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SandBoxGeom1

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Editor: Kurt Pagani Time: 2014/11/23 02:23:48 GMT+0
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changed:
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\begin{spad}
)abbrev domain RSPACE RealSpace
++ Author: Kurt Pagani
++ Date Created: Thu Nov 06 03:51:56 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations: 
++ Related Domains: 
++ Also See:
++ AMS Classifications:
++ Keywords: 
++ Examples:
++ References:
++
++ Description:
++ 
++
RealSpace(n) : Exports == Implementation where
  
  NNI ==> NonNegativeInteger
  PI  ==> PositiveInteger
  I   ==> Integer
  R   ==> Expression Integer -- Real
  n:NNI
  
  Exports == Join(VectorSpace R , CoercibleTo OutputForm) with
    
    coerce : % -> OutputForm
    coerce : List R -> %
    coerce : Vector R -> %
    coerce : List Symbol -> %
    coerce : R -> %
    coerce : % -> List R
    coerce : % -> Vector R
      
    dot : (%,%) -> R
    dim : % -> NNI
    
    "+" : (%,%) -> %
    "-" : (%,%) -> %
    "*" : (R,%) -> %
    "*" : (NNI,%) -> %
    "*" : (PI,%)  -> %
    "*" : (I,%)   -> %
    "/" : (%,R)   -> %
    
    "#" : % -> NNI
    "-" : % -> %
    "=" : (%,%) -> Boolean
    "~=": (%,%) -> Boolean
    
    unitVector : PI -> %
    elt : (%,I) -> R
    
    eval: (%, Equation R) -> %
    eval: (%, List Equation R) -> % 
    
    every?: (R -> Boolean, %) -> Boolean
    map: (R -> R, %) -> %  -- strange, to check
    member?: (R, %) -> Boolean
    any?: (R -> Boolean, %) -> Boolean
    copy: % -> %
    D: (%, Symbol) -> %
    D: (%, Symbol, NonNegativeInteger) -> %
    D: (%, List Symbol) -> %
    D: (%, List Symbol, List NonNegativeInteger) -> %
    
    dist : (%,%) -> R
    norm : % -> R
    tprod : (%,%) -> Matrix(R)   -- tensor prod as Matrix Real
    -- _*_* : (%,%) -> RealSpace(n^2) 
    cross : List(%) -> %

   
   
  Implementation == DirectProduct(n,R) add
    
    Rep := DirectProduct(n,R)
    
    coerce(l:List R):% == 
      #l=n => directProduct((vector l)::Vector(R))$Rep
      
    coerce(l:Vector R):% == 
      #l=n => directProduct(l)$Rep 
      
    coerce(l:List Symbol):% ==
      #l=n => coerce([r::R for r in l])
        
    coerce(x:R):% == x*unitVector(1)$Rep
    
    coerce(v:%):List R == [v(j) for j in 1..#v]
   
    coerce(v:%):Vector R == vector [v(j) for j in 1..#v]
    
    dim(x:%):NNI == #(x::List R) -- n
    dist(x:%,y:%):R == sqrt dot(x-y,x-y)
    norm(x:%):R == sqrt dot(x,x)
    
    tprod(x:%,y:%):Matrix(R) == 
      matrix [[x.i*(y.j)::R for j in 1..n] for i in 1..n]
      
    --wprod(x:%,y:%):Matrix(R) == (tprod(x,y)-tprod(y,x))/2::R
    
    --_*_*(x:%,y:%):RealSpace(n^2) == 
    --  X:List Real:=[]
    --  for i in 1..n repeat
    --    X:=append(X,[x.i*(y.j)::Real for j in 1..n])  
    --  X::RealSpace(n^2)
    
    -- local ; better coerce in ??? 
    vec2list(v:Vector(R)):List(R) == [v.j for j in 1..#v]
    
    cross(X:List %):% ==
      #X ~= n-1 => error "Wrong number of arguments."
      L:List(List R):=[r::List(R) for r in X]
      M:Matrix(R):= transpose matrix L
      S:List(List R):=[vec2list(row(M,i)) for i in 1..n]
      LS:List(Matrix R):=[matrix([S.i for i in 1..n | i~=j]) for j in 1..n]
      r:List R:=[(-1)^(i-1)::R*determinant(LS.i) for i in 1..n]
      r::Rep
           
)abbrev package INTVAL Interval
++ Author: kfp
++ Date Created: Thu Nov 06 02:46:09 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations: 
++ Related Domains: 
++ Also See:
++ AMS Classifications:
++ Keywords: 
++ Examples:
++ References:
++
++ Description:
++ 
++
Interval(F:Field) : Exports == Implementation where

  OF  ==> OutputForm
  PI  ==> PositiveInteger
  NNI ==> NonNegativeInteger
  Q   ==> Fraction Integer
  
  Exports ==  Join(CoercibleTo OutputForm) with
    
    coerce : Tuple F -> %
    coerce : List F -> %
    coerce : % -> List F
    coerce : % -> OutputForm
    
    closedInterval    : (F,F) -> %
    openInterval      : (F,F) -> %
    leftOpenInterval  : (F,F) -> %
    rightOpenInterval : (F,F) -> %
    
    lb : % -> F
    ub : % -> F
    topology : % -> OutputForm 
    measure  : % -> F
    
    --"*" : (%,%) -> Cell(F,2)
    "*" : (F,%) -> %
    "+" : (F,%) -> %
	
  Implementation == add 
    
    Rep:= Record(lb:F,ub:F,tp:String)
    
    --check(x:%):% == 
      --if number?(x.lb) and number?(x.ub) then
        --(x.ub)::Q < (x.lb)::Q => error "a>b"
      --x
 
    coerce(l:Tuple F):% == -- does not work ????
      length l ~=2 => error "Form expected: (a,b)"
      [select(l,0), select(l,1),"open"]::Rep
      
    coerce(l:List F):% == 
      #l ~=2 => error "Form expected: [a,b]"
      [l.1,l.2,"closed"]::Rep
      
    coerce(x:%):List F == [x.lb,x.ub]
  
    coerce(x:%):OF ==
      x.tp = "closed"    => hconcat ["[",x.lb::OF,",",x.ub::OF,"]"]
      x.tp = "open"      => hconcat ["(",x.lb::OF,",",x.ub::OF,")"]
      x.tp = "leftOpen"  => hconcat ["(",x.lb::OF,",",x.ub::OF,"]"]
      x.tp = "rightOpen" => hconcat ["[",x.lb::OF,",",x.ub::OF,")"]
   
    lb(x) == x.lb
    ub(x) == x.ub
    topology(x:%):OF == x.tp::Symbol::OF
    measure(x:%):F == x.ub - x.lb

    closedInterval(a,b)    == [a,b,"closed"]::Rep
    openInterval(a,b)      == [a,b,"open"]::Rep
    leftOpenInterval(a,b)  == [a,b,"leftOpen"]::Rep
    rightOpenInterval(a,b) == [a,b,"rightOpen"]::Rep
    
    --x:% * y:% == 
      --m:Matrix(F):=matrix([[lb x,ub x],[lb y,ub y]])
      --m::Cell(F,2)

    a:F + x:% == [a+x.lb,a+x.ub,x.tp]::Rep
    a:F * x:% == [a*x.lb,a*x.ub,x.tp]::Rep
      

)abbrev domain CELL Cell
++ Author: kfp
++ Date Created: Thu Nov 06 02:39:44 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations: 
++ Related Domains: 
++ Also See:
++ AMS Classifications:
++ Keywords: 
++ Examples:
++ References:
++
++ Description:
++ 
++
Cell(k,F) : Exports == Implementation where
  
  NNI ==> NonNegativeInteger
  OF  ==> OutputForm
  
  F: Field
  k: NNI 
  
  Exports == Join(CoercibleTo OutputForm) with
    
   coerce : Matrix F -> % 
   coerce : % -> List Interval F   --  ex. c::List(Interval RR)
   coerce : List Interval F -> %
   coerce : % -> OutputForm

   measure : % -> F
	
  Implementation ==  Matrix F add 

    Rep := Matrix F
    
    coerce(c:%):List Interval F == 
      lc:List(List F):=listOfLists(c)
      #lc ~= k => error "Wrong dimension"
      [i::Interval(F) for i in lc]
      
    coerce(l:List Interval F):% ==
      matrix [i::List(F) for i in l]
    
    coerce(m:Matrix F):% ==
      ncols m ~= 2   => error "Expecting (k x 2) matrix"
      nrows m ~= k   => error "Expecting (k x 2) matrix"
      m
   
    coerce(c:%):OF == (c::List Interval F)::OF
      
   
    measure(c:%):F == 
      L:=c::List Interval F
      mL:=[measure(v) for v in L]
      determinant(diagonalMatrix(mL))


    -- pi:=%pi$RR 
    
)abbrev domain SURF Surface
++ Author: kfp
++ Date Created: Tue Nov 11 21:57:52 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations: 
++ Related Domains: 
++ Also See:
++ AMS Classifications:
++ Keywords: 
++ Examples:
++ References:
++
++ Description:
++ k-surface
++
Surface(k,n) : Exports == Implementation where
  
  PYI   ==> Polynomial(Integer)
  NNI   ==> NonNegativeInteger
  SYM   ==> Symbol
  PI    ==> PositiveInteger
  OF    ==> OutputForm
  X     ==> RealSpace(k) -> RealSpace(n) 
  R     ==> Expression Integer 
  k:PI
  n:PI
  --V:Vectorspace(Real)
  
  Exports == Join(CoercibleTo OutputForm) with
    
    coerce : % -> OutputForm
    coerce : X -> % 
    dim    : % -> PI
    codim  : % -> PI
    eval   : (%, List R) -> RealSpace(n)
    eval   : (%, List Symbol) -> RealSpace(n)
	
    JacobianMatrix : (%, List Symbol) -> Matrix(R)
    TangentSpace : (%, List Symbol) -> List RealSpace(n) 
    CoTangentSpace : (%, List Symbol) -> List RealSpace(n)
    normal : (%, List Symbol) -> RealSpace(n)
	
  Implementation == add 

    Rep := Record(kk:PI,nn:PI,ff:X)
                  
    coerce(f:X):% == n>k and k>0 => [k,n,f]::Rep

    coerce(S:%):OF == hconcat [S.kk::OF,"-surface in R^"::SYM::OF,S.nn::OF]
   
    dim(S:%):PI == S.kk
    codim(S:%):PI == (S.nn - S.kk)::PI 
    
    eval(S:%,x:List R) == (S.ff)(x::RealSpace(k)) 
    eval(S:%,x:List Symbol) == (S.ff)(x::RealSpace(k))
  
    JacobianMatrix(S:%, x:List Symbol):Matrix(R) ==
      transpose matrix [D((S.ff)(x::RealSpace(k)),u)::List(R) for u in x]
     
    -- local ; better coerce in ??? 
    vec2list(v:Vector(R)):List(R) == [v.j for j in 1..#v]
      
     
    TangentSpace(S:%, x:List Symbol):List RealSpace(n) ==
      J:=JacobianMatrix(S,x)
      [vec2list(column(J,j))::RealSpace(n) for j in 1..ncols(J)]
      
    CoTangentSpace(S:%, x:List Symbol):List RealSpace(n) ==  
      ts:=TangentSpace(S,x)
      rd:=[[0::R for i in 1..n]::RealSpace(n) for j in k+1..n]
      Q:=append(ts,rd)
      QR:=[v::List R for v in Q]
      M:=(matrix QR)::Matrix(R)
      [vec2list(y)::RealSpace(n) for y in nullSpace(M)]
      
     
    normal(S:%, x:List Symbol):RealSpace(n) ==
      cs:=CoTangentSpace(S,x)
      #cs ~= 1 => error "Not a hypersurface"
      v:=cs.1
      r:= sqrt(dot(v,v))
      r = 0 => error "No normal; zero vector"
      v/r          

\end{spad}

\begin{axiom}
R ==> EXPR INT
R2 ==> RealSpace(2)
R3 ==> RealSpace(3)
f: R2 -> R3
f(X) == [(1+X.2/2*cos(X.1/2))*cos(X.1),(1+X.2/2*cos(X.1/2))*sin(X.1),X.2/2*sin(X.1/2)::R]::R3
f
typeOf f
\end{axiom}

Example
  Möbius strip as 2-surface

\begin{axiom}
f([u,v::Symbol]::R2)
-- proxy error without tex off !!
)set output tex off
)set output algebra on
S:=f::Surface(2,3)
dim S
eval(S,[u,v])
ts:=TangentSpace(S,[u,v])
cs:=CoTangentSpace(S,[u,v])
ns:=normal(S,[u,v])
\end{axiom}


**Cross product in R^n**

\begin{axiom}
X:=[a,b,c,d::R]::RealSpace(4)
Y:=[p,q,r,s::R]::RealSpace(4)
Z:=[t,u,v,w::R]::RealSpace(4)
C:=cross [X,Y,Z]
[dot(C,v) for v in [X,Y,Z]]
\end{axiom}

spad

)abbrev domain RSPACE RealSpace
++ Author: Kurt Pagani
++ Date Created: Thu Nov 06 03:51:56 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations: 
++ Related Domains: 
++ Also See:
++ AMS Classifications:
++ Keywords: 
++ Examples:
++ References:
++
++ Description:
++ 
++
RealSpace(n) : Exports == Implementation where

  NNI ==> NonNegativeInteger
  PI  ==> PositiveInteger
  I   ==> Integer
  R   ==> Expression Integer -- Real
  n:NNI

  Exports == Join(VectorSpace R , CoercibleTo OutputForm) with

    coerce : % -> OutputForm
    coerce : List R -> %
    coerce : Vector R -> %
    coerce : List Symbol -> %
    coerce : R -> %
    coerce : % -> List R
    coerce : % -> Vector R

    dot : (%,%) -> R
    dim : % -> NNI

    "+" : (%,%) -> %
    "-" : (%,%) -> %
    "*" : (R,%) -> %
    "*" : (NNI,%) -> %
    "*" : (PI,%)  -> %
    "*" : (I,%)   -> %
    "/" : (%,R)   -> %

    "#" : % -> NNI
    "-" : % -> %
    "=" : (%,%) -> Boolean
    "~=": (%,%) -> Boolean

    unitVector : PI -> %
    elt : (%,I) -> R

    eval: (%, Equation R) -> %
    eval: (%, List Equation R) -> % 

    every?: (R -> Boolean, %) -> Boolean
    map: (R -> R, %) -> %  -- strange, to check
    member?: (R, %) -> Boolean
    any?: (R -> Boolean, %) -> Boolean
    copy: % -> %
    D: (%, Symbol) -> %
    D: (%, Symbol, NonNegativeInteger) -> %
    D: (%, List Symbol) -> %
    D: (%, List Symbol, List NonNegativeInteger) -> %

    dist : (%,%) -> R
    norm : % -> R
    tprod : (%,%) -> Matrix(R)   -- tensor prod as Matrix Real
    -- _*_* : (%,%) -> RealSpace(n^2) 
    cross : List(%) -> %

  Implementation == DirectProduct(n,R) add

    Rep := DirectProduct(n,R)

    coerce(l:List R):% == 
      #l=n => directProduct((vector l)::Vector(R))$Rep

    coerce(l:Vector R):% == 
      #l=n => directProduct(l)$Rep 

    coerce(l:List Symbol):% ==
      #l=n => coerce([r::R for r in l])

    coerce(x:R):% == x*unitVector(1)$Rep

    coerce(v:%):List R == [v(j) for j in 1..#v]

    coerce(v:%):Vector R == vector [v(j) for j in 1..#v]

    dim(x:%):NNI == #(x::List R) -- n
    dist(x:%,y:%):R == sqrt dot(x-y,x-y)
    norm(x:%):R == sqrt dot(x,x)

    tprod(x:%,y:%):Matrix(R) == 
      matrix [[x.i*(y.j)::R for j in 1..n] for i in 1..n]

    --wprod(x:%,y:%):Matrix(R) == (tprod(x,y)-tprod(y,x))/2::R

    --_*_*(x:%,y:%):RealSpace(n^2) == 
    --  X:List Real:=[]
    --  for i in 1..n repeat
    --    X:=append(X,[x.i*(y.j)::Real for j in 1..n])  
    --  X::RealSpace(n^2)

    -- local ; better coerce in ??? 
    vec2list(v:Vector(R)):List(R) == [v.j for j in 1..#v]

    cross(X:List %):% ==
      #X ~= n-1 => error "Wrong number of arguments."
      L:List(List R):=[r::List(R) for r in X]
      M:Matrix(R):= transpose matrix L
      S:List(List R):=[vec2list(row(M,i)) for i in 1..n]
      LS:List(Matrix R):=[matrix([S.i for i in 1..n | i~=j]) for j in 1..n]
      r:List R:=[(-1)^(i-1)::R*determinant(LS.i) for i in 1..n]
      r::Rep

)abbrev package INTVAL Interval
++ Author: kfp
++ Date Created: Thu Nov 06 02:46:09 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations: 
++ Related Domains: 
++ Also See:
++ AMS Classifications:
++ Keywords: 
++ Examples:
++ References:
++
++ Description:
++ 
++
Interval(F:Field) : Exports == Implementation where

  OF  ==> OutputForm
  PI  ==> PositiveInteger
  NNI ==> NonNegativeInteger
  Q   ==> Fraction Integer

  Exports ==  Join(CoercibleTo OutputForm) with

    coerce : Tuple F -> %
    coerce : List F -> %
    coerce : % -> List F
    coerce : % -> OutputForm

    closedInterval    : (F,F) -> %
    openInterval      : (F,F) -> %
    leftOpenInterval  : (F,F) -> %
    rightOpenInterval : (F,F) -> %

    lb : % -> F
    ub : % -> F
    topology : % -> OutputForm 
    measure  : % -> F

    --"*" : (%,%) -> Cell(F,2)
    "*" : (F,%) -> %
    "+" : (F,%) -> %

  Implementation == add 

    Rep:= Record(lb:F,ub:F,tp:String)

    --check(x:%):% == 
      --if number?(x.lb) and number?(x.ub) then
        --(x.ub)::Q < (x.lb)::Q => error "a>b"
      --x

    coerce(l:Tuple F):% == -- does not work ????
      length l ~=2 => error "Form expected: (a,b)"
      [select(l,0), select(l,1),"open"]::Rep

    coerce(l:List F):% == 
      #l ~=2 => error "Form expected: [a,b]"
      [l.1,l.2,"closed"]::Rep

    coerce(x:%):List F == [x.lb,x.ub]

    coerce(x:%):OF ==
      x.tp = "closed"    => hconcat ["[",x.lb::OF,",",x.ub::OF,"]"]
      x.tp = "open"      => hconcat ["(",x.lb::OF,",",x.ub::OF,")"]
      x.tp = "leftOpen"  => hconcat ["(",x.lb::OF,",",x.ub::OF,"]"]
      x.tp = "rightOpen" => hconcat ["[",x.lb::OF,",",x.ub::OF,")"]

    lb(x) == x.lb
    ub(x) == x.ub
    topology(x:%):OF == x.tp::Symbol::OF
    measure(x:%):F == x.ub - x.lb

    closedInterval(a,b)    == [a,b,"closed"]::Rep
    openInterval(a,b)      == [a,b,"open"]::Rep
    leftOpenInterval(a,b)  == [a,b,"leftOpen"]::Rep
    rightOpenInterval(a,b) == [a,b,"rightOpen"]::Rep

    --x:% * y:% == 
      --m:Matrix(F):=matrix([[lb x,ub x],[lb y,ub y]])
      --m::Cell(F,2)

    a:F + x:% == [a+x.lb,a+x.ub,x.tp]::Rep
    a:F * x:% == [a*x.lb,a*x.ub,x.tp]::Rep

)abbrev domain CELL Cell
++ Author: kfp
++ Date Created: Thu Nov 06 02:39:44 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations: 
++ Related Domains: 
++ Also See:
++ AMS Classifications:
++ Keywords: 
++ Examples:
++ References:
++
++ Description:
++ 
++
Cell(k,F) : Exports == Implementation where

  NNI ==> NonNegativeInteger
  OF  ==> OutputForm

  F: Field
  k: NNI 

  Exports == Join(CoercibleTo OutputForm) with

   coerce : Matrix F -> % 
   coerce : % -> List Interval F   --  ex. c::List(Interval RR)
   coerce : List Interval F -> %
   coerce : % -> OutputForm

   measure : % -> F

  Implementation ==  Matrix F add 

    Rep := Matrix F

    coerce(c:%):List Interval F == 
      lc:List(List F):=listOfLists(c)
      #lc ~= k => error "Wrong dimension"
      [i::Interval(F) for i in lc]

    coerce(l:List Interval F):% ==
      matrix [i::List(F) for i in l]

    coerce(m:Matrix F):% ==
      ncols m ~= 2   => error "Expecting (k x 2) matrix"
      nrows m ~= k   => error "Expecting (k x 2) matrix"
      m

    coerce(c:%):OF == (c::List Interval F)::OF

    measure(c:%):F == 
      L:=c::List Interval F
      mL:=[measure(v) for v in L]
      determinant(diagonalMatrix(mL))

    -- pi:=%pi$RR 

)abbrev domain SURF Surface
++ Author: kfp
++ Date Created: Tue Nov 11 21:57:52 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations: 
++ Related Domains: 
++ Also See:
++ AMS Classifications:
++ Keywords: 
++ Examples:
++ References:
++
++ Description:
++ k-surface
++
Surface(k,n) : Exports == Implementation where

  PYI   ==> Polynomial(Integer)
  NNI   ==> NonNegativeInteger
  SYM   ==> Symbol
  PI    ==> PositiveInteger
  OF    ==> OutputForm
  X     ==> RealSpace(k) -> RealSpace(n) 
  R     ==> Expression Integer 
  k:PI
  n:PI
  --V:Vectorspace(Real)

  Exports == Join(CoercibleTo OutputForm) with

    coerce : % -> OutputForm
    coerce : X -> % 
    dim    : % -> PI
    codim  : % -> PI
    eval   : (%, List R) -> RealSpace(n)
    eval   : (%, List Symbol) -> RealSpace(n)

    JacobianMatrix : (%, List Symbol) -> Matrix(R)
    TangentSpace : (%, List Symbol) -> List RealSpace(n) 
    CoTangentSpace : (%, List Symbol) -> List RealSpace(n)
    normal : (%, List Symbol) -> RealSpace(n)

  Implementation == add 

    Rep := Record(kk:PI,nn:PI,ff:X)

    coerce(f:X):% == n>k and k>0 => [k,n,f]::Rep

    coerce(S:%):OF == hconcat [S.kk::OF,"-surface in R^"::SYM::OF,S.nn::OF]

    dim(S:%):PI == S.kk
    codim(S:%):PI == (S.nn - S.kk)::PI 

    eval(S:%,x:List R) == (S.ff)(x::RealSpace(k)) 
    eval(S:%,x:List Symbol) == (S.ff)(x::RealSpace(k))

    JacobianMatrix(S:%, x:List Symbol):Matrix(R) ==
      transpose matrix [D((S.ff)(x::RealSpace(k)),u)::List(R) for u in x]

    -- local ; better coerce in ??? 
    vec2list(v:Vector(R)):List(R) == [v.j for j in 1..#v]

    TangentSpace(S:%, x:List Symbol):List RealSpace(n) ==
      J:=JacobianMatrix(S,x)
      [vec2list(column(J,j))::RealSpace(n) for j in 1..ncols(J)]

    CoTangentSpace(S:%, x:List Symbol):List RealSpace(n) ==  
      ts:=TangentSpace(S,x)
      rd:=[[0::R for i in 1..n]::RealSpace(n) for j in k+1..n]
      Q:=append(ts,rd)
      QR:=[v::List R for v in Q]
      M:=(matrix QR)::Matrix(R)
      [vec2list(y)::RealSpace(n) for y in nullSpace(M)]

    normal(S:%, x:List Symbol):RealSpace(n) ==
      cs:=CoTangentSpace(S,x)
      #cs ~= 1 => error "Not a hypersurface"
      v:=cs.1
      r:= sqrt(dot(v,v))
      r = 0 => error "No normal; zero vector"
      v/r

spad

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6168575904420507084-25px001.spad
      using old system compiler.
   RSPACE abbreviates domain RealSpace 
------------------------------------------------------------------------
   initializing NRLIB RSPACE for RealSpace 
   compiling into NRLIB RSPACE 
   compiling exported coerce : List Expression Integer -> $
Time: 0.06 SEC.

   compiling exported coerce : Vector Expression Integer -> $
Time: 0 SEC.

   compiling exported coerce : List Symbol -> $
Time: 0.01 SEC.

   compiling exported coerce : Expression Integer -> $
Time: 0.02 SEC.

   compiling exported coerce : $ -> List Expression Integer
Time: 0.01 SEC.

   compiling exported coerce : $ -> Vector Expression Integer
Time: 0 SEC.

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

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

   compiling exported norm : $ -> Expression Integer
Time: 0 SEC.

   compiling exported tprod : ($,$) -> Matrix Expression Integer
Time: 0.02 SEC.

   compiling local vec2list : Vector Expression Integer -> List Expression Integer
Time: 0 SEC.

   compiling exported cross : List $ -> $
Time: 0.06 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |RealSpace| REDEFINED

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

   Cumulative Statistics for Constructor RealSpace
      Time: 0.22 seconds

--------------non extending category----------------------
.. RealSpace(#1) of cat 
(|Join| (|VectorSpace| (|Expression| (|Integer|)))
        (|CoercibleTo| (|OutputForm|))
        (CATEGORY |domain| (SIGNATURE |coerce| ((|OutputForm|) $))
         (SIGNATURE |coerce| ($ (|List| (|Expression| (|Integer|)))))
         (SIGNATURE |coerce| ($ (|Vector| (|Expression| (|Integer|)))))
         (SIGNATURE |coerce| ($ (|List| (|Symbol|))))
         (SIGNATURE |coerce| ($ (|Expression| (|Integer|))))
         (SIGNATURE |coerce| ((|List| (|Expression| (|Integer|))) $))
         (SIGNATURE |coerce| ((|Vector| (|Expression| (|Integer|))) $))
         (SIGNATURE |dot| ((|Expression| (|Integer|)) $ $))
         (SIGNATURE |dim| ((|NonNegativeInteger|) $)) (SIGNATURE + ($ $ $))
         (SIGNATURE - ($ $ $)) (SIGNATURE * ($ (|Expression| (|Integer|)) $))
         (SIGNATURE * ($ (|NonNegativeInteger|) $))
         (SIGNATURE * ($ (|PositiveInteger|) $))
         (SIGNATURE * ($ (|Integer|) $))
         (SIGNATURE / ($ $ (|Expression| (|Integer|))))
         (SIGNATURE |#| ((|NonNegativeInteger|) $)) (SIGNATURE - ($ $))
         (SIGNATURE = ((|Boolean|) $ $)) (SIGNATURE ~= ((|Boolean|) $ $))
         (SIGNATURE |unitVector| ($ (|PositiveInteger|)))
         (SIGNATURE |elt| ((|Expression| (|Integer|)) $ (|Integer|)))
         (SIGNATURE |eval| ($ $ (|Equation| (|Expression| (|Integer|)))))
         (SIGNATURE |eval|
          ($ $ (|List| (|Equation| (|Expression| (|Integer|))))))
         (SIGNATURE |every?|
          ((|Boolean|) (|Mapping| (|Boolean|) (|Expression| (|Integer|))) $))
         (SIGNATURE |map|
          ($ (|Mapping| (|Expression| (|Integer|)) (|Expression| (|Integer|)))
           $))
         (SIGNATURE |member?| ((|Boolean|) (|Expression| (|Integer|)) $))
         (SIGNATURE |any?|
          ((|Boolean|) (|Mapping| (|Boolean|) (|Expression| (|Integer|))) $))
         (SIGNATURE |copy| ($ $)) (SIGNATURE D ($ $ (|Symbol|)))
         (SIGNATURE D ($ $ (|Symbol|) (|NonNegativeInteger|)))
         (SIGNATURE D ($ $ (|List| (|Symbol|))))
         (SIGNATURE D
          ($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|))))
         (SIGNATURE |dist| ((|Expression| (|Integer|)) $ $))
         (SIGNATURE |norm| ((|Expression| (|Integer|)) $))
         (SIGNATURE |tprod| ((|Matrix| (|Expression| (|Integer|))) $ $))
         (SIGNATURE |cross| ($ (|List| $)))))   has no 
(|DirectProductCategory| |#1| (|Expression| (|Integer|)))    finalizing NRLIB RSPACE 
   Processing RealSpace for Browser database:
--------constructor---------
--->-->RealSpace((coerce ((OutputForm) %))): Not documented!!!!
--->-->RealSpace((coerce (% (List (Expression (Integer)))))): Not documented!!!!
--->-->RealSpace((coerce (% (Vector (Expression (Integer)))))): Not documented!!!!
--->-->RealSpace((coerce (% (List (Symbol))))): Not documented!!!!
--->-->RealSpace((coerce (% (Expression (Integer))))): Not documented!!!!
--->-->RealSpace((coerce ((List (Expression (Integer))) %))): Not documented!!!!
--->-->RealSpace((coerce ((Vector (Expression (Integer))) %))): Not documented!!!!
--->-->RealSpace((dot ((Expression (Integer)) % %))): Not documented!!!!
--->-->RealSpace((dim ((NonNegativeInteger) %))): Not documented!!!!
--->-->RealSpace((+ (% % %))): Not documented!!!!
--->-->RealSpace((- (% % %))): Not documented!!!!
--->-->RealSpace((* (% (Expression (Integer)) %))): Not documented!!!!
--->-->RealSpace((* (% (NonNegativeInteger) %))): Not documented!!!!
--->-->RealSpace((* (% (PositiveInteger) %))): Not documented!!!!
--->-->RealSpace((* (% (Integer) %))): Not documented!!!!
--->-->RealSpace((/ (% % (Expression (Integer))))): Not documented!!!!
--->-->RealSpace((# ((NonNegativeInteger) %))): Not documented!!!!
--->-->RealSpace((- (% %))): Not documented!!!!
--->-->RealSpace((= ((Boolean) % %))): Not documented!!!!
--->-->RealSpace((~= ((Boolean) % %))): Not documented!!!!
--->-->RealSpace((unitVector (% (PositiveInteger)))): Not documented!!!!
--->-->RealSpace((elt ((Expression (Integer)) % (Integer)))): Not documented!!!!
--->-->RealSpace((eval (% % (Equation (Expression (Integer)))))): Not documented!!!!
--->-->RealSpace((eval (% % (List (Equation (Expression (Integer))))))): Not documented!!!!
--->-->RealSpace((every? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->RealSpace((map (% (Mapping (Expression (Integer)) (Expression (Integer))) %))): Not documented!!!!
--->-->RealSpace((member? ((Boolean) (Expression (Integer)) %))): Not documented!!!!
--->-->RealSpace((any? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->RealSpace((copy (% %))): Not documented!!!!
--->-->RealSpace((D (% % (Symbol)))): Not documented!!!!
--->-->RealSpace((D (% % (Symbol) (NonNegativeInteger)))): Not documented!!!!
--->-->RealSpace((D (% % (List (Symbol))))): Not documented!!!!
--->-->RealSpace((D (% % (List (Symbol)) (List (NonNegativeInteger))))): Not documented!!!!
--->-->RealSpace((dist ((Expression (Integer)) % %))): Not documented!!!!
--->-->RealSpace((norm ((Expression (Integer)) %))): Not documented!!!!
--->-->RealSpace((tprod ((Matrix (Expression (Integer))) % %))): Not documented!!!!
--->-->RealSpace((cross (% (List %)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/RSPACE.NRLIB/RSPACE.lsp" (written 23 NOV 2014 02:23:48 AM):

; /var/aw/var/LatexWiki/RSPACE.NRLIB/RSPACE.fasl written
; compilation finished in 0:00:00.069
------------------------------------------------------------------------
   RealSpace is now explicitly exposed in frame initial 
   RealSpace will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/RSPACE.NRLIB/RSPACE

   INTVAL abbreviates package Interval 
------------------------------------------------------------------------
   initializing NRLIB INTVAL for Interval 
   compiling into NRLIB INTVAL 
Local variable Rep type redefined: (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: lb F) (: ub F) (: tp (String))) F F (String))) (SIGNATURE ~= ((Boolean) (Record (: lb F) (: ub F) (: tp (String))) (Record (: lb F) (: ub F) (: tp (String))))) (SIGNATURE coerce ((OutputForm) (Record (: lb F) (: ub F) (: tp (String))))) (SIGNATURE elt (F (Record (: lb F) (: ub F) (: tp (String))) lb)) (SIGNATURE elt (F (Record (: lb F) (: ub F) (: tp (String))) ub)) (SIGNATURE elt ((String) (Record (: lb F) (: ub F) (: tp (String))) tp)) (SIGNATURE setelt! (F (Record (: lb F) (: ub F) (: tp (String))) lb F)) (SIGNATURE setelt! (F (Record (: lb F) (: ub F) (: tp (String))) ub F)) (SIGNATURE setelt! ((String) (Record (: lb F) (: ub F) (: tp (String))) tp (String))) (SIGNATURE copy ((Record (: lb F) (: ub F) (: tp (String))) (Record (: lb F) (: ub F) (: tp (String))))))) to (DirectProductCategory n (Expression (Integer)))
   compiling exported coerce : Tuple F -> $
Time: 0 SEC.

   compiling exported coerce : List F -> $
Time: 0.01 SEC.

   compiling exported coerce : $ -> List F
Time: 0 SEC.

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

   compiling exported lb : $ -> F
      INTVAL;lb;$F;5 is replaced by QVELTx0 
Time: 0 SEC.

   compiling exported ub : $ -> F
      INTVAL;ub;$F;6 is replaced by QVELTx1 
Time: 0 SEC.

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

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

   compiling exported closedInterval : (F,F) -> $
      INTVAL;closedInterval;2F$;9 is replaced by VECTORabclosed 
Time: 0 SEC.

   compiling exported openInterval : (F,F) -> $
      INTVAL;openInterval;2F$;10 is replaced by VECTORabopen 
Time: 0 SEC.

   compiling exported leftOpenInterval : (F,F) -> $
      INTVAL;leftOpenInterval;2F$;11 is replaced by VECTORableftOpen 
Time: 0 SEC.

   compiling exported rightOpenInterval : (F,F) -> $
      INTVAL;rightOpenInterval;2F$;12 is replaced by VECTORabrightOpen 
Time: 0 SEC.

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

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

(time taken in buildFunctor:  0)

;;;     ***       |Interval| REDEFINED

;;;     ***       |Interval| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] coerce:  tp has no value
      [2] topology:  tp has no value
      [3] +:  tp has no value
      [4] *:  tp has no value

   Cumulative Statistics for Constructor Interval
      Time: 0.03 seconds

   finalizing NRLIB INTVAL 
   Processing Interval for Browser database:
--------constructor---------
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((OutputForm) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (List (Expression (Integer)))))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (Vector (Expression (Integer)))))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (List (Symbol))))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (Expression (Integer))))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((List (Expression (Integer))) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((Vector (Expression (Integer))) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((dot ((Expression (Integer)) % %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((dim ((NonNegativeInteger) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((+ (% % %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((- (% % %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% (Expression (Integer)) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% (NonNegativeInteger) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% (PositiveInteger) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% (Integer) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((/ (% % (Expression (Integer))))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((# ((NonNegativeInteger) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((- (% %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((= ((Boolean) % %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((~= ((Boolean) % %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((unitVector (% (PositiveInteger)))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((elt ((Expression (Integer)) % (Integer)))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((eval (% % (Equation (Expression (Integer)))))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((eval (% % (List (Equation (Expression (Integer))))))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((every? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((map (% (Mapping (Expression (Integer)) (Expression (Integer))) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((member? ((Boolean) (Expression (Integer)) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((any? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((copy (% %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((D (% % (Symbol)))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((D (% % (Symbol) (NonNegativeInteger)))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((D (% % (List (Symbol))))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((D (% % (List (Symbol)) (List (NonNegativeInteger))))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((dist ((Expression (Integer)) % %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((norm ((Expression (Integer)) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((tprod ((Matrix (Expression (Integer))) % %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((cross (% (List %)))): Not documented!!!!
--------constructor---------
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (Tuple F)))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (List F)))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((List F) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((OutputForm) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((closedInterval (% F F))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((openInterval (% F F))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((leftOpenInterval (% F F))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((rightOpenInterval (% F F))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((lb (F %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((ub (F %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((topology ((OutputForm) %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((measure (F %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% F %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((+ (% F %))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval(): Spurious comments: \blankline \blankline
; compiling file "/var/aw/var/LatexWiki/INTVAL.NRLIB/INTVAL.lsp" (written 23 NOV 2014 02:23:48 AM):

; /var/aw/var/LatexWiki/INTVAL.NRLIB/INTVAL.fasl written
; compilation finished in 0:00:00.041
------------------------------------------------------------------------
   Interval is now explicitly exposed in frame initial 
   Interval will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/INTVAL.NRLIB/INTVAL

   CELL abbreviates domain Cell 
------------------------------------------------------------------------
   initializing NRLIB CELL for Cell 
   compiling into NRLIB CELL 
Local variable Rep type redefined: (Join (MatrixCategory F (Vector F) (Vector F)) (CATEGORY domain (SIGNATURE diagonalMatrix ($ (Vector F))) (IF (has F (ConvertibleTo (InputForm))) (ATTRIBUTE (ConvertibleTo (InputForm))) noBranch) (IF (has F (Field)) (SIGNATURE inverse ((Union $ failed) $)) noBranch))) to (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: lb F) (: ub F) (: tp (String))) F F (String))) (SIGNATURE ~= ((Boolean) (Record (: lb F) (: ub F) (: tp (String))) (Record (: lb F) (: ub F) (: tp (String))))) (SIGNATURE coerce ((OutputForm) (Record (: lb F) (: ub F) (: tp (String))))) (SIGNATURE elt (F (Record (: lb F) (: ub F) (: tp (String))) lb)) (SIGNATURE elt (F (Record (: lb F) (: ub F) (: tp (String))) ub)) (SIGNATURE elt ((String) (Record (: lb F) (: ub F) (: tp (String))) tp)) (SIGNATURE setelt! (F (Record (: lb F) (: ub F) (: tp (String))) lb F)) (SIGNATURE setelt! (F (Record (: lb F) (: ub F) (: tp (String))) ub F)) (SIGNATURE setelt! ((String) (Record (: lb F) (: ub F) (: tp (String))) tp (String))) (SIGNATURE copy ((Record (: lb F) (: ub F) (: tp (String))) (Record (: lb F) (: ub F) (: tp (String)))))))
   compiling exported coerce : $ -> List Interval F
Time: 0.01 SEC.

   compiling exported coerce : List Interval F -> $
Time: 0.01 SEC.

   compiling exported coerce : Matrix F -> $
Time: 0 SEC.

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

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

(time taken in buildFunctor:  0)

;;;     ***       |Cell| REDEFINED

;;;     ***       |Cell| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor Cell
      Time: 0.02 seconds

--------------non extending category----------------------
.. Cell(#1,#2) of cat 
(|Join| (|CoercibleTo| (|OutputForm|))
        (CATEGORY |domain| (SIGNATURE |coerce| ($ (|Matrix| |#2|)))
         (SIGNATURE |coerce| ((|List| (|Interval| |#2|)) $))
         (SIGNATURE |coerce| ($ (|List| (|Interval| |#2|))))
         (SIGNATURE |coerce| ((|OutputForm|) $))
         (SIGNATURE |measure| (|#2| $))))   has no 
(|MatrixCategory| |#2| (|Vector| |#2|) (|Vector| |#2|))    finalizing NRLIB CELL 
   Processing Cell for Browser database:
--------constructor---------
--->-->Cell((coerce ((OutputForm) %))): Not documented!!!!
--->-->Cell((coerce (% (List (Expression (Integer)))))): Not documented!!!!
--->-->Cell((coerce (% (Vector (Expression (Integer)))))): Not documented!!!!
--->-->Cell((coerce (% (List (Symbol))))): Not documented!!!!
--->-->Cell((coerce (% (Expression (Integer))))): Not documented!!!!
--->-->Cell((coerce ((List (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((coerce ((Vector (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((dot ((Expression (Integer)) % %))): Not documented!!!!
--->-->Cell((dim ((NonNegativeInteger) %))): Not documented!!!!
--->-->Cell((+ (% % %))): Not documented!!!!
--->-->Cell((- (% % %))): Not documented!!!!
--->-->Cell((* (% (Expression (Integer)) %))): Not documented!!!!
--->-->Cell((* (% (NonNegativeInteger) %))): Not documented!!!!
--->-->Cell((* (% (PositiveInteger) %))): Not documented!!!!
--->-->Cell((* (% (Integer) %))): Not documented!!!!
--->-->Cell((/ (% % (Expression (Integer))))): Not documented!!!!
--->-->Cell((# ((NonNegativeInteger) %))): Not documented!!!!
--->-->Cell((- (% %))): Not documented!!!!
--->-->Cell((= ((Boolean) % %))): Not documented!!!!
--->-->Cell((~= ((Boolean) % %))): Not documented!!!!
--->-->Cell((unitVector (% (PositiveInteger)))): Not documented!!!!
--->-->Cell((elt ((Expression (Integer)) % (Integer)))): Not documented!!!!
--->-->Cell((eval (% % (Equation (Expression (Integer)))))): Not documented!!!!
--->-->Cell((eval (% % (List (Equation (Expression (Integer))))))): Not documented!!!!
--->-->Cell((every? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((map (% (Mapping (Expression (Integer)) (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((member? ((Boolean) (Expression (Integer)) %))): Not documented!!!!
--->-->Cell((any? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((copy (% %))): Not documented!!!!
--->-->Cell((D (% % (Symbol)))): Not documented!!!!
--->-->Cell((D (% % (Symbol) (NonNegativeInteger)))): Not documented!!!!
--->-->Cell((D (% % (List (Symbol))))): Not documented!!!!
--->-->Cell((D (% % (List (Symbol)) (List (NonNegativeInteger))))): Not documented!!!!
--->-->Cell((dist ((Expression (Integer)) % %))): Not documented!!!!
--->-->Cell((norm ((Expression (Integer)) %))): Not documented!!!!
--->-->Cell((tprod ((Matrix (Expression (Integer))) % %))): Not documented!!!!
--->-->Cell((cross (% (List %)))): Not documented!!!!
--------constructor---------
--->-->Cell((coerce (% (Tuple F)))): Not documented!!!!
--->-->Cell((coerce (% (List F)))): Not documented!!!!
--->-->Cell((coerce ((List F) %))): Not documented!!!!
--->-->Cell((coerce ((OutputForm) %))): Not documented!!!!
--->-->Cell((closedInterval (% F F))): Not documented!!!!
--->-->Cell((openInterval (% F F))): Not documented!!!!
--->-->Cell((leftOpenInterval (% F F))): Not documented!!!!
--->-->Cell((rightOpenInterval (% F F))): Not documented!!!!
--->-->Cell((lb (F %))): Not documented!!!!
--->-->Cell((ub (F %))): Not documented!!!!
--->-->Cell((topology ((OutputForm) %))): Not documented!!!!
--->-->Cell((measure (F %))): Not documented!!!!
--->-->Cell((* (% F %))): Not documented!!!!
--->-->Cell((+ (% F %))): Not documented!!!!
--------constructor---------
--->-->Cell((coerce (% (Matrix F)))): Not documented!!!!
--->-->Cell((coerce ((List (Interval F)) %))): Not documented!!!!
--->-->Cell((coerce (% (List (Interval F))))): Not documented!!!!
--->-->Cell((coerce ((OutputForm) %))): Not documented!!!!
--->-->Cell((measure (F %))): Not documented!!!!
--->-->Cell(): Spurious comments: \blankline \blankline
--->-->Cell(): Spurious comments: \blankline \blankline
; compiling file "/var/aw/var/LatexWiki/CELL.NRLIB/CELL.lsp" (written 23 NOV 2014 02:23:48 AM):

; /var/aw/var/LatexWiki/CELL.NRLIB/CELL.fasl written
; compilation finished in 0:00:00.023
------------------------------------------------------------------------
   Cell is now explicitly exposed in frame initial 
   Cell will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CELL.NRLIB/CELL

   SURF abbreviates domain Surface 
------------------------------------------------------------------------
   initializing NRLIB SURF for Surface 
   compiling into NRLIB SURF 
Local variable Rep type redefined: (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) (PositiveInteger) (PositiveInteger) (Mapping (RealSpace n) (RealSpace k)))) (SIGNATURE ~= ((Boolean) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))))) (SIGNATURE coerce ((OutputForm) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))))) (SIGNATURE elt ((PositiveInteger) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) kk)) (SIGNATURE elt ((PositiveInteger) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) nn)) (SIGNATURE elt ((Mapping (RealSpace n) (RealSpace k)) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) ff)) (SIGNATURE setelt! ((PositiveInteger) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) kk (PositiveInteger))) (SIGNATURE setelt! ((PositiveInteger) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) nn (PositiveInteger))) (SIGNATURE setelt! ((Mapping (RealSpace n) (RealSpace k)) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) ff (Mapping (RealSpace n) (RealSpace k)))) (SIGNATURE copy ((Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))))))) to (Join (MatrixCategory F (Vector F) (Vector F)) (CATEGORY domain (SIGNATURE diagonalMatrix ($ (Vector F))) (IF (has F (ConvertibleTo (InputForm))) (ATTRIBUTE (ConvertibleTo (InputForm))) noBranch) (IF (has F (Field)) (SIGNATURE inverse ((Union $ failed) $)) noBranch)))
   compiling exported coerce : RealSpace k -> RealSpace n -> $
Time: 0.01 SEC.

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

   compiling exported dim : $ -> PositiveInteger
      SURF;dim;$Pi;3 is replaced by QVELTS0 
Time: 0 SEC.

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

   compiling exported eval : ($,List Expression Integer) -> RealSpace n
Time: 0.01 SEC.

   compiling exported eval : ($,List Symbol) -> RealSpace n
Time: 0 SEC.

   compiling exported JacobianMatrix : ($,List Symbol) -> Matrix Expression Integer
Time: 0.02 SEC.

   compiling local vec2list : Vector Expression Integer -> List Expression Integer
Time: 0 SEC.

   compiling exported TangentSpace : ($,List Symbol) -> List RealSpace n
Time: 0.01 SEC.

   compiling exported CoTangentSpace : ($,List Symbol) -> List RealSpace n
Time: 0.05 SEC.

   compiling exported normal : ($,List Symbol) -> RealSpace n
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |Surface| REDEFINED

;;;     ***       |Surface| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] coerce:  kk has no value
      [2] coerce:  nn has no value
      [3] dim:  kk has no value
      [4] codim:  nn has no value
      [5] codim:  kk has no value
      [6] eval:  ff has no value
      [7] JacobianMatrix:  ff has no value

   Cumulative Statistics for Constructor Surface
      Time: 0.10 seconds

   finalizing NRLIB SURF 
   Processing Surface for Browser database:
--------constructor---------
--->-->Surface((coerce ((OutputForm) %))): Not documented!!!!
--->-->Surface((coerce (% (List (Expression (Integer)))))): Not documented!!!!
--->-->Surface((coerce (% (Vector (Expression (Integer)))))): Not documented!!!!
--->-->Surface((coerce (% (List (Symbol))))): Not documented!!!!
--->-->Surface((coerce (% (Expression (Integer))))): Not documented!!!!
--->-->Surface((coerce ((List (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((coerce ((Vector (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((dot ((Expression (Integer)) % %))): Not documented!!!!
--->-->Surface((dim ((NonNegativeInteger) %))): Not documented!!!!
--->-->Surface((+ (% % %))): Not documented!!!!
--->-->Surface((- (% % %))): Not documented!!!!
--->-->Surface((* (% (Expression (Integer)) %))): Not documented!!!!
--->-->Surface((* (% (NonNegativeInteger) %))): Not documented!!!!
--->-->Surface((* (% (PositiveInteger) %))): Not documented!!!!
--->-->Surface((* (% (Integer) %))): Not documented!!!!
--->-->Surface((/ (% % (Expression (Integer))))): Not documented!!!!
--->-->Surface((# ((NonNegativeInteger) %))): Not documented!!!!
--->-->Surface((- (% %))): Not documented!!!!
--->-->Surface((= ((Boolean) % %))): Not documented!!!!
--->-->Surface((~= ((Boolean) % %))): Not documented!!!!
--->-->Surface((unitVector (% (PositiveInteger)))): Not documented!!!!
--->-->Surface((elt ((Expression (Integer)) % (Integer)))): Not documented!!!!
--->-->Surface((eval (% % (Equation (Expression (Integer)))))): Not documented!!!!
--->-->Surface((eval (% % (List (Equation (Expression (Integer))))))): Not documented!!!!
--->-->Surface((every? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((map (% (Mapping (Expression (Integer)) (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((member? ((Boolean) (Expression (Integer)) %))): Not documented!!!!
--->-->Surface((any? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((copy (% %))): Not documented!!!!
--->-->Surface((D (% % (Symbol)))): Not documented!!!!
--->-->Surface((D (% % (Symbol) (NonNegativeInteger)))): Not documented!!!!
--->-->Surface((D (% % (List (Symbol))))): Not documented!!!!
--->-->Surface((D (% % (List (Symbol)) (List (NonNegativeInteger))))): Not documented!!!!
--->-->Surface((dist ((Expression (Integer)) % %))): Not documented!!!!
--->-->Surface((norm ((Expression (Integer)) %))): Not documented!!!!
--->-->Surface((tprod ((Matrix (Expression (Integer))) % %))): Not documented!!!!
--->-->Surface((cross (% (List %)))): Not documented!!!!
--------constructor---------
--->-->Surface((coerce (% (Tuple F)))): Not documented!!!!
--->-->Surface((coerce (% (List F)))): Not documented!!!!
--->-->Surface((coerce ((List F) %))): Not documented!!!!
--->-->Surface((coerce ((OutputForm) %))): Not documented!!!!
--->-->Surface((closedInterval (% F F))): Not documented!!!!
--->-->Surface((openInterval (% F F))): Not documented!!!!
--->-->Surface((leftOpenInterval (% F F))): Not documented!!!!
--->-->Surface((rightOpenInterval (% F F))): Not documented!!!!
--->-->Surface((lb (F %))): Not documented!!!!
--->-->Surface((ub (F %))): Not documented!!!!
--->-->Surface((topology ((OutputForm) %))): Not documented!!!!
--->-->Surface((measure (F %))): Not documented!!!!
--->-->Surface((* (% F %))): Not documented!!!!
--->-->Surface((+ (% F %))): Not documented!!!!
--------constructor---------
--->-->Surface((coerce (% (Matrix F)))): Not documented!!!!
--->-->Surface((coerce ((List (Interval F)) %))): Not documented!!!!
--->-->Surface((coerce (% (List (Interval F))))): Not documented!!!!
--->-->Surface((coerce ((OutputForm) %))): Not documented!!!!
--->-->Surface((measure (F %))): Not documented!!!!
--------constructor---------
--->-->Surface((coerce ((OutputForm) %))): Not documented!!!!
--->-->Surface((coerce (% (Mapping (RealSpace n) (RealSpace k))))): Not documented!!!!
--->-->Surface((dim ((PositiveInteger) %))): Not documented!!!!
--->-->Surface((codim ((PositiveInteger) %))): Not documented!!!!
--->-->Surface((eval ((RealSpace n) % (List (Expression (Integer)))))): Not documented!!!!
--->-->Surface((eval ((RealSpace n) % (List (Symbol))))): Not documented!!!!
--->-->Surface((JacobianMatrix ((Matrix (Expression (Integer))) % (List (Symbol))))): Not documented!!!!
--->-->Surface((TangentSpace ((List (RealSpace n)) % (List (Symbol))))): Not documented!!!!
--->-->Surface((CoTangentSpace ((List (RealSpace n)) % (List (Symbol))))): Not documented!!!!
--->-->Surface((normal ((RealSpace n) % (List (Symbol))))): Not documented!!!!
--->-->Surface(): Spurious comments: \blankline \blankline
--->-->Surface(): Spurious comments: \blankline \blankline
--->-->Surface(): Spurious comments: \spad{k}-surface \blankline
; compiling file "/var/aw/var/LatexWiki/SURF.NRLIB/SURF.lsp" (written 23 NOV 2014 02:23:48 AM):

; /var/aw/var/LatexWiki/SURF.NRLIB/SURF.fasl written
; compilation finished in 0:00:00.053
------------------------------------------------------------------------
   Surface is now explicitly exposed in frame initial 
   Surface will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/SURF.NRLIB/SURF

fricas

R ==> EXPR INT

Type: Void

fricas

R2 ==> RealSpace(2)

Type: Void

fricas

R3 ==> RealSpace(3)

Type: Void

fricas

f: R2 -> R3

Type: Void

fricas

f(X) == [(1+X.2/2*cos(X.1/2))*cos(X.1),(1+X.2/2*cos(X.1/2))*sin(X.1),X.2/2*sin(X.1/2)::R]::R3

Type: Void

fricas

$\label{eq1}\begin{array}{@{}l} \displaystyle f \ X \ = = \ { \begin{array}{@{}l} \displaystyle {\left({ \begin{array}{@{}l} \displaystyle \left[{{\left(1 +{{{X \left({2}\right)}\over 2}\ {\cos \left({{X \left({1}\right)}\over 2}\right)}}\right)}\ {\cos \left({X \left({1}\right)}\right)}}, \: \right. \ \ \displaystyle \left.{{\left(1 +{{{X \left({2}\right)}\over 2}\ {\cos \left({{X \left({1}\right)}\over 2}\right)}}\right)}\ {\sin \left({X \left({1}\right)}\right)}}, \: \right. \ \ \displaystyle \left.{{{X \left({2}\right)}\over 2}\ {\left({\left(\sin \left({{X \left({1}\right)}\over 2}\right) \right)}\ : : \ \hbox{\axiomType{EXPR}\ } (\hbox{\axiomType{INT}\ }) \right)}}\right]$

(1)

Type: FunctionCalled?(f)

fricas

typeOf f

$\label{eq2}\mbox{\rm (\hbox{\axiomType{RealSpace}\ } (2) - > \hbox{\axiomType{RealSpace}\ } (3))}$

(2)

Type: Type

Example Möbius strip as 2-surface

fricas

f([u,v::Symbol]::R2)

fricas

Compiling function f with type RealSpace(2) -> RealSpace(3)

$\label{eq3}\left[{{{\left({v \ {\cos \left({u \over 2}\right)}}+ 2 \right)}\ {\cos \left({u}\right)}}\over 2}, \:{{{\left({v \ {\cos \left({u \over 2}\right)}}+ 2 \right)}\ {\sin \left({u}\right)}}\over 2}, \:{{v \ {\sin \left({u \over 2}\right)}}\over 2}\right]$

(3)

Type: RealSpace?(3)

fricas

-- proxy error without tex off !!

fricas

)set output tex off

fricas

)set output algebra on

S:=f::Surface(2,3)

   (9)  2-surface in R^3

Type: Surface(2,3)

fricas

dim S

   (10)  2

Type: PositiveInteger?

fricas

eval(S,[u,v])

                 u                    u                   u
          (v cos(-) + 2)cos(u) (v cos(-) + 2)sin(u) v sin(-)
                 2                    2                   2
   (11)  [--------------------,--------------------,--------]
                    2                    2              2

Type: RealSpace?(3)

fricas

ts:=TangentSpace(S,[u,v])

   (12)
   [
                u                           u
      (- 2v cos(-) - 4)sin(u) - v cos(u)sin(-)
                2                           2
     [----------------------------------------,
                          4
              u                  u                    u
      - v sin(-)sin(u) + (2v cos(-) + 4)cos(u)  v cos(-)
              2                  2                    2
      ----------------------------------------, --------]
                          4                         4
     ,
         u            u            u
     cos(-)cos(u) cos(-)sin(u) sin(-)
         2            2            2
    [------------,------------,------]]
           2            2         2

Type: List(RealSpace?(3))

fricas

cs:=CoTangentSpace(S,[u,v])

   (13)
   [
             u 2         u 2                    u                u
      (v sin(-)  + v cos(-) )sin(u) + (- 2v cos(-) - 4)cos(u)sin(-)
             2           2                      2                2
     [-------------------------------------------------------------,
              u 2        u        2           u 2        u        2
      (2v cos(-)  + 4cos(-))sin(u)  + (2v cos(-)  + 4cos(-))cos(u)
              2          2                    2          2
                u          u                      u 2         u 2
      (- 2v cos(-) - 4)sin(-)sin(u) - v cos(u)sin(-)  - v cos(-) cos(u)
                2          2                      2           2
      -----------------------------------------------------------------, 1]
                u 2        u        2           u 2        u        2
        (2v cos(-)  + 4cos(-))sin(u)  + (2v cos(-)  + 4cos(-))cos(u)
                2          2                    2          2
     ]

Type: List(RealSpace?(3))

fricas

ns:=normal(S,[u,v])

   (14)
   [
              u 2         u 2                    u                u
       (v sin(-)  + v cos(-) )sin(u) + (- 2v cos(-) - 4)cos(u)sin(-)
              2           2                      2                2
    /
                  u 2        u        2           u 2        u        2
         ((2v cos(-)  + 4cos(-))sin(u)  + (2v cos(-)  + 4cos(-))cos(u) )
                  2          2                    2          2
      *
         ROOT
                   2    u 4           u 3         u 2       2    2    u 4
                (4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)  + v sin(-)
                        2             2           2                   2
              + 
                   2    u 2           u           u 2
                (6v cos(-)  + 16v cos(-) + 16)sin(-)
                        2             2           2
              + 
                   2    u 4           u 3         u 2       2    2    u 4
                (4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)  + v cos(-)
                        2             2           2                   2
           /
                   2    u 4           u 3         u 2       2
                (4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)
                        2             2           2
              + 
                   2    u 4           u 3         u 2       2
                (4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)
                        2             2           2
     ,

                 u          u                      u 2         u 2
       (- 2v cos(-) - 4)sin(-)sin(u) - v cos(u)sin(-)  - v cos(-) cos(u)
                 2          2                      2           2
    /
                  u 2        u        2           u 2        u        2
         ((2v cos(-)  + 4cos(-))sin(u)  + (2v cos(-)  + 4cos(-))cos(u) )
                  2          2                    2          2
      *
         ROOT
                   2    u 4           u 3         u 2       2    2    u 4
                (4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)  + v sin(-)
                        2             2           2                   2
              + 
                   2    u 2           u           u 2
                (6v cos(-)  + 16v cos(-) + 16)sin(-)
                        2             2           2
              + 
                   2    u 4           u 3         u 2       2    2    u 4
                (4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)  + v cos(-)
                        2             2           2                   2
           /
                   2    u 4           u 3         u 2       2
                (4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)
                        2             2           2
              + 
                   2    u 4           u 3         u 2       2
                (4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)
                        2             2           2
     ,

       1
    /
       ROOT
                 2    u 4           u 3         u 2       2    2    u 4
              (4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)  + v sin(-)
                      2             2           2                   2
            + 
                 2    u 2           u           u 2
              (6v cos(-)  + 16v cos(-) + 16)sin(-)
                      2             2           2
            + 
                 2    u 4           u 3         u 2       2    2    u 4
              (4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)  + v cos(-)
                      2             2           2                   2
         /
                 2    u 4           u 3         u 2       2
              (4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)
                      2             2           2
            + 
                 2    u 4           u 3         u 2       2
              (4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)
                      2             2           2
     ]

Type: RealSpace?(3)

Cross product in R^n

fricas

X:=[a,b,c,d::R]::RealSpace(4)

   (15)  [a,b,c,d]

Type: RealSpace?(4)

fricas

Y:=[p,q,r,s::R]::RealSpace(4)

   (16)  [p,q,r,s]

Type: RealSpace?(4)

fricas

Z:=[t,u,v,w::R]::RealSpace(4)

   (17)  [t,u,v,w]

Type: RealSpace?(4)

fricas

C:=cross [X,Y,Z]

   (18)
   [(b r - c q)w + (- b s + d q)v + (c s - d r)u,
    (- a r + c p)w + (a s - d p)v + (- c s + d r)t,
    (a q - b p)w + (- a s + d p)u + (b s - d q)t,
    (- a q + b p)v + (a r - c p)u + (- b r + c q)t]

Type: RealSpace?(4)

fricas

[dot(C,v) for v in [X,Y,Z]]

   (19)  [0,0,0]

Type: List(Expression(Integer))