Edit detail for LinearOperator revision 31 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 = 2     outputs

Lines (edges) in the graph represent vectors, nodes represent operators. Horizontal juxtaposition represents product. Vertical juxtaposition represents 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 == Monoid with
    dom: % -> R
      ++ domain
    cod: % -> R
      ++ co-domain
    _/: (%,%) -> %
      ++ vertical composition f/g = g*f
    apply:(%,%) -> %
      ++ horizontal composition
--  add
--    (f:% / g:%):% == apply(g,f)

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/2643402771659714163-25px001.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} = \\spad{g*f}"
--------(apply (% % %))---------
--->-->Monoidal(constructor): Not documented!!!!
--->-->Monoidal(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL.lsp" (written 12 APR 2011 12:42:26 PM):

; /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL.fasl written
; compilation finished in 0:00:00.271
------------------------------------------------------------------------
   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).

)abbrev domain PROP Prop
Prop(): Exports == Implementation where
  R ==> NonNegativeInteger

  Exports ==> Join(Monoidal R, CoercibleTo OutputForm) with
    _/:(R,R) -> %

  Implementation ==> add
    Rep == Record(domain:R,codomain:R)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    dom(f:%):R == rep(f).domain
    cod(f:%):R == rep(f).codomain

    coerce(f:%):OutputForm == dom(f)::OutputForm / cod(f)::OutputForm
    (f:R / g:R):% == per [f,g]
    apply(gg:%,ff:%):% ==
      -- partial application from the right
      n:=subtractIfCan(dom ff,cod gg)
      f:=ff; n case R => f:=1^n*ff
      m:=subtractIfCan(cod gg, dom ff)
      g:=gg; m case R => g:=1^m*gg
      f/g
    (ff:% / gg:%):% ==
      -- partial application from the left
      n:=subtractIfCan(dom ff,cod gg)
      f:=ff; n case R => f:=ff*1^n
      m:=subtractIfCan(cod gg, dom ff)
      g:=gg; m case R => g:=gg*1^m
      --dom(g) ~= cod(f) => error "arity"
      per [dom f,cod g]
    1:% == per [0,0]
    (f:% * g:%):% == per [dom(f)+dom(g),cod(f)+cod(g)]
    0:% == per [0,0]
    -- preserves arity
    (f:% + g:%):% ==
      dom(f)~=dom(g) or cod(g) ~= cod(g) => error "arity"
      f

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/3365105071098393884-25px002.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: 0.03 SEC.

   compiling local per : Record(domain: NonNegativeInteger,codomain: NonNegativeInteger) -> $
      PROP;per is replaced by x 
Time: 0.01 SEC.

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

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

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

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

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

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

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

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

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

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

(time taken in buildFunctor:  10)

;;;     ***       |Prop| REDEFINED

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

   Cumulative Statistics for Constructor Prop
      Time: 0.11 seconds

   finalizing NRLIB PROP 
   Processing Prop for Browser database:
--->-->Prop((/ (% R R))): Not documented!!!!
--->-->Prop(constructor): Not documented!!!!
--->-->Prop(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/PROP.NRLIB/PROP.lsp" (written 12 APR 2011 12:42:27 PM):

; /var/zope2/var/LatexWiki/PROP.NRLIB/PROP.fasl written
; compilation finished in 0:00:00.229
------------------------------------------------------------------------
   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?. It is the free algebra over Prop with n-generators.

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
LinearOperator(dim:NNI,gen:OrderedFinite,K:CommutativeRing): Exports == Implementation where
  NNI ==> NonNegativeInteger
  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,%) -> %
    ravel: % -> List K
    unravel: (Prop,List K) -> %
    coerce:(x:List PositiveInteger) -> %
      ++ identity for composition and permutations of its products
    coerce:(x:List None) -> %
      ++ [] = 1

  Implementation ==> add
    import List NNI
    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 == dom(f)/cod(f)

    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]

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

    0 == per [0,0,0]

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

    -- evaluation:
    --
    -- g f : A^n -> A^p = g:A^m -> A^p * f:A^n -> A^m
    apply(gg:%,ff:%):% ==
      -- partial application from the right
      n:=subtractIfCan(dom ff,cod gg)
      f:=ff; n case NNI => f:=1^n*ff
      m:=subtractIfCan(cod gg, dom ff)
      g:=gg; m case NNI => g:=1^m*gg
      f/g

    -- f/g : A^n -> A^p = f:A^n -> A^m / g:A^m -> A^p
    (ff:% / gg:%):% ==
      -- partial application from the left
      n:=subtractIfCan(dom ff,cod gg)
      f:=ff; n case NNI => f:=ff*1^n
      m:=subtractIfCan(cod gg, dom ff)
      g:=gg; m case NNI => g:=gg*1^m
      --dom(g) ~= cod(f) => error "arity"
      r:T := product(dat f, dat g)
      g1:Integer:=dom(f)+1
      f1:Integer:=dom(f)+cod(f)+1
      for i in 0..cod(f)-1 repeat
        r := contract(r,g1,f1-i)
      per [dom(f),cod(g),r]

    -- permutations and identities
    coerce(p:List PositiveInteger):% ==
      r:=per([1,1,kroneckerDelta()$T])^#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]
    1:% == per [0,0,1]
    coerce(x:K):% == 1*x

    -- product
    (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
      per [dom f,cod f,1]

    -- inherited from Ring
    (x:% = y:%):Boolean ==
      dom(x) ~= dom(y) or cod(x) ~= cod(y) => error "arity"
      dat(x) = dat(y)
    (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 [[hconcat(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 [[hconcat(outs.k,gens.j) for j in 1..dim] for k in 1..#outs]
      -- term list
      bases:List OutputForm:=[]
      coeffs:=ravel dat(x)
      if #inps > 0 and #outs > 0 then
        expon:List List OutputForm:=concat([[[i,j] for j in outs] for i in inps])
        bases:=[scripts('e::OutputForm,ij) for ij in expon]
      else if #inps > 0 then
        bases:=[sub('e::OutputForm,i) for i in inps]
      else if #outs > 0 then
        bases:=[super('e::OutputForm,j) for j in outs]
      terms:=[(k=1 => base;k::OutputForm*base) for base in bases for k in coeffs | k~=0]
      empty? terms => return 0::OutputForm
      return reduce(_+,terms)

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/2820722638285457770-25px003.spad using 
      old system compiler.
   LIN abbreviates domain LinearOperator 
------------------------------------------------------------------------
   initializing NRLIB LIN for LinearOperator 
   compiling into NRLIB LIN 
   importing List NonNegativeInteger
   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 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 SEC.

   compiling exported retract : $ -> K
Time: 0.01 SEC.

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

   compiling exported basisForms : () -> List $
Time: 0.02 SEC.

   compiling exported map : (K -> K,$) -> $
Time: 0.12 SEC.

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

   compiling exported unravel : (Prop,List K) -> $
Time: 0.05 SEC.

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

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

   compiling exported - : ($,$) -> $
Time: 0.02 SEC.

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

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

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

   compiling exported / : ($,$) -> $
Time: 0.02 SEC.

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

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

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

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

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

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

   compiling exported = : ($,$) -> Boolean
Time: 0.01 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: 0.04 SEC.

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

   compiling exported out : List $ -> $
Time: 0.07 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.11 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |LinearOperator| REDEFINED

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

   Warnings: 
      [1] coerce:  bases has no value

   Cumulative Statistics for Constructor LinearOperator
      Time: 0.69 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 (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(constructor): Not documented!!!!
--->-->LinearOperator(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/LIN.NRLIB/LIN.lsp" (written 12 APR 2011 12:42:28 PM):

; /var/zope2/var/LatexWiki/LIN.NRLIB/LIN.fasl written
; compilation finished in 0:00:01.468
------------------------------------------------------------------------
   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.

Basis

dim:=2

(1)

L:=LinearOperator(dim,OVAR [x,y],FRAC POLY INT)

(2)

Dx:=basisVectors()$L

(3)

dx:=basisForms()$L

(4)

matrix [[dx.i * Dx.j for j in 1..dim] for i in 1..dim]

(5)

matrix [[Dx.i / dx.j for j in 1..dim] for i in 1..dim]

(6)

Conveniences

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

macro /\(x,s)==superscript(x,s)

macro \/(x,s)==subscript(x,s)

Construction

A1:L := Σ(a1\/[i],dx,i)

(7)

arity A1

(8)

A2:L := Σ(a2\/[i],dx,i)

(9)

A:L := inp[A1,A2]

(10)

arity A

(11)

B1:L := Σ(b1/\[i],Dx,i)

(12)

arity B1

(13)

B2:L := Σ(b2/\[i],Dx,i)

(14)

B:L := out[B1,B2]

(15)

arity B

(16)

BB:L := Σ(Σ(b/\[i,j],Dx,i),Dx,j)

(17)

W:L := Σ(Σ(Σ(w/\[i,j,k],dx,i),dx,j),Dx,k)

(18)

Composition (evaluation)

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

(19)

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

(20)

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

(21)

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

(22)

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

(23)

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

(24)

Partial Evaluation

BBA1 := B/A1

(25)

BBA2 := B A1

(26)

BBA3 := A1/B

(27)

BBA4 := A1 B

(28)

Powers

AB3:=(AB1*AB1)*AB1;

arity(AB3)

(29)

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

(30)

test(AB3=AB1^3)

(31)

Sums

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

(32)

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

(33)

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

(34)

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

(35)

Multiplication

A3s:=(A1+A1)+A1

(36)

arity(A3s)

(37)

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

(38)

test(A3s=A1*3)

(39)

B3s:=(B1+B1)+B1

(40)

arity(B3s)

(41)

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

(42)

test(B3s=B1*3)

(43)

Product

AB11:=A1*B1

(44)

arity(AB11)

(45)

BA11:= B1*A1

(46)

arity(BA11)

(47)

AB := A*B

(48)

arity(AB)

(49)

BA := B*A

(50)

arity(BA)

(51)

WB1:=W*B1

(52)

arity(WB1)

(53)

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

(54)

XB21:=X2*B1

(55)

arity(XB21)

(56)

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

(57)

XY22:=X2*Y2

(58)

arity(XY22)

(59)

YX22:=Y2*X2

(60)

arity(XY22)

(61)

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

(62)

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

(63)

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

(64)

Permutations and Identities

H:L:=[1,2]

(65)

X:L:=[2,1]

(66)

test(X/X=H)

(67)

Manipulations

tensor AB

(68)

ravel YU

   There are 2 exposed and 0 unexposed library operations named ravel 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op ravel
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named ravel
      with argument type(s) 
                                Variable(YU)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
test(unravel(arity YU,ravel YU)$L=YU)

   There are 2 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named arity
      with argument type(s) 
                                Variable(YU)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
map(x+->x+1,YU)$L

(69)

Examples

Another kind of diagram::

__ _ __ ) = \ / __ )

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

(70)

arity Y

(71)

tensor Y

(72)

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

(73)

arity U

(74)

YU:=(Y*[1])/U

(75)

arity YU

(76)

UY:=([1]*Y)/U

(77)

arity UY

(78)

Handle

          
 _/ _\__ 

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]

(79)

arity λ

(80)

tensor λ

(81)

Φ := λ/Y

(82)

arity Φ

(83)

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)

(84)

Unfortunately:

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

(85)

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)

(86)