limited and seem oddly focused on number theory (finite fields). It is a good thing however that DirectProduct? is a conditional member of this cateogry.

DirectProduct(2,FRAC INT) has VectorSpace(FRAC INT)

(102)

Unfortunately:

DirectProduct(2,DirectProduct(2,FRAC INT)) has VectorSpace(FRAC INT)

(103)

The problem with VectorSpace? is that the domain [Vector]? is not a member of this category instead it satisfies VectoryCategory?.

Is it possible to treat tensors from CartesianTensor? as maps from VectorSpace? to VectorSpace?? We would like for example:

  T:DirectProduct(dim,DirectProduct(dim,FRAC INT)) -> DirectProduct(dim,FRAC INT))

To be a linear operator with two inputs and one output.

FreeModule(Fraction Integer,OrderedVariableList [e1,e1]) has VectorSpace(Fraction Integer)

(104)