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	FriCAS Algebra
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LinearOperator

last edited 2 years ago by Bill Page

Edit detail for LinearOperator revision 5 of 63

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Editor: Bill Page Time: 2011/03/08 16:43:14 GMT-8
Note: documentation and tests

added:
Linear transformations (operators) over n-dimensional cartesian vector spaces oveer a commutative ring $K$. Members of this domain are morphisms $K^n \to K^m$. Products, co-products and composition (grafting) of morphisms is 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.


added:
      ++ incoming vector

added:
      ++ output vector

added:


added:


added:


added:


added:


added:
    ((x:%)^(p:NonNegativeInteger)):% ==
       q:=subtractIfCan(p,1)
       q case NonNegativeInteger => x^q * x
       1$%


added:

    (x:% ** p:NonNegativeInteger):% ==
      q:=subtractIfCan(p,1)
      q case NonNegativeInteger => x**q + x
      0$%


added:


added:


added:


changed:
-    id():% == per [1,1,1]

    id():% == per [1,1,kroneckerDelta()$T]


added:


added:


added:


added:


added:
Construction operators: input and output

changed:
-X1:L:=inp[script(x,[[i]]) for i in 1..2]
-arity X1
-Y1:L:=out[script(y,[[],[i]]) for i in 1..2]
-arity Y1
-Y11:L:=inp[Y1,Y1]
-arity(Y11)
-\end{axiom}
-
-\begin{axiom}
-XX1:=X1*X1
-arity(XX1)
-\end{axiom}
-\begin{axiom}
-YY1:=Y1*Y1
-arity(YY1)
-\end{axiom}
-\begin{axiom}
-Y1X1:=Y1*X1
-arity(Y1X1)
-\end{axiom}
-
-\begin{axiom}
-XY11:=X1 Y1
-arity(XY11)
-YX11:= Y1 X1
-arity(YX11)
A1:L:=inp[script(a,[[1,i]]) for i in 1..2]
arity A1
A2:L:=inp[script(a,[[2,i]]) for i in 1..2]
A:L:=inp[A1,A2]
arity A
B1:L:=out[script(b,[[],[1,i]]) for i in 1..2]
arity B1
B2:L:=out[script(b,[[],[2,i]]) for i in 1..2]
B:L:=out[B1,B2]
arity B
\end{axiom}

Powers
\begin{axiom}
A3p:=(A1*A1)*A1
arity(A3p)
test(A3p=A1*(A1*A1))
test(A3p=A1^3)
\end{axiom}

\begin{axiom}
B3p:=(B1*B1)*B1
arity(B3p)
test(B3p=B1*(B1*B1))
test(B3p=B1^3)
\end{axiom}

Sums
\begin{axiom}
A3s:=(A1+A1)+A1
arity(A3s)
test(A3s=A1*(A1*A1))
test(A3s=A1^3)
\end{axiom}

\begin{axiom}
B3s:=(B1+B1)+B1
arity(B3s)
test(B3s=B1*(B1*B1))
test(B3s=B1^3)
\end{axiom}

Expected error
\begin{axiom}
B1A1:=B1*A1
arity(B1A1)
\end{axiom}

Composition
\begin{axiom}
AB11:=A1 B1
arity(AB11)
BA11:= B1 A1
arity(BA11)
AB := A B
arity(AB)
BA := B A
arity(BA)

changed:
-arity(X2)
-XX21:=X2 X1
-arity(XX21)
XB21:=X2 B1
arity(XB21)

removed:
-arity(Y2)

added:
\end{axiom}

Multiple inputs and outputs
\begin{axiom}

changed:
-WX1:=W X1
-arity(WX1)
-\end{axiom}
WB1:=W B1
arity(WB1)
\end{axiom}

Linear transformations (operators) over n-dimensional cartesian vector spaces oveer a commutative ring . Members of this domain are morphisms $K^n \to K^m$ . Products, co-products and composition (grafting) of morphisms is 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.

spad

)abbrev domain LIN LinearOperator
LinearOperator(dim:NonNegativeInteger,K:CommutativeRing): Join(Ring,BiModule(K,K)) with
    arity: % -> DirectProduct(2,NonNegativeInteger)
    elt: (%,%) -> %
    id: ()->%
    inp: List K -> %
      ++ incoming vector
    inp: List % -> %
    out: List K -> %
      ++ output vector
    out: List % -> %
    coerce: SquareMatrix(dim,K) -> %
  == add
    import List NonNegativeInteger
    T == CartesianTensor(1,dim,K)
    Rep == Record(n:NonNegativeInteger, m:NonNegativeInteger, t:T)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    arity(x:%):DirectProduct(2,NonNegativeInteger) == directProduct [rep(x).n,rep(x).m]

    0 == per [0,0,0]

    (x:% + y:%):% ==
      rep(x).t=0 => per [rep(y).n,rep(y).m,rep(y).t]
      rep(y).t=0 => per [rep(x).n,rep(x).m,rep(x).t]
      rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity"
      per [rep(x).n,rep(x).m,rep(x).t+rep(y).t]

    (x:% - y:%):% ==
      rep(x).t=0 => per [rep(y).n,rep(y).m,-rep(y).t]
      rep(y).t=0 => per [rep(x).n,rep(x).m,rep(x).t]
      rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity"
      per [rep(x).n,rep(x).m,rep(x).t-rep(y).t]

    1 == per [0,0,1]

    (x:% * y:%):% ==
      rep(x).n ~= rep(y).n => error "arity"
      r:=product(rep(x).t, rep(y).t)
      u:=1$DirectProduct(dim,K)::T
      ud:=product(u,kroneckerDelta()$T)
      for i in 1..rep(x).n repeat
        --output("rank",rank(r)::OutputForm)$OutputPackage
        --output("n",rep(x).n::OutputForm)$OutputPackage
        r:=contract(contract(ud,1,r,rep(x).n+1),1,rep(x).n+1)
        --output("rank",rank(r)::OutputForm)$OutputPackage
      per [rep(x).n,rep(x).m+rep(y).m,r]
    ((x:%)^(p:NonNegativeInteger)):% ==
       q:=subtractIfCan(p,1)
       q case NonNegativeInteger => x^q * x
       1$%

    (x:% + y:%):% ==
      rep(x).m ~= rep(y).m  => error "arity"
      r:=product(rep(x).t, rep(y).t)
      u:=1$DirectProduct(dim,K)::T
      du:=product(kroneckerDelta()$T,u)
      for i in 1..rep(y).m repeat
        --output("rank",rank(r)::OutputForm)$OutputPackage
        --output("m",rep(y).m::OutputForm)$OutputPackage
        r:=contract(contract(du,1,r,rep(y).m+1),1,rep(y).m+1)
        --output("rank",rank(r)::OutputForm)$OutputPackage
      per [rep(x).n+rep(y).m,rep(y).m,r]

    (x:% ** p:NonNegativeInteger):% ==
      q:=subtractIfCan(p,1)
      q case NonNegativeInteger => x**q + x
      0$%

    (x:% = y:%):Boolean ==
      rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity"
      rep(x).t = rep(y).t

    (x:K * y:%):% == per [rep(y).n,rep(y).m,x*rep(y).t]

    (x:% * y:K):% == per [rep(x).n,rep(x).m,rep(x).t*y]

    elt(x:%,y:%):% ==
      r:=product(rep(x).t,rep(y).t)
      yn:=rep(y).n  -- outputs of y
      xm:=rep(x).m  -- inputs of x
      while yn>0 and xm>0 repeat
        output("yn",yn::OutputForm)$OutputPackage
        output("xm",xm::OutputForm)$OutputPackage
        output("rank",rank(r)::OutputForm)$OutputPackage
        r:=contract(r,rep(y).n+xm,yn+rep(y).m)
        yn:=subtractIfCan(yn,1)::NonNegativeInteger
        xm:=subtractIfCan(xm,1)::NonNegativeInteger
      per [rep(x).n+xm,yn+rep(y).m,r]

    id():% == per [1,1,kroneckerDelta()$T]

    inp(x:List K):% == per [1,0,entries(x)::T]

    inp(x:List %):% ==
      #removeDuplicates([rep(y).n for y in x])~=1 or
        #removeDuplicates([rep(y).m for y in x])~=1 => error "arity"
      per [rep(first x).n+1,rep(first x).m,[rep(y).t for y in x]::T]$Rep

    out(x:List K):% == per [0,1,entries(x)::T]

    out(x:List %):% ==
      #removeDuplicates([rep(y).n for y in x])~=1 or
        #removeDuplicates([rep(y).m for y in x])~=1 => error "arity"
      per [rep(first x).n,rep(first x).m+1,[rep(y).t for y in x]::T]$Rep

    coerce(x:%):OutputForm == (rep(x).t)::OutputForm

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/74143022208027506-25px001.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(n: NonNegativeInteger,m: NonNegativeInteger,t: CartesianTensor(One,dim,K))
      LIN;rep is replaced by x 
Time: 0.09 SEC.

   compiling local per : Record(n: NonNegativeInteger,m: NonNegativeInteger,t: CartesianTensor(One,dim,K)) -> $
      LIN;per is replaced by x 
Time: 0 SEC.

   compiling exported arity : $ -> DirectProduct(2,NonNegativeInteger)
Time: 0.21 SEC.

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

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

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

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

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

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

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

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

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

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

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

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

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

   compiling exported inp : List K -> $
Time: 0.07 SEC.

   compiling exported inp : List $ -> $
Time: 0.10 SEC.

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

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

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

(time taken in buildFunctor:  10)

;;;     ***       |LinearOperator| REDEFINED

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

   Cumulative Statistics for Constructor LinearOperator
      Time: 0.72 seconds

   finalizing NRLIB LIN 
   Processing LinearOperator for Browser database:
--->-->LinearOperator((arity ((DirectProduct 2 (NonNegativeInteger)) %))): Not documented!!!!
--->-->LinearOperator((elt (% % %))): Not documented!!!!
--->-->LinearOperator((id (%))): Not documented!!!!
--------(inp (% (List K)))---------
--->-->LinearOperator((inp (% (List %)))): Not documented!!!!
--------(out (% (List K)))---------
--->-->LinearOperator((out (% (List %)))): Not documented!!!!
--->-->LinearOperator((coerce (% (SquareMatrix dim K)))): Not documented!!!!
--->-->LinearOperator(constructor): Not documented!!!!
--->-->LinearOperator(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/LIN.NRLIB/LIN.lsp" (written 08 MAR 2011 04:42:58 PM):
; compiling (/VERSIONCHECK 2)
; compiling (PUT (QUOTE |LIN;rep|) ...)
; compiling (DEFUN |LIN;rep| ...)
; compiling (PUT (QUOTE |LIN;per|) ...)
; compiling (DEFUN |LIN;per| ...)
; compiling (DEFUN |LIN;arity;$Dp;3| ...)
; compiling (DEFUN |LIN;Zero;$;4| ...)
; compiling (DEFUN |LIN;+;3$;5| ...)
; compiling (DEFUN |LIN;-;3$;6| ...)
; compiling (DEFUN |LIN;One;$;7| ...)
; compiling (DEFUN |LIN;*;3$;8| ...)
; compiling (DEFUN |LIN;^;$Nni$;9| ...)
; compiling (DEFUN |LIN;+;3$;10| ...)
; compiling (DEFUN |LIN;**| ...)
; compiling (DEFUN |LIN;=;2$B;12| ...)
; compiling (DEFUN |LIN;*;K2$;13| ...)
; compiling (DEFUN |LIN;*;$K$;14| ...)
; compiling (DEFUN |LIN;elt;3$;15| ...)
; compiling (DEFUN |LIN;id;$;16| ...)
; compiling (DEFUN |LIN;inp;L$;17| ...)
; compiling (DEFUN |LIN;inp;L$;18| ...)
; compiling (DEFUN |LIN;out;L$;19| ...)
; compiling (DEFUN |LIN;out;L$;20| ...)
; compiling (DEFUN |LIN;coerce;$Of;21| ...)
; compiling (DEFUN |LinearOperator| ...)
; compiling (DEFUN |LinearOperator;| ...)
; compiling (MAKEPROP (QUOTE |LinearOperator|) ...)

; /var/zope2/var/LatexWiki/LIN.NRLIB/LIN.fasl written
; compilation finished in 0:00:00.672
------------------------------------------------------------------------
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN

Construction operators: input and output

axiom

L:=LIN(2,FRAC POLY INT)

$\label{eq1}\hbox{\axiomType{LinearOperator}\ } (2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))$

(1)

Type: Domain

axiom

A1:L:=inp[script(a,[[1,i]]) for i in 1..2]

$\label{eq2}\left[{a_{1, \: 1}}, \:{a_{1, \: 2}}\right]$

(2)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity A1

$\label{eq3}\left[ 1, \: 0 \right]$

(3)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

A2:L:=inp[script(a,[[2,i]]) for i in 1..2]

$\label{eq4}\left[{a_{2, \: 1}}, \:{a_{2, \: 2}}\right]$

(4)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

A:L:=inp[A1,A2]

$\label{eq5}\left[ \begin{array}{cc} {a_{1, \: 1}}&{a_{1, \: 2}} \ {a_{2, \: 1}}&{a_{2, \: 2}}$

(5)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity A

$\label{eq6}\left[ 2, \: 0 \right]$

(6)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

B1:L:=out[script(b,[[],[1,i]]) for i in 1..2]

$\label{eq7}\left[{b^{1, \: 1}}, \:{b^{1, \: 2}}\right]$

(7)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity B1

$\label{eq8}\left[ 0, \: 1 \right]$

(8)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

B2:L:=out[script(b,[[],[2,i]]) for i in 1..2]

$\label{eq9}\left[{b^{2, \: 1}}, \:{b^{2, \: 2}}\right]$

(9)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

B:L:=out[B1,B2]

$\label{eq10}\left[ \begin{array}{cc} {b^{1, \: 1}}&{b^{1, \: 2}} \ {b^{2, \: 1}}&{b^{2, \: 2}}$

(10)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity B

$\label{eq11}\left[ 0, \: 2 \right]$

(11)

Type: DirectProduct?(2,NonNegativeInteger?)

Powers

axiom

A3p:=(A1*A1)*A1

$\label{eq12}\begin{array}{@{}l} \displaystyle \left[{{4 \ {a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}+{8 \ {{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}+{4 \ {{a_{1, \: 1}}^3}}}, \: \right. \ \ \displaystyle \left.{{4 \ {{a_{1, \: 2}}^3}}+{8 \ {a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}+{4 \ {{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}}\right]$

(12)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(A3p)

$\label{eq13}\left[ 1, \: 0 \right]$

(13)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

test(A3p=A1*(A1*A1))

$\label{eq14} \mbox{\rm true}$

(14)

Type: Boolean

axiom

test(A3p=A1^3)

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

(15)

Type: Boolean

axiom

B3p:=(B1*B1)*B1

$\label{eq16}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {{b^{1, \: 1}}^3}&{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}} \ {{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}&{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}$

(16)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(B3p)

$\label{eq17}\left[ 0, \: 3 \right]$

(17)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

test(B3p=B1*(B1*B1))

$\label{eq18} \mbox{\rm true}$

(18)

Type: Boolean

axiom

test(B3p=B1^3)

$\label{eq19} \mbox{\rm true}$

(19)

Type: Boolean

Sums

axiom

A3s:=(A1+A1)+A1

$\label{eq20}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {{a_{1, \: 1}}^3}&{{{a_{1, \: 1}}^2}\ {a_{1, \: 2}}} \ {{{a_{1, \: 1}}^2}\ {a_{1, \: 2}}}&{{a_{1, \: 1}}\ {{a_{1, \: 2}}^2}}$

(20)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(A3s)

$\label{eq21}\left[ 1, \: 0 \right]$

(21)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

test(A3s=A1*(A1*A1))

$\label{eq22} \mbox{\rm false}$

(22)

Type: Boolean

axiom

test(A3s=A1^3)

$\label{eq23} \mbox{\rm false}$

(23)

Type: Boolean

axiom

B3s:=(B1+B1)+B1

$\label{eq24}\begin{array}{@{}l} \displaystyle \left[{{{b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}+{2 \ {{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}+{{b^{1, \: 1}}^3}}, \: \right. \ \ \displaystyle \left.{{{b^{1, \: 2}}^3}+{2 \ {b^{1, \: 1}}\ {{b^{1, \: 2}}^2}}+{{{b^{1, \: 1}}^2}\ {b^{1, \: 2}}}}\right]$

(24)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(B3s)

$\label{eq25}\left[ 2, \: 1 \right]$

(25)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

test(B3s=B1*(B1*B1))

   >> Error detected within library code:
   arity

Expected error

axiom

B1A1:=B1*A1

   >> Error detected within library code:
   arity

Composition

axiom

AB11:=A1 B1

$\label{eq26}\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{1, \: 1}}\ {a_{1, \: 2}}}&{{b^{1, \: 2}}\ {a_{1, \: 2}}}$

(26)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(AB11)

$\label{eq27}\left[ 1, \: 1 \right]$

(27)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

BA11:= B1 A1

   yn 1
   xm 1
   rank 2

$\label{eq28}{{b^{1, \: 2}}\ {a_{1, \: 2}}}+{{b^{1, \: 1}}\ {a_{1, \: 1}}}$

(28)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(BA11)

$\label{eq29}\left[ 0, \: 0 \right]$

(29)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

AB := A B

$\label{eq30}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {a_{1, \: 1}}}&{{b^{1, \: 2}}\ {a_{1, \: 1}}} \ {{b^{2, \: 1}}\ {a_{1, \: 1}}}&{{b^{2, \: 2}}\ {a_{1, \: 1}}}$

(30)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(AB)

$\label{eq31}\left[ 2, \: 2 \right]$

(31)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

BA := B A

   yn 2
   xm 2
   rank 4
   yn 1
   xm 1
   rank 2

   >> Error detected within library code:
   Improper index for contraction

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]

$\label{eq32}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {w_{1, \: 1}^{1}}&{w_{1, \: 2}^{1}} \ {w_{2, \: 1}^{1}}&{w_{2, \: 2}^{1}}$

(32)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

WB1:=W B1

$\label{eq33}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{b^{1, \: 1}}\ {w_{1, \: 1}^{1}}}&{{b^{1, \: 2}}\ {w_{1, \: 1}^{1}}} \ {{b^{1, \: 1}}\ {w_{1, \: 2}^{1}}}&{{b^{1, \: 2}}\ {w_{1, \: 2}^{1}}}$

(33)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(WB1)

$\label{eq34}\left[ 3, \: 1 \right]$

(34)

Type: DirectProduct?(2,NonNegativeInteger?)