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	FriCAS Algebra
FreeModule ←	↑	→ OrderedVariableList

LinearOperator

last edited 1 year ago by Bill Page

Edit detail for LinearOperator revision 28 of 63

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Editor: Bill Page Time: 2011/04/11 21:07:04 GMT-7
Note: Prop missing One

added:
    1:% = per [0,0]

Introduction

Bi-graded linear operators (transformations) over n-dimensional vector spaces on a commutative ring . 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/8030329591618487444-25px001.spad using 
      old system compiler.
   MONAL abbreviates category Monoidal 
------------------------------------------------------------------------
   initializing NRLIB MONAL for Monoidal 
   compiling into NRLIB MONAL 

;;;     ***       |Monoidal| REDEFINED
Time: 0.01 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 (% % %))---------
--------(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 09:06:47 PM):

; /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL.fasl written
; compilation finished in 0:00:00.043
------------------------------------------------------------------------
   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:%):% == f/g
    (f:% / g:%):% ==
      cod(f) ~= dom(g) => error "arity"
      per [dom f,cod g]
    (f:% * g:%):% == per [dom(f)+dom(g),cod(f)+cod(g)]
    1:% = per [0,0]
    -- 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/5074464407714417206-25px002.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: 0.03 SEC.

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

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

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

   Semantic Errors: 
      [1] compColonInside: colon inside expressions is unsupported

****** comp fails at level 2 with expression: ******
(= | << | (|:| 1 $) | >> | (|per| (|construct| 0 0)))
****** level 2  ******
$x:= (: (One) $)
$m:= $EmptyMode
$f:=
((((|per| #)) ((|per| #) (|rep| #))
  ((|rep| #) (|copy| #) (|setelt| #) (|codomain| #) ...)))

   >> Apparent user error:
   cannot compile (= (: (One) $) (per (construct (Zero) (Zero))))

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, RetractableTo K) with
    inp: List K -> %
      ++ incoming vector
    inp: List % -> %
    out: List K -> %
      ++ output vector
    out: List % -> %
    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)

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

    -- 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:%):% == f/g
    --apply(g:%,f:%):% ==
    (f:% / g:%):% ==
      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)

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/8009333309120175973-25px003.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: 0.01 SEC.

   compiling exported arity : $ -> Prop NonNegativeInteger
****** comp fails at level 1 with expression: ******
error in function arity 

((/ (|dom| |f|) (|cod| |f|)))
****** level 1  ******
$x:= (/ (dom f) (cod f))
$m:= (Prop (NonNegativeInteger))
$f:=
((((|f| # #) (|dat| #)) ((|dat| #) (* #) (+ #) (- #) ...) ((|per| #) (|rep| #))
  ((|rep| #) (|copy| #) (|setelt| #) (|data| #) ...)))

   >> Apparent user error:
   cannot compile (/ (dom f) (cod f))

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)

   LinearOperator is an unknown constructor and so is unavailable. Did 
      you mean to use -> but type something different instead?
Dx:=basisVectors()$L                    

   The function basisVectors is not implemented in NIL .
dx:=basisForms()$L                      

   The function basisForms is not implemented in NIL .
matrix [[dx.i * Dx.j for j in 1..dim] for i in 1..dim]

   There are no library operations named dx 
      Use HyperDoc Browse or issue
                                 )what op dx
      to learn if there is any operation containing " dx " in its name.
   Cannot find a definition or applicable library operation named dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named dx 
      Use HyperDoc Browse or issue
                                 )what op dx
      to learn if there is any operation containing " dx " in its name.

   Cannot find a definition or applicable library operation named dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
matrix [[Dx.i / dx.j for j in 1..dim] for i in 1..dim]

   There are no library operations named Dx 
      Use HyperDoc Browse or issue
                                 )what op Dx
      to learn if there is any operation containing " Dx " in its name.
   Cannot find a definition or applicable library operation named Dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named Dx 
      Use HyperDoc Browse or issue
                                 )what op Dx
      to learn if there is any operation containing " Dx " in its name.

   Cannot find a definition or applicable library operation named Dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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)

   There are no library operations named dx 
      Use HyperDoc Browse or issue
                                 )what op dx
      to learn if there is any operation containing " dx " in its name.
   Cannot find a definition or applicable library operation named dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named dx 
      Use HyperDoc Browse or issue
                                 )what op dx
      to learn if there is any operation containing " dx " in its name.

   Cannot find a definition or applicable library operation named dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
arity A1

   There are 1 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(A1)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
A2:L := Σ(a2\/[i],dx,i)

   There are no library operations named dx 
      Use HyperDoc Browse or issue
                                 )what op dx
      to learn if there is any operation containing " dx " in its name.
   Cannot find a definition or applicable library operation named dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named dx 
      Use HyperDoc Browse or issue
                                 )what op dx
      to learn if there is any operation containing " dx " in its name.

   Cannot find a definition or applicable library operation named dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
A:L := inp[A1,A2]

$\label{eq2}inp_{A 1, \: A 2}$

(2)

Type: Symbol

axiom

arity A

   There are 1 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) 
                                   Symbol

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
B1:L := Σ(b1/\[i],Dx,i)

   There are no library operations named Dx 
      Use HyperDoc Browse or issue
                                 )what op Dx
      to learn if there is any operation containing " Dx " in its name.
   Cannot find a definition or applicable library operation named Dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named Dx 
      Use HyperDoc Browse or issue
                                 )what op Dx
      to learn if there is any operation containing " Dx " in its name.

   Cannot find a definition or applicable library operation named Dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
arity B1

   There are 1 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(B1)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
B2:L := Σ(b2/\[i],Dx,i)

   There are no library operations named Dx 
      Use HyperDoc Browse or issue
                                 )what op Dx
      to learn if there is any operation containing " Dx " in its name.
   Cannot find a definition or applicable library operation named Dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named Dx 
      Use HyperDoc Browse or issue
                                 )what op Dx
      to learn if there is any operation containing " Dx " in its name.

   Cannot find a definition or applicable library operation named Dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
B:L := out[B1,B2]

$\label{eq3}out_{B 1, \: B 2}$

(3)

Type: Symbol

axiom

arity B

   There are 1 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) 
                                   Symbol

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
B:L := Σ(Σ(b/\[i,j],Dx,i),Dx,j)

   There are no library operations named Dx 
      Use HyperDoc Browse or issue
                                 )what op Dx
      to learn if there is any operation containing " Dx " in its name.
   Cannot find a definition or applicable library operation named Dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named Dx 
      Use HyperDoc Browse or issue
                                 )what op Dx
      to learn if there is any operation containing " Dx " in its name.

   Cannot find a definition or applicable library operation named Dx 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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

$\label{eq4}{A 2 \over B 2}={A 2 \over B 2}$

(4)

Type: Equation(OutputForm?)

axiom

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

   There are 1 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) 
                        Fraction(Polynomial(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
BA1 := B1 / A1; BA1::OutputForm = B1::OutputForm / A1::OutputForm

$\label{eq5}{B 1 \over A 1}={B 1 \over A 1}$

(5)

Type: Equation(OutputForm?)

axiom

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

   There are 1 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) 
                        Fraction(Polynomial(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
AB1 := A1 / B1; AB1::OutputForm = A1::OutputForm / B1::OutputForm

$\label{eq6}{A 1 \over B 1}={A 1 \over B 1}$

(6)

Type: Equation(OutputForm?)

axiom

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

   There are 1 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) 
                        Fraction(Polynomial(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Powers

axiom

AB3:=(AB1*AB1)*AB1;

Type: Fraction(Polynomial(Integer))

axiom

arity(AB3)

   There are 1 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) 
                        Fraction(Polynomial(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
test(AB3=AB1*(AB1*AB1))

$\label{eq7} \mbox{\rm true}$

(7)

Type: Boolean

axiom

test(AB3=AB1^3)

$\label{eq8} \mbox{\rm true}$

(8)

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

$\label{eq9}{A 2 + A 1}={A 1 + A 2}$

(9)

Type: Equation(OutputForm?)

axiom

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

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
B12s := B1 + B2; B12s::OutputForm = B1::OutputForm + B2::OutputForm

$\label{eq10}{B 2 + B 1}={B 1 + B 2}$

(10)

Type: Equation(OutputForm?)

axiom

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

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Multiplication

axiom

A3s:=(A1+A1)+A1

$\label{eq11}3 \ A 1$

(11)

Type: Polynomial(Integer)

axiom

arity(A3s)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
test(A3s=A1+(A1+A1))

$\label{eq12} \mbox{\rm true}$

(12)

Type: Boolean

axiom

test(A3s=A1*3)

$\label{eq13} \mbox{\rm true}$

(13)

Type: Boolean

axiom

B3s:=(B1+B1)+B1

$\label{eq14}3 \ B 1$

(14)

Type: Polynomial(Integer)

axiom

arity(B3s)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
test(B3s=B1+(B1+B1))

$\label{eq15} \mbox{\rm true}$

(15)

Type: Boolean

axiom

test(B3s=B1*3)

$\label{eq16} \mbox{\rm true}$

(16)

Type: Boolean

Product

axiom

AB11:=A1*B1

$\label{eq17}A 1 \ B 1$

(17)

Type: Polynomial(Integer)

axiom

arity(AB11)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
BA11:= B1*A1

$\label{eq18}A 1 \ B 1$

(18)

Type: Polynomial(Integer)

axiom

arity(BA11)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
AB := A*B

$\label{eq19}{inp_{A 1, \: A 2}}\ {out_{B 1, \: B 2}}$

(19)

Type: Polynomial(Integer)

axiom

arity(AB)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
BA := B*A

$\label{eq20}{inp_{A 1, \: A 2}}\ {out_{B 1, \: B 2}}$

(20)

Type: Polynomial(Integer)

axiom

arity(BA)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
X2:L:=inp[inp([script(x,[[i,j]]) for j in 1..2])$L for i in 1..2]

   The function inp is not implemented in NIL .
XB21:=X2*B1

$\label{eq21}B 1 \ X 2$

(21)

Type: Polynomial(Integer)

axiom

arity(XB21)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
Y2:L:=out[out([script(y,[[],[i,j]]) for j in 1..2])$L for i in 1..2]

   The function out is not implemented in NIL .
XY22:=X2*Y2

$\label{eq22}X 2 \ Y 2$

(22)

Type: Polynomial(Integer)

axiom

arity(XY22)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
YX22:=Y2*X2

$\label{eq23}X 2 \ Y 2$

(23)

Type: Polynomial(Integer)

axiom

arity(XY22)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
test((A*A)*A=A*(A*A))

$\label{eq24} \mbox{\rm true}$

(24)

Type: Boolean

axiom

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

$\label{eq25} \mbox{\rm true}$

(25)

Type: Boolean

axiom

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

$\label{eq26} \mbox{\rm true}$

(26)

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]

   The function inp is not implemented in NIL .
WB1:=W*B1

$\label{eq27}B 1 \ W$

(27)

Type: Polynomial(Integer)

axiom

arity(WB1)

   There are 1 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) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Permutations and Identities

axiom

H:L:=[1,2]

$\label{eq28}\left[ 1, \: 2 \right]$

(28)

Type: List(PositiveInteger?)

axiom

X:L:=[2,1]

$\label{eq29}\left[ 2, \: 1 \right]$

(29)

Type: List(PositiveInteger?)

axiom

test(X/X=H)

   There are 13 exposed and 12 unexposed library operations named / 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op /
      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 / 
      with argument type(s) 
                            List(PositiveInteger)
                            List(PositiveInteger)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Tensor

axiom

tensor AB

   There are 1 exposed and 1 unexposed library operations named tensor 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                             )display op tensor
      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 
      tensor with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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]

   The function inp is not implemented in NIL .
arity Y

   There are 1 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(Y)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
tensor Y

   There are 1 exposed and 1 unexposed library operations named tensor 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                             )display op tensor
      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 
      tensor with argument type(s) 
                                 Variable(Y)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
U:=inp[inp([script(u,[[i,j],[]]) for j in 1..2])$L for i in 1..2]

   The function inp is not implemented in NIL .
arity U

   There are 1 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(U)

      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 35 exposed and 24 unexposed library operations named * 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op *
      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 * 
      with argument type(s) 
                                 Variable(Y)
                            List(PositiveInteger)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
arity YU

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

   There are 35 exposed and 24 unexposed library operations named * 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op *
      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 * 
      with argument type(s) 
                            List(PositiveInteger)
                                 Variable(Y)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
arity UY

   There are 1 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(UY)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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]

   The function out is not implemented in NIL .
arity λ

   There are 1 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(λ)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
tensor λ

   There are 1 exposed and 1 unexposed library operations named tensor 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                             )display op tensor
      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 
      tensor with argument type(s) 
                                 Variable(λ)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
Φ := λ/Y

$\label{eq30}�� \over Y$

(30)

Type: Fraction(Polynomial(Integer))

axiom

arity Φ

   There are 1 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) 
                        Fraction(Polynomial(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Manipulations

axiom

ravel YU

   There are 1 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)

   The function unravel is not implemented in NIL .
map(x+->x+1,YU)$L

   The function map is not implemented in NIL .

Back to the top.

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)

$\label{eq31} \mbox{\rm true}$

(31)

Type: Boolean

Unfortunately:

axiom

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

$\label{eq32} \mbox{\rm false}$

(32)

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)

$\label{eq33} \mbox{\rm true}$

(33)

Type: Boolean