(1) -> <spad>

)abbrev domain RSPACE RealSpace
++ Author: kfp
++ 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/6605988090913086661-25px001.spad
      using old system compiler.
   RSPACE abbreviates domain RealSpace 
------------------------------------------------------------------------
   initializing NRLIB RSPACE for RealSpace 
   compiling into NRLIB RSPACE 
****** comp fails at level 1 with expression: ******
((|VectorSpace| (|Expression| (|Integer|))))
****** level 1  ******
$x:= (VectorSpace (Expression (Integer)))
$m:= $EmptyMode
$f:=
((((|n| # . #1=#) (|RealSpace| #) (|n| . #1#) (* #) ...)))

   >> Apparent user error:
   cannot compile (VectorSpace (Expression (Integer)))

R ==> EXPR INT

R2 ==> RealSpace(2)

R3 ==> RealSpace(3)

f: R2 -> R3

   RealSpace is an unknown constructor and so is unavailable. Did you 
      mean to use -> but type something different instead?

Example Möbius strip as 2-surface

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

   RealSpace is an unknown constructor and so is unavailable. Did you 
      mean to use -> but type something different instead?

Cross product in R^n

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

   RealSpace is an unknown constructor and so is unavailable. Did you 
      mean to use -> but type something different instead?