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FriCAS Algebra
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LinearOperator
  
last edited 2 years ago by Bill Page

Edit detail for LinearOperator revision 24 of 63
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Editor: Bill Page
Time: 2011/04/11 15:35:46 GMT-7
Note: bug in permutations and identities 
added:
      p=0 => per [dom f,cod f,0]
      p=1 => f

changed:
-      0
      per [0,0,0]

changed:
-      #p = 1 => return r
      #p = 1 and p.1 = 1 => return r

added:
      p=0 => per [dom f,cod f,1]
      p=1 => f

changed:
-      1
      per [0,0,1]

Introduction

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

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

      n = 3     inputs

      \  |  /         \  /   /         \              |
       \ | /           \/   /           \            / \
        \|/             \  /           / \          /   \
        / \              \/           /   \         \   /
       /   \              \          /   / \         \ /
      /     \              \        /   /   \         |

      m = 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)

spad
)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
    coerce:(x:List PositiveInteger) -> %
      ++ identity for composition and permutations of its products
    coerce:(x:List None) -> %
      ++ [] = 1
  add
    (f:% / g:%):% == apply(g,f)
spad
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/5155383802378491972-25px001.spad using 
      old system compiler.
   MONAL abbreviates category Monoidal 
------------------------------------------------------------------------
   initializing NRLIB MONAL for Monoidal 
   compiling into NRLIB MONAL 

;;;     ***       |Monoidal| REDEFINED
Time: 0.03 SEC.

   MONAL- abbreviates domain Monoidal& 
------------------------------------------------------------------------
   initializing NRLIB MONAL- for Monoidal& 
   compiling into NRLIB MONAL- 
   compiling exported / : (S,S) -> S
Time: 0.18 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |Monoidal&| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor Monoidal&
      Time: 0.19 seconds

   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 (% % %))---------
--------(coerce (% (List (PositiveInteger))))---------
--->-->Monoidal&((coerce (% (List (PositiveInteger))))): Improper first word in comments: identity
"identity for composition and permutations of its products"
--------(coerce (% (List (None))))---------
--->-->Monoidal&((coerce (% (List (None))))): Improper first word in comments: []
"[] = 1"
--->-->Monoidal&(constructor): Not documented!!!!
--->-->Monoidal&(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/MONAL-.NRLIB/MONAL-.lsp" (written 11 APR 2011 03:35:20 PM):

; /var/zope2/var/LatexWiki/MONAL-.NRLIB/MONAL-.fasl written
; compilation finished in 0:00:00.065
------------------------------------------------------------------------
   Monoidal& is now explicitly exposed in frame initial 
   Monoidal& will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MONAL-.NRLIB/MONAL-
   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 (% % %))---------
--------(coerce (% (List (PositiveInteger))))---------
--->-->Monoidal((coerce (% (List (PositiveInteger))))): Improper first word in comments: identity
"identity for composition and permutations of its products"
--------(coerce (% (List (None))))---------
--->-->Monoidal((coerce (% (List (None))))): Improper first word in comments: []
"[] = 1"
--->-->Monoidal(constructor): Not documented!!!!
--->-->Monoidal(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL.lsp" (written 11 APR 2011 03:35:20 PM):

; /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL.fasl written
; compilation finished in 0:00:00.023
------------------------------------------------------------------------
   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).

spad
)abbrev domain PROP Prop
Prop(R:CommutativeRing): Join(Monoidal R, CoercibleTo OutputForm) with
    _/:(R,R) -> %
  == 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(g:%, f:%):% ==
      cod(f) ~= dom(g) => error "arity"
      per [dom f,cod g]
    (f:% * g:%):% == per [dom(f)+dom(g),cod(f)+cod(g)]
    -- preserves arity
    (f:% + g:%):% ==
      dom(f)~=dom(g) or cod(g) ~= cod(g) => error "arity"
      f
    coerce(p:List PositiveInteger):% == per [#p::R,#p::R]
    coerce(p:List None):% == per [0,0]
spad
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/785743822053201310-25px002.spad using 
      old system compiler.
   PROP abbreviates domain Prop 
------------------------------------------------------------------------
   initializing NRLIB PROP for Prop 
   compiling into NRLIB PROP 
   compiling local rep : $ -> Record(domain: R,codomain: R)
      PROP;rep is replaced by x 
Time: 0.04 SEC.

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

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

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

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

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

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

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

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

   compiling exported coerce : List PositiveInteger -> $
Time: 0.06 SEC.

   compiling exported coerce : List None -> $
Time: 0 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |Prop| REDEFINED

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

   Cumulative Statistics for Constructor Prop
      Time: 0.14 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 11 APR 2011 03:35:21 PM):

; /var/zope2/var/LatexWiki/PROP.NRLIB/PROP.fasl written
; compilation finished in 0:00:00.121
------------------------------------------------------------------------
   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

spad
)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, CoercibleTo Prop NNI) with
    inp: List K -> %
      ++ incoming vector
    inp: List % -> %
    out: List K -> %
      ++ output vector
    out: List % -> %
    coerce: K -> %
    arity: % -> Prop(NNI)
    apply: (%,%) -> %
    basisVectors: () -> List %
    basisForms: () -> List %
    tensor: % -> T
    map: (K->K,%) -> %
    ravel: % -> List K
    unravel: (Prop NNI,List K) -> %

  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 NNI == dom(f)/cod(f)
    coerce(f:%):Prop NNI == arity f

    -- 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))$T]
    ravel(g:%):List K == ravel dat g
    unravel(p:Prop NNI,r:List K):% ==
      dim^(dom(p)+cod(p)) ~= #r => error "failed"
      per [dom(p),cod(p),unravel(r)$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:%):% ==
      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=0 => per [dom f,cod f,0]
      p=1 => f
      q:=subtractIfCan(p,1)
      q case NNI => q*f + f
      per [0,0,0]

    -- evaluation:
    --
    -- f/g : A^n -> A^p = f:A^n -> A^m / g:A^m -> A^p
    -- g f : A^n -> A^p = g:A^m -> A^p * f:A^n -> A^m
    --
    apply(g:%,f:%):% ==
      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=0 => per [dom f,cod f,1]
      p=1 => f
      cod(f) ~= dom(f) => error "arity"
      q:=subtractIfCan(p,1)
      q case NNI => f^q * f
      per [0,0,1]

    -- 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)
spad
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/9130945086552056883-25px003.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: 0.05 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: 0.01 SEC.

   compiling exported arity : $ -> Prop NonNegativeInteger
Time: 0 SEC.

   compiling exported coerce : $ -> Prop NonNegativeInteger
Time: 0 SEC.

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

   compiling exported basisForms : () -> List $
Time: 0.15 SEC.

   compiling exported map : (K -> K,$) -> $
Time: 0.05 SEC.

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

   compiling exported unravel : (Prop NonNegativeInteger,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 SEC.

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

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

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

   compiling exported coerce : List PositiveInteger -> $
Time: 0.04 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: 0.03 SEC.

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

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

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

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

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

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

   compiling exported inp : List $ -> $
Time: 0.05 SEC.

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

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

   compiling exported coerce : $ -> OutputForm
Time: 0.16 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |LinearOperator| REDEFINED

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

   Warnings: 
      [1] not known that (CommutativeRing) is of mode (CATEGORY domain (SIGNATURE quo ($ $ $)) (SIGNATURE rem ($ $ $)) (SIGNATURE gcd ($ $ $)) (SIGNATURE divide ((Record (: quotient $) (: remainder $)) $ $)) (SIGNATURE exquo ((Union $ failed) $ $)) (SIGNATURE shift ($ $ (Integer))) (SIGNATURE random ($ $)))
      [2] arity: not known that (CommutativeRing) is of mode (CATEGORY domain (SIGNATURE quo ($ $ $)) (SIGNATURE rem ($ $ $)) (SIGNATURE gcd ($ $ $)) (SIGNATURE divide ((Record (: quotient $) (: remainder $)) $ $)) (SIGNATURE exquo ((Union $ failed) $ $)) (SIGNATURE shift ($ $ (Integer))) (SIGNATURE random ($ $)))
      [3] coerce:  bases has no value

   Cumulative Statistics for Constructor LinearOperator
      Time: 0.70 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((coerce (% K))): Not documented!!!!
--->-->LinearOperator((arity ((Prop NNI) %))): Not documented!!!!
--->-->LinearOperator((apply (% % %))): 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 NNI) (List K)))): Not documented!!!!
--->-->LinearOperator(constructor): Not documented!!!!
--->-->LinearOperator(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/LIN.NRLIB/LIN.lsp" (written 11 APR 2011 03:35:23 PM):

; /var/zope2/var/LatexWiki/LIN.NRLIB/LIN.fasl written
; compilation finished in 0:00:01.091
------------------------------------------------------------------------
   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

axiom
dim:=2
\label{eq1}2(1)
Type: PositiveInteger?
axiom
L:=LinearOperator(dim,OVAR [x,y],FRAC POLY INT)
\label{eq2}\hbox{\axiomType{LinearOperator}\ } (2, \hbox{\axiomType{OrderedVariableList}\ } ([ x , y ]) , \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))(2)
Type: Type
axiom
Dx:=basisVectors()$L
\label{eq3}\left[{e_{\ }^{x}}, \:{e_{\ }^{y}}\right](3)
Type: List(LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer))))
axiom
dx:=basisForms()$L
\label{eq4}\left[{e_{x}}, \:{e_{y}}\right](4)
Type: List(LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer))))
axiom
matrix [[dx.i * Dx.j for j in 1..dim] for i in 1..dim]
\label{eq5}\left[ \begin{array}{cc} {e_{x}^{x}}&{e_{x}^{y}} \ {e_{y}^{x}}&{e_{y}^{y}} (5)
Type: Matrix(LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer))))
axiom
matrix [[Dx.i / dx.j for j in 1..dim] for i in 1..dim]
\label{eq6}\left[ \begin{array}{cc} 1 & 0 \ 0 & 1 (6)
Type: Matrix(LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer))))

Conveniences

axiom
macro Σ(x,b,i)==reduce(+,[x*b.i for i in 1..dim])
Type: Void
axiom
macro /\(x,s)==superscript(x,s)
Type: Void
axiom
macro \/(x,s)==subscript(x,s)
Type: Void

Construction

axiom
A1:L := Σ(a1\/[i],dx,i)
\label{eq7}{{a 1_{1}}\ {e_{x}}}+{{a 1_{2}}\ {e_{y}}}(7)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity A1
\label{eq8}1 \over 0(8)
Type: Prop(NonNegativeInteger?)
axiom
A2:L := Σ(a2\/[i],dx,i)
\label{eq9}{{a 2_{1}}\ {e_{x}}}+{{a 2_{2}}\ {e_{y}}}(9)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
A:L := inp[A1,A2]
\label{eq10}{{a 1_{1}}\ {e_{xx}}}+{{a 1_{2}}\ {e_{xy}}}+{{a 2_{1}}\ {e_{yx}}}+{{a 2_{2}}\ {e_{yy}}}(10)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity A
\label{eq11}2 \over 0(11)
Type: Prop(NonNegativeInteger?)
axiom
B1:L := Σ(b1/\[i],Dx,i)
\label{eq12}{{b 1^{1}}\ {e_{\ }^{x}}}+{{b 1^{2}}\ {e_{\ }^{y}}}(12)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity B1
\label{eq13}0 \over 1(13)
Type: Prop(NonNegativeInteger?)
axiom
B2:L := Σ(b2/\[i],Dx,i)
\label{eq14}{{b 2^{1}}\ {e_{\ }^{x}}}+{{b 2^{2}}\ {e_{\ }^{y}}}(14)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
B:L := out[B1,B2]
\label{eq15}{{b 1^{1}}\ {e_{\ }^{xx}}}+{{b 1^{2}}\ {e_{\ }^{xy}}}+{{b 2^{1}}\ {e_{\ }^{yx}}}+{{b 2^{2}}\ {e_{\ }^{yy}}}(15)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity B
\label{eq16}0 \over 2(16)
Type: Prop(NonNegativeInteger?)
axiom
B:L := Σ(Σ(b/\[i,j],Dx,i),Dx,j)
\label{eq17}{{b^{1, \: 1}}\ {e_{\ }^{xx}}}+{{b^{1, \: 2}}\ {e_{\ }^{xy}}}+{{b^{2, \: 1}}\ {e_{\ }^{yx}}}+{{b^{2, \: 2}}\ {e_{\ }^{yy}}}(17)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))

Composition

g/f : A^n \to A^{m+p} = f:A^n \to A^m <em> g:A^n \to A^p  
axiom
AB2 := A2 / B2; AB2::OutputForm = A2::OutputForm / B2::OutputForm
\label{eq18}\begin{array}{@{}l} \displaystyle {{{b 2^{1}}\ {a 2_{1}}\ {e_{x}^{x}}}+{{b 2^{2}}\ {a 2_{1}}\ {e_{x}^{y}}}+{{b 2^{1}}\ {a 2_{2}}\ {e_{y}^{x}}}+{{b 2^{2}}\ {a 2_{2}}\ {e_{y}^{y}}}}= \ \ \displaystyle {{{{a 2_{1}}\ {e_{x}}}+{{a 2_{2}}\ {e_{y}}}}\over{{{b 2^{1}}\ {e_{\ }^{x}}}+{{b 2^{2}}\ {e_{\ }^{y}}}}} (18)
Type: Equation(OutputForm?)
axiom
arity(AB2)::OutputForm = arity(A2)::OutputForm / arity(B2)::OutputForm
\label{eq19}{1 \over 1}={{1 \over 0}\over{0 \over 1}}(19)
Type: Equation(OutputForm?)
axiom
BA1 := B1 / A1; BA1::OutputForm = B1::OutputForm / A1::OutputForm
\label{eq20}{{{b 1^{2}}\ {a 1_{2}}}+{{b 1^{1}}\ {a 1_{1}}}}={{{{b 1^{1}}\ {e_{\ }^{x}}}+{{b 1^{2}}\ {e_{\ }^{y}}}}\over{{{a 1_{1}}\ {e_{x}}}+{{a 1_{2}}\ {e_{y}}}}}(20)
Type: Equation(OutputForm?)
axiom
arity(BA1)::OutputForm = arity(B1)::OutputForm / arity(A1)::OutputForm
\label{eq21}{0 \over 0}={{0 \over 1}\over{1 \over 0}}(21)
Type: Equation(OutputForm?)
axiom
AB1 := A1 / B1; AB1::OutputForm = A1::OutputForm / B1::OutputForm
\label{eq22}\begin{array}{@{}l} \displaystyle {{{b 1^{1}}\ {a 1_{1}}\ {e_{x}^{x}}}+{{b 1^{2}}\ {a 1_{1}}\ {e_{x}^{y}}}+{{b 1^{1}}\ {a 1_{2}}\ {e_{y}^{x}}}+{{b 1^{2}}\ {a 1_{2}}\ {e_{y}^{y}}}}= \ \ \displaystyle {{{{a 1_{1}}\ {e_{x}}}+{{a 1_{2}}\ {e_{y}}}}\over{{{b 1^{1}}\ {e_{\ }^{x}}}+{{b 1^{2}}\ {e_{\ }^{y}}}}} (22)
Type: Equation(OutputForm?)
axiom
arity(AB1)::OutputForm = arity(A1)::OutputForm / arity(B1)::OutputForm
\label{eq23}{1 \over 1}={{1 \over 0}\over{0 \over 1}}(23)
Type: Equation(OutputForm?)

Powers

axiom
AB3:=(AB1*AB1)*AB1;
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(AB3)
\label{eq24}3 \over 3(24)
Type: Prop(NonNegativeInteger?)
axiom
test(AB3=AB1*(AB1*AB1))
\label{eq25} \mbox{\rm true} (25)
Type: Boolean
axiom
test(AB3=AB1^3)
\label{eq26} \mbox{\rm true} (26)
Type: Boolean

Sums

f+g : A^n \to A^m = f:A^n \to A^m + g:A^n \to A^m  
axiom
A12s := A1 + A2; A12s::OutputForm = A1::OutputForm + A2::OutputForm
\label{eq27}\begin{array}{@{}l} \displaystyle {{{\left({a 2_{1}}+{a 1_{1}}\right)}\ {e_{x}}}+{{\left({a 2_{2}}+{a 1_{2}}\right)}\ {e_{y}}}}= \ \ \displaystyle {{{a 1_{1}}\ {e_{x}}}+{{a 1_{2}}\ {e_{y}}}+{{a 2_{1}}\ {e_{x}}}+{{a 2_{2}}\ {e_{y}}}} (27)
Type: Equation(OutputForm?)
axiom
arity(A12s)::OutputForm = arity(A1)::OutputForm + arity(A2)::OutputForm
\label{eq28}{1 \over 0}={{1 \over 0}+{1 \over 0}}(28)
Type: Equation(OutputForm?)
axiom
B12s := B1 + B2; B12s::OutputForm = B1::OutputForm + B2::OutputForm
\label{eq29}\begin{array}{@{}l} \displaystyle {{{\left({b 2^{1}}+{b 1^{1}}\right)}\ {e_{\ }^{x}}}+{{\left({b 2^{2}}+{b 1^{2}}\right)}\ {e_{\ }^{y}}}}= \ \ \displaystyle {{{b 1^{1}}\ {e_{\ }^{x}}}+{{b 1^{2}}\ {e_{\ }^{y}}}+{{b 2^{1}}\ {e_{\ }^{x}}}+{{b 2^{2}}\ {e_{\ }^{y}}}} (29)
Type: Equation(OutputForm?)
axiom
arity(B12s)::OutputForm = arity(B1)::OutputForm + arity(B2)::OutputForm
\label{eq30}{0 \over 1}={{0 \over 1}+{0 \over 1}}(30)
Type: Equation(OutputForm?)

Multiplication

axiom
A3s:=(A1+A1)+A1
\label{eq31}{3 \ {a 1_{1}}\ {e_{x}}}+{3 \ {a 1_{2}}\ {e_{y}}}(31)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(A3s)
\label{eq32}1 \over 0(32)
Type: Prop(NonNegativeInteger?)
axiom
test(A3s=A1+(A1+A1))
\label{eq33} \mbox{\rm true} (33)
Type: Boolean
axiom
test(A3s=A1*3)
\label{eq34} \mbox{\rm true} (34)
Type: Boolean

axiom
B3s:=(B1+B1)+B1
\label{eq35}{3 \ {b 1^{1}}\ {e_{\ }^{x}}}+{3 \ {b 1^{2}}\ {e_{\ }^{y}}}(35)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(B3s)
\label{eq36}0 \over 1(36)
Type: Prop(NonNegativeInteger?)
axiom
test(B3s=B1+(B1+B1))
\label{eq37} \mbox{\rm true} (37)
Type: Boolean
axiom
test(B3s=B1*3)
\label{eq38} \mbox{\rm true} (38)
Type: Boolean

Product

axiom
AB11:=A1*B1
\label{eq39}{{b 1^{1}}\ {a 1_{1}}\ {e_{x}^{x}}}+{{b 1^{2}}\ {a 1_{1}}\ {e_{x}^{y}}}+{{b 1^{1}}\ {a 1_{2}}\ {e_{y}^{x}}}+{{b 1^{2}}\ {a 1_{2}}\ {e_{y}^{y}}}(39)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(AB11)
\label{eq40}1 \over 1(40)
Type: Prop(NonNegativeInteger?)
axiom
BA11:= B1*A1
\label{eq41}{{b 1^{1}}\ {a 1_{1}}\ {e_{x}^{x}}}+{{b 1^{2}}\ {a 1_{1}}\ {e_{x}^{y}}}+{{b 1^{1}}\ {a 1_{2}}\ {e_{y}^{x}}}+{{b 1^{2}}\ {a 1_{2}}\ {e_{y}^{y}}}(41)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(BA11)
\label{eq42}1 \over 1(42)
Type: Prop(NonNegativeInteger?)
axiom
AB := A*B
\label{eq43}\begin{array}{@{}l} \displaystyle {{b^{1, \: 1}}\ {a 1_{1}}\ {e_{xx}^{xx}}}+{{b^{1, \: 2}}\ {a 1_{1}}\ {e_{xx}^{xy}}}+{{b^{2, \: 1}}\ {a 1_{1}}\ {e_{xx}^{yx}}}+ \ \ \displaystyle {{b^{2, \: 2}}\ {a 1_{1}}\ {e_{xx}^{yy}}}+{{b^{1, \: 1}}\ {a 1_{2}}\ {e_{xy}^{xx}}}+{{b^{1, \: 2}}\ {a 1_{2}}\ {e_{xy}^{xy}}}+ \ \ \displaystyle {{b^{2, \: 1}}\ {a 1_{2}}\ {e_{xy}^{yx}}}+{{b^{2, \: 2}}\ {a 1_{2}}\ {e_{xy}^{yy}}}+{{b^{1, \: 1}}\ {a 2_{1}}\ {e_{yx}^{xx}}}+ \ \ \displaystyle {{b^{1, \: 2}}\ {a 2_{1}}\ {e_{yx}^{xy}}}+{{b^{2, \: 1}}\ {a 2_{1}}\ {e_{yx}^{yx}}}+{{b^{2, \: 2}}\ {a 2_{1}}\ {e_{yx}^{yy}}}+ \ \ \displaystyle {{b^{1, \: 1}}\ {a 2_{2}}\ {e_{yy}^{xx}}}+{{b^{1, \: 2}}\ {a 2_{2}}\ {e_{yy}^{xy}}}+{{b^{2, \: 1}}\ {a 2_{2}}\ {e_{yy}^{yx}}}+ \ \ \displaystyle {{b^{2, \: 2}}\ {a 2_{2}}\ {e_{yy}^{yy}}} (43)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(AB)
\label{eq44}2 \over 2(44)
Type: Prop(NonNegativeInteger?)
axiom
BA := B*A
\label{eq45}\begin{array}{@{}l} \displaystyle {{b^{1, \: 1}}\ {a 1_{1}}\ {e_{xx}^{xx}}}+{{b^{1, \: 2}}\ {a 1_{1}}\ {e_{xx}^{xy}}}+{{b^{2, \: 1}}\ {a 1_{1}}\ {e_{xx}^{yx}}}+ \ \ \displaystyle {{b^{2, \: 2}}\ {a 1_{1}}\ {e_{xx}^{yy}}}+{{b^{1, \: 1}}\ {a 1_{2}}\ {e_{xy}^{xx}}}+{{b^{1, \: 2}}\ {a 1_{2}}\ {e_{xy}^{xy}}}+ \ \ \displaystyle {{b^{2, \: 1}}\ {a 1_{2}}\ {e_{xy}^{yx}}}+{{b^{2, \: 2}}\ {a 1_{2}}\ {e_{xy}^{yy}}}+{{b^{1, \: 1}}\ {a 2_{1}}\ {e_{yx}^{xx}}}+ \ \ \displaystyle {{b^{1, \: 2}}\ {a 2_{1}}\ {e_{yx}^{xy}}}+{{b^{2, \: 1}}\ {a 2_{1}}\ {e_{yx}^{yx}}}+{{b^{2, \: 2}}\ {a 2_{1}}\ {e_{yx}^{yy}}}+ \ \ \displaystyle {{b^{1, \: 1}}\ {a 2_{2}}\ {e_{yy}^{xx}}}+{{b^{1, \: 2}}\ {a 2_{2}}\ {e_{yy}^{xy}}}+{{b^{2, \: 1}}\ {a 2_{2}}\ {e_{yy}^{yx}}}+ \ \ \displaystyle {{b^{2, \: 2}}\ {a 2_{2}}\ {e_{yy}^{yy}}} (45)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(BA)
\label{eq46}2 \over 2(46)
Type: Prop(NonNegativeInteger?)
axiom
X2:L:=inp[inp([script(x,[[i,j]]) for j in 1..2])$L for i in 1..2]
\label{eq47}{{x_{1, \: 1}}\ {e_{xx}}}+{{x_{1, \: 2}}\ {e_{xy}}}+{{x_{2, \: 1}}\ {e_{yx}}}+{{x_{2, \: 2}}\ {e_{yy}}}(47)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
XB21:=X2*B1
\label{eq48}\begin{array}{@{}l} \displaystyle {{b 1^{1}}\ {x_{1, \: 1}}\ {e_{xx}^{x}}}+{{b 1^{2}}\ {x_{1, \: 1}}\ {e_{xx}^{y}}}+{{b 1^{1}}\ {x_{1, \: 2}}\ {e_{xy}^{x}}}+ \ \ \displaystyle {{b 1^{2}}\ {x_{1, \: 2}}\ {e_{xy}^{y}}}+{{b 1^{1}}\ {x_{2, \: 1}}\ {e_{yx}^{x}}}+{{b 1^{2}}\ {x_{2, \: 1}}\ {e_{yx}^{y}}}+ \ \ \displaystyle {{b 1^{1}}\ {x_{2, \: 2}}\ {e_{yy}^{x}}}+{{b 1^{2}}\ {x_{2, \: 2}}\ {e_{yy}^{y}}} (48)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(XB21)
\label{eq49}2 \over 1(49)
Type: Prop(NonNegativeInteger?)
axiom
Y2:L:=out[out([script(y,[[],[i,j]]) for j in 1..2])$L for i in 1..2]
\label{eq50}{{y^{1, \: 1}}\ {e_{\ }^{xx}}}+{{y^{1, \: 2}}\ {e_{\ }^{xy}}}+{{y^{2, \: 1}}\ {e_{\ }^{yx}}}+{{y^{2, \: 2}}\ {e_{\ }^{yy}}}(50)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
XY22:=X2*Y2
\label{eq51}\begin{array}{@{}l} \displaystyle {{y^{1, \: 1}}\ {x_{1, \: 1}}\ {e_{xx}^{xx}}}+{{y^{1, \: 2}}\ {x_{1, \: 1}}\ {e_{xx}^{xy}}}+{{y^{2, \: 1}}\ {x_{1, \: 1}}\ {e_{xx}^{yx}}}+ \ \ \displaystyle {{y^{2, \: 2}}\ {x_{1, \: 1}}\ {e_{xx}^{yy}}}+{{y^{1, \: 1}}\ {x_{1, \: 2}}\ {e_{xy}^{xx}}}+{{y^{1, \: 2}}\ {x_{1, \: 2}}\ {e_{xy}^{xy}}}+ \ \ \displaystyle {{y^{2, \: 1}}\ {x_{1, \: 2}}\ {e_{xy}^{yx}}}+{{y^{2, \: 2}}\ {x_{1, \: 2}}\ {e_{xy}^{yy}}}+{{y^{1, \: 1}}\ {x_{2, \: 1}}\ {e_{yx}^{xx}}}+ \ \ \displaystyle {{y^{1, \: 2}}\ {x_{2, \: 1}}\ {e_{yx}^{xy}}}+{{y^{2, \: 1}}\ {x_{2, \: 1}}\ {e_{yx}^{yx}}}+{{y^{2, \: 2}}\ {x_{2, \: 1}}\ {e_{yx}^{yy}}}+ \ \ \displaystyle {{y^{1, \: 1}}\ {x_{2, \: 2}}\ {e_{yy}^{xx}}}+{{y^{1, \: 2}}\ {x_{2, \: 2}}\ {e_{yy}^{xy}}}+{{y^{2, \: 1}}\ {x_{2, \: 2}}\ {e_{yy}^{yx}}}+ \ \ \displaystyle {{y^{2, \: 2}}\ {x_{2, \: 2}}\ {e_{yy}^{yy}}} (51)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(XY22)
\label{eq52}2 \over 2(52)
Type: Prop(NonNegativeInteger?)
axiom
YX22:=Y2*X2
\label{eq53}\begin{array}{@{}l} \displaystyle {{y^{1, \: 1}}\ {x_{1, \: 1}}\ {e_{xx}^{xx}}}+{{y^{1, \: 2}}\ {x_{1, \: 1}}\ {e_{xx}^{xy}}}+{{y^{2, \: 1}}\ {x_{1, \: 1}}\ {e_{xx}^{yx}}}+ \ \ \displaystyle {{y^{2, \: 2}}\ {x_{1, \: 1}}\ {e_{xx}^{yy}}}+{{y^{1, \: 1}}\ {x_{1, \: 2}}\ {e_{xy}^{xx}}}+{{y^{1, \: 2}}\ {x_{1, \: 2}}\ {e_{xy}^{xy}}}+ \ \ \displaystyle {{y^{2, \: 1}}\ {x_{1, \: 2}}\ {e_{xy}^{yx}}}+{{y^{2, \: 2}}\ {x_{1, \: 2}}\ {e_{xy}^{yy}}}+{{y^{1, \: 1}}\ {x_{2, \: 1}}\ {e_{yx}^{xx}}}+ \ \ \displaystyle {{y^{1, \: 2}}\ {x_{2, \: 1}}\ {e_{yx}^{xy}}}+{{y^{2, \: 1}}\ {x_{2, \: 1}}\ {e_{yx}^{yx}}}+{{y^{2, \: 2}}\ {x_{2, \: 1}}\ {e_{yx}^{yy}}}+ \ \ \displaystyle {{y^{1, \: 1}}\ {x_{2, \: 2}}\ {e_{yy}^{xx}}}+{{y^{1, \: 2}}\ {x_{2, \: 2}}\ {e_{yy}^{xy}}}+{{y^{2, \: 1}}\ {x_{2, \: 2}}\ {e_{yy}^{yx}}}+ \ \ \displaystyle {{y^{2, \: 2}}\ {x_{2, \: 2}}\ {e_{yy}^{yy}}} (53)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(XY22)
\label{eq54}2 \over 2(54)
Type: Prop(NonNegativeInteger?)
axiom
test((A*A)*A=A*(A*A))
\label{eq55} \mbox{\rm true} (55)
Type: Boolean
axiom
test((B*B)*B=B*(B*B))
\label{eq56} \mbox{\rm true} (56)
Type: Boolean
axiom
test((A*B)*A=A*(B*A))
\label{eq57} \mbox{\rm true} (57)
Type: Boolean

Multiple inputs and outputs

axiom
W:L:=out[inp[inp([script(w,[[i,j],[k]]) for j in 1..2])$L for i in 1..2] for k in 1..2]
\label{eq58}\begin{array}{@{}l} \displaystyle {{w_{1, \: 1}^{1}}\ {e_{xx}^{x}}}+{{w_{1, \: 2}^{1}}\ {e_{xx}^{y}}}+{{w_{2, \: 1}^{1}}\ {e_{xy}^{x}}}+{{w_{2, \: 2}^{1}}\ {e_{xy}^{y}}}+ \ \ \displaystyle {{w_{1, \: 1}^{2}}\ {e_{yx}^{x}}}+{{w_{1, \: 2}^{2}}\ {e_{yx}^{y}}}+{{w_{2, \: 1}^{2}}\ {e_{yy}^{x}}}+{{w_{2, \: 2}^{2}}\ {e_{yy}^{y}}} (58)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
WB1:=W*B1
\label{eq59}\begin{array}{@{}l} \displaystyle {{b 1^{1}}\ {w_{1, \: 1}^{1}}\ {e_{xx}^{xx}}}+{{b 1^{2}}\ {w_{1, \: 1}^{1}}\ {e_{xx}^{xy}}}+{{b 1^{1}}\ {w_{1, \: 2}^{1}}\ {e_{xx}^{yx}}}+ \ \ \displaystyle {{b 1^{2}}\ {w_{1, \: 2}^{1}}\ {e_{xx}^{yy}}}+{{b 1^{1}}\ {w_{2, \: 1}^{1}}\ {e_{xy}^{xx}}}+{{b 1^{2}}\ {w_{2, \: 1}^{1}}\ {e_{xy}^{xy}}}+ \ \ \displaystyle {{b 1^{1}}\ {w_{2, \: 2}^{1}}\ {e_{xy}^{yx}}}+{{b 1^{2}}\ {w_{2, \: 2}^{1}}\ {e_{xy}^{yy}}}+{{b 1^{1}}\ {w_{1, \: 1}^{2}}\ {e_{yx}^{xx}}}+ \ \ \displaystyle {{b 1^{2}}\ {w_{1, \: 1}^{2}}\ {e_{yx}^{xy}}}+{{b 1^{1}}\ {w_{1, \: 2}^{2}}\ {e_{yx}^{yx}}}+{{b 1^{2}}\ {w_{1, \: 2}^{2}}\ {e_{yx}^{yy}}}+ \ \ \displaystyle {{b 1^{1}}\ {w_{2, \: 1}^{2}}\ {e_{yy}^{xx}}}+{{b 1^{2}}\ {w_{2, \: 1}^{2}}\ {e_{yy}^{xy}}}+{{b 1^{1}}\ {w_{2, \: 2}^{2}}\ {e_{yy}^{yx}}}+ \ \ \displaystyle {{b 1^{2}}\ {w_{2, \: 2}^{2}}\ {e_{yy}^{yy}}} (59)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity(WB1)
\label{eq60}2 \over 2(60)
Type: Prop(NonNegativeInteger?)

Permutations and Identities

axiom
H:L:=[1,2]
\label{eq61}{e_{xx}^{xx}}+{e_{xy}^{xy}}+{e_{yx}^{yx}}+{e_{yy}^{yy}}(61)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
X:L:=[2,1]
\label{eq62}{e_{xx}^{xx}}+{e_{xy}^{yx}}+{e_{yx}^{xy}}+{e_{yy}^{yy}}(62)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
test(X/X=H)
\label{eq63} \mbox{\rm true} (63)
Type: Boolean

Tensor

axiom
tensor AB
\label{eq64}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a 1_{1}}}&{{b^{1, \: 2}}\ {a 1_{1}}} \ {{b^{2, \: 1}}\ {a 1_{1}}}&{{b^{2, \: 2}}\ {a 1_{1}}} (64)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

Another kind of diagram: 

  ___ _       _     
  ___ _)  =   _\_ _ 
  _/          ___ _)

axiom
Y:=out[inp[inp([script(y,[[i,j],[k]]) for k in 1..2])$L for j in 1..2] for i in 1..2]
\label{eq65}\begin{array}{@{}l} \displaystyle {{y_{1, \: 1}^{1}}\ {e_{xx}^{x}}}+{{y_{1, \: 1}^{2}}\ {e_{xx}^{y}}}+{{y_{1, \: 2}^{1}}\ {e_{xy}^{x}}}+{{y_{1, \: 2}^{2}}\ {e_{xy}^{y}}}+ \ \ \displaystyle {{y_{2, \: 1}^{1}}\ {e_{yx}^{x}}}+{{y_{2, \: 1}^{2}}\ {e_{yx}^{y}}}+{{y_{2, \: 2}^{1}}\ {e_{yy}^{x}}}+{{y_{2, \: 2}^{2}}\ {e_{yy}^{y}}} (65)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity Y
\label{eq66}2 \over 1(66)
Type: Prop(NonNegativeInteger?)
axiom
tensor Y
\label{eq67}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {y_{1, \: 1}^{1}}&{y_{1, \: 1}^{2}} \ {y_{1, \: 2}^{1}}&{y_{1, \: 2}^{2}} (67)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))
axiom
U:=inp[inp([script(u,[[i,j],[]]) for j in 1..2])$L for i in 1..2]
\label{eq68}{{u_{1, \: 1}}\ {e_{xx}}}+{{u_{1, \: 2}}\ {e_{xy}}}+{{u_{2, \: 1}}\ {e_{yx}}}+{{u_{2, \: 2}}\ {e_{yy}}}(68)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity U
\label{eq69}2 \over 0(69)
Type: Prop(NonNegativeInteger?)
axiom
YU:=(Y*[1])/U
\label{eq70}\begin{array}{@{}l} \displaystyle {{\left({{y_{1, \: 1}^{2}}\ {u_{2, \: 1}}}+{{y_{1, \: 1}^{1}}\ {u_{1, \: 1}}}\right)}\ {e_{{xx}x}}}+ \ \ \displaystyle {{\left({{y_{1, \: 1}^{2}}\ {u_{2, \: 2}}}+{{y_{1, \: 1}^{1}}\ {u_{1, \: 2}}}\right)}\ {e_{{xx}y}}}+ \ \ \displaystyle {{\left({{y_{1, \: 2}^{2}}\ {u_{2, \: 1}}}+{{y_{1, \: 2}^{1}}\ {u_{1, \: 1}}}\right)}\ {e_{{xy}x}}}+ \ \ \displaystyle {{\left({{y_{1, \: 2}^{2}}\ {u_{2, \: 2}}}+{{y_{1, \: 2}^{1}}\ {u_{1, \: 2}}}\right)}\ {e_{{xy}y}}}+ \ \ \displaystyle {{\left({{y_{2, \: 1}^{2}}\ {u_{2, \: 1}}}+{{y_{2, \: 1}^{1}}\ {u_{1, \: 1}}}\right)}\ {e_{{yx}x}}}+ \ \ \displaystyle {{\left({{y_{2, \: 1}^{2}}\ {u_{2, \: 2}}}+{{y_{2, \: 1}^{1}}\ {u_{1, \: 2}}}\right)}\ {e_{{yx}y}}}+ \ \ \displaystyle {{\left({{y_{2, \: 2}^{2}}\ {u_{2, \: 1}}}+{{y_{2, \: 2}^{1}}\ {u_{1, \: 1}}}\right)}\ {e_{{yy}x}}}+ \ \ \displaystyle {{\left({{y_{2, \: 2}^{2}}\ {u_{2, \: 2}}}+{{y_{2, \: 2}^{1}}\ {u_{1, \: 2}}}\right)}\ {e_{{yy}y}}} (70)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity YU
\label{eq71}3 \over 0(71)
Type: Prop(NonNegativeInteger?)
axiom
UY:=([1]*Y)/U
\label{eq72}\begin{array}{@{}l} \displaystyle {{\left({{y_{1, \: 1}^{2}}\ {u_{1, \: 2}}}+{{y_{1, \: 1}^{1}}\ {u_{1, \: 1}}}\right)}\ {e_{{xx}x}}}+ \ \ \displaystyle {{\left({{y_{1, \: 2}^{2}}\ {u_{1, \: 2}}}+{{y_{1, \: 2}^{1}}\ {u_{1, \: 1}}}\right)}\ {e_{{xx}y}}}+ \ \ \displaystyle {{\left({{y_{2, \: 1}^{2}}\ {u_{1, \: 2}}}+{{y_{2, \: 1}^{1}}\ {u_{1, \: 1}}}\right)}\ {e_{{xy}x}}}+ \ \ \displaystyle {{\left({{y_{2, \: 2}^{2}}\ {u_{1, \: 2}}}+{{y_{2, \: 2}^{1}}\ {u_{1, \: 1}}}\right)}\ {e_{{xy}y}}}+ \ \ \displaystyle {{\left({{y_{1, \: 1}^{2}}\ {u_{2, \: 2}}}+{{y_{1, \: 1}^{1}}\ {u_{2, \: 1}}}\right)}\ {e_{{yx}x}}}+ \ \ \displaystyle {{\left({{y_{1, \: 2}^{2}}\ {u_{2, \: 2}}}+{{y_{1, \: 2}^{1}}\ {u_{2, \: 1}}}\right)}\ {e_{{yx}y}}}+ \ \ \displaystyle {{\left({{y_{2, \: 1}^{2}}\ {u_{2, \: 2}}}+{{y_{2, \: 1}^{1}}\ {u_{2, \: 1}}}\right)}\ {e_{{yy}x}}}+ \ \ \displaystyle {{\left({{y_{2, \: 2}^{2}}\ {u_{2, \: 2}}}+{{y_{2, \: 2}^{1}}\ {u_{2, \: 1}}}\right)}\ {e_{{yy}y}}} (72)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity UY
\label{eq73}3 \over 0(73)
Type: Prop(NonNegativeInteger?)

Handle

          
 _/ _\__

axiom
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]
\label{eq74}\begin{array}{@{}l} \displaystyle {{y_{1}^{1, \: 1}}\ {e_{x}^{xx}}}+{{y_{1}^{1, \: 2}}\ {e_{x}^{xy}}}+{{y_{1}^{2, \: 1}}\ {e_{x}^{yx}}}+{{y_{1}^{2, \: 2}}\ {e_{x}^{yy}}}+ \ \ \displaystyle {{y_{2}^{1, \: 1}}\ {e_{y}^{xx}}}+{{y_{2}^{1, \: 2}}\ {e_{y}^{xy}}}+{{y_{2}^{2, \: 1}}\ {e_{y}^{yx}}}+{{y_{2}^{2, \: 2}}\ {e_{y}^{yy}}} (74)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity λ
\label{eq75}1 \over 2(75)
Type: Prop(NonNegativeInteger?)
axiom
tensor λ
\label{eq76}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {y_{1}^{1, \: 1}}&{y_{1}^{1, \: 2}} \ {y_{1}^{2, \: 1}}&{y_{1}^{2, \: 2}} (76)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))
axiom
Φ := λ/Y
\label{eq77}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{y_{1}^{2, \: 2}}\ {y_{2, \: 2}^{1}}}+{{y_{1}^{2, \: 1}}\ {y_{2, \: 1}^{1}}}+{{y_{1}^{1, \: 2}}\ {y_{1, \: 2}^{1}}}+ \ \ \displaystyle {{y_{1}^{1, \: 1}}\ {y_{1, \: 1}^{1}}} (77)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))
axiom
arity Φ
\label{eq78}1 \over 1(78)
Type: Prop(NonNegativeInteger?)

Manipulations

axiom
ravel YU
\label{eq79}\begin{array}{@{}l} \displaystyle \left[{{{y_{1, \: 1}^{2}}\ {u_{2, \: 1}}}+{{y_{1, \: 1}^{1}}\ {u_{1, \: 1}}}}, \:{{{y_{1, \: 1}^{2}}\ {u_{2, \: 2}}}+{{y_{1, \: 1}^{1}}\ {u_{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1, \: 2}^{2}}\ {u_{2, \: 1}}}+{{y_{1, \: 2}^{1}}\ {u_{1, \: 1}}}}, \:{{{y_{1, \: 2}^{2}}\ {u_{2, \: 2}}}+{{y_{1, \: 2}^{1}}\ {u_{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{2, \: 1}^{2}}\ {u_{2, \: 1}}}+{{y_{2, \: 1}^{1}}\ {u_{1, \: 1}}}}, \:{{{y_{2, \: 1}^{2}}\ {u_{2, \: 2}}}+{{y_{2, \: 1}^{1}}\ {u_{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{2, \: 2}^{2}}\ {u_{2, \: 1}}}+{{y_{2, \: 2}^{1}}\ {u_{1, \: 1}}}}, \:{{{y_{2, \: 2}^{2}}\ {u_{2, \: 2}}}+{{y_{2, \: 2}^{1}}\ {u_{1, \: 2}}}}\right] (79)
Type: List(Fraction(Polynomial(Integer)))
axiom
test(unravel(arity YU,ravel YU)$L=YU)
\label{eq80} \mbox{\rm true} (80)
Type: Boolean
axiom
map(x+->x+1,YU)$L
\label{eq81}\begin{array}{@{}l} \displaystyle {{\left({{y_{1, \: 1}^{2}}\ {u_{2, \: 1}}}+{{y_{1, \: 1}^{1}}\ {u_{1, \: 1}}}+ 1 \right)}\ {e_{{xx}x}}}+ \ \ \displaystyle {{\left({{y_{1, \: 1}^{2}}\ {u_{2, \: 2}}}+{{y_{1, \: 1}^{1}}\ {u_{1, \: 2}}}+ 1 \right)}\ {e_{{xx}y}}}+ \ \ \displaystyle {{\left({{y_{1, \: 2}^{2}}\ {u_{2, \: 1}}}+{{y_{1, \: 2}^{1}}\ {u_{1, \: 1}}}+ 1 \right)}\ {e_{{xy}x}}}+ \ \ \displaystyle {{\left({{y_{1, \: 2}^{2}}\ {u_{2, \: 2}}}+{{y_{1, \: 2}^{1}}\ {u_{1, \: 2}}}+ 1 \right)}\ {e_{{xy}y}}}+ \ \ \displaystyle {{\left({{y_{2, \: 1}^{2}}\ {u_{2, \: 1}}}+{{y_{2, \: 1}^{1}}\ {u_{1, \: 1}}}+ 1 \right)}\ {e_{{yx}x}}}+ \ \ \displaystyle {{\left({{y_{2, \: 1}^{2}}\ {u_{2, \: 2}}}+{{y_{2, \: 1}^{1}}\ {u_{1, \: 2}}}+ 1 \right)}\ {e_{{yx}y}}}+ \ \ \displaystyle {{\left({{y_{2, \: 2}^{2}}\ {u_{2, \: 1}}}+{{y_{2, \: 2}^{1}}\ {u_{1, \: 1}}}+ 1 \right)}\ {e_{{yy}x}}}+ \ \ \displaystyle {{\left({{y_{2, \: 2}^{2}}\ {u_{2, \: 2}}}+{{y_{2, \: 2}^{1}}\ {u_{1, \: 2}}}+ 1 \right)}\ {e_{{yy}y}}} (81)
Type: LinearOperator?(2,OrderedVariableList?([x,y]),Fraction(Polynomial(Integer)))

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Comments

Please leave comments and suggestions.

Thanks

Bill Page

Linear operators as morphisms --Bill Page, Thu, 10 Mar 2011 09:42:33 -0800 reply
If linear operators really are to be morphisms (in the sense of category theory) then they must have a domain and a co-domain that are vector spaces, not just an in-degree and out-degree. E.g.: 
  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.

axiom
DirectProduct(2,FRAC INT) has VectorSpace(FRAC INT)
\label{eq82} \mbox{\rm true} (82)
Type: Boolean

Unfortunately:

axiom
DirectProduct(2,DirectProduct(2,FRAC INT)) has VectorSpace(FRAC INT)
\label{eq83} \mbox{\rm false} (83)
Type: Boolean

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.

Modules --Bill Page, Thu, 10 Mar 2011 16:03:13 -0800 reply
A [FreeModule]? over a [Field]? is a VectorSpace? unfortunately this is not currently understood by Axiom:
axiom
FreeModule(Fraction Integer,OrderedVariableList [e1,e1]) has VectorSpace(Fraction Integer)
\label{eq84} \mbox{\rm true} (84)
Type: Boolean