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 Topics FrontPage SandBox SandBoxSurfaceComplex <-- You are here. fricas)abbrev domain CMAP CellMap ++ CellMap(R,n) : Exports == Implementation where R: Join(Ring,Comparable) n: NonNegativeInteger X ==> Expression R DP ==> DirectProduct OF ==> OutputForm NNI ==> NonNegativeInteger MAP ==> List X -> List X DOM ==> List(Segment X) Exports == Join(CoercibleTo OF,SetCategory,Evalable X) with _= : (%,%) -> Boolean ++ f1=f2 checks if two given cell maps are equal, that is if they have ++ the same domain D and the same mapping from D into X^n. cellMap : (DOM,MAP) -> % ++ cellMap(D,f) is the constructor. Usually one has to specify the ++ dimension of the target space. For example, let Q=[a..b,c..d], then ++ cellMap(Q,Z+->[sin(Z.1),cos(Z.2),Z.1*Z.2])$CMAP(INT,3) defines a ++ 2-surface in X^3. getDom : % -> DOM ++ getDom(f) extracts the domain of f. getMap : % -> MAP ++ getMap(f) extracts the map of f. faces : % -> List List(%) ++ faces(f) returns the faces of f, that means the images of the boundary ++ of the domain. Note: the returned list contains pairs of faces ++ corresponding to the endpoints of intervals. coords : (Symbol,PositiveInteger) -> List X ++ coords(s,m) provides a sample of coordinates s[1],..,s[m] as a list. coordSymbols : (Symbol,PositiveInteger) -> List Symbol ++ coordSymbols(s,m) provides a sample of coordinates s[1],..,s[m] as a ++ list of symbols. jacobianMatrix : % -> (List X -> Matrix X) ++ jacobianMatrix(f) returns the Jacobian matrix as a marix valued ++ function defined on the same cell as the cellMap. tangentSpace : % -> (List(X) -> List(Vector X)) ++ tangentSpace(f) returns a coerce : % -> OutputForm ++ coerce(f) gives the output representation. Implementation == add Rep := Record(d:DOM,f:MAP) (x:% = y:%):Boolean == l:NNI:=min(#(x.d),#(y.d)) v:List X for j in 1..l repeat s:X:=subscript('z,[j::OF])::X v:=concat(v,s::X) x.d =y.d and (x.f) v = (y.f) v => true false cellMap(dd:DOM,ff:MAP):% == #dd > n => error concat("#DOM > ",string n) v:List X:=[1::X for j in 1..#dd] ~test(#ff(v)=n) => error concat("#Range ~= ", string n) construct(dd,ff) faceLoHi(x:%,i:NNI,lo:Boolean):% == l:NNI:=#(x.d) v:List X for j in 1..l repeat if j=i then if lo then s:X:=lo(x.d.i) else s:X:=hi(x.d.i) else if j>i then s:X:=subscript('%,[(j-1)::OF])::X else s:X:=subscript('%,[j::OF])::X v:=concat(v,s::X) vv:=delete(v,i..i) dd:List(Segment X):=delete(x.d,i..i) ff:MAP:=vv+->(x.f) v cellMap(dd,ff) faces(x:%):List List(%) == l:NNI:=#(x.d) [[faceLoHi(x,j,true), faceLoHi(x,j,false)] for j in 1..l] getDom(x) == x.d getMap(x) == x.f coordSymbols(s:Symbol,m:PositiveInteger):List Symbol == [subscript(s,[j::OF]) for j in 1..m] coords(s:Symbol,m:PositiveInteger):List X == xs:=[subscript(s,[j::OF]) for j in 1..m] [coerce(xs.j)$X for j in 1..#xs] jacobianMatrix(S:%):List(X) -> Matrix(X) == --xs:List Symbol:=v:=[subscript('x,[j::OF]) for j in 1..#(getDom S)] --x:List X:=[coerce(xs.j)$X for j in 1..#xs] xs:List Symbol:=coordSymbols('x,#(getDom S)::PositiveInteger) x:List X:=coords('x,#xs::PositiveInteger) F:List X:=(getMap S) x J:Matrix(X):=matrix [[D(ff,u) for u in xs] for ff in F] if Matrix(X) has Join(SetCategory,Evalable(X)) then (y:List X):Matrix(X)+-> eval(J,x,y) else (y:List X):Matrix(X)+-> J tangentSpace(S:%):List(X) -> List(Vector X) == J:=jacobianMatrix(S) x:List X:=coords('x,#(getDom S)::PositiveInteger) if Vector(X) has Join(SetCategory,Evalable(X)) then if X has EuclideanDomain then cs:List(Vector X):=columnSpace(J x) (y:List X):List Vector(X)+-> [eval(t,x,y) for t in cs] coerce(x) == v:List X for j in 1..#(x.d) repeat s:X:=subscript('%,[j::OF])::X v:=concat(v,s::X) r:List X:=(x.f) v hconcat ["|",x.d::OF," -> ",r::OF,"|"]  fricas)abbrev domain SCMPLX SurfaceComplex ++ SurfaceComplex(R,n) : Exports == Implementation where NNI ==> NonNegativeInteger INT ==> Integer n : NNI R : Join(Ring,Comparable) CMAP ==> CellMap(R,n) CTOF ==> CoercibleTo OutputForm X ==> Expression R OF ==> OutputForm MAP ==> List X -> List X DOM ==> List(Segment X) Exports == Join(AbelianGroup ,CTOF, RetractableTo CMAP) with bdry : % -> % ++ bdry(S) computes the boundary of the surface complex S. size : % -> NNI ++ size(S) returns the number of "pieces" of the surface complex S. nthCoef : (%,Integer) -> Integer ++ nthCoef(x, n) returns the coefficient of the n^th term of x. nthFactor : (%,Integer) -> CMAP ++ nthFactor(x, n) returns the factor of the n^th term of x. zero? : % -> Boolean ++ zero?(S) returns true if S is the empty surface complex. _= : (%,%) -> Boolean ++ S=S' checks if the surface complexes S and S' are equal. terms : % -> List(Record(gen: CMAP,exp: Integer)) ++ terms(S) returns all terms of S as a record. mapGen : ((CMAP -> CMAP),%) -> % ++ mapGen(f, e1 a1 +...+ en an) returns ++ \spad{e1 f(a1) +...+ en f(an)}. mapCoef : ((Integer -> Integer),%) -> % ++ mapCoef(f, e1 a1 +...+ en an) returns ++ \spad{f(e1) a1 +...+ f(en) an}. construct : (DOM,MAP) -> % ++ construct(d,f) constructs a term (piece) of a k-surface, where ++ d is the domain (a k-cell) and f is a mapping from d to a vector ++ space of dimension n. --coerce : % -> OutputForm Implementation == FreeAbelianGroup(CMAP) add Rep:=FreeAbelianGroup(CMAP) bdry(c:%):% == if size(c) = 1 then s:=nthFactor(c,1) l:=faces(s) fs:=[(a.2::Rep-a.1::Rep) for a in l] sgn:=(j:INT):INT+->if even? (j-1) then 1 else -1 nthCoef(c,1)*reduce("+",[sgn(j)*fs.j::Rep for j in 1..#fs]) else ct:=[(nthCoef(c,j)*nthFactor(c,j))::Rep for j in 1..size(c)] reduce("+",map(bdry,ct)) construct(d:DOM,f:MAP):% == cellMap(d,f)$CMAP::% fricasCompiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/9220523246559970704-25px.001.spad using old system compiler. CMAP abbreviates domain CellMap ------------------------------------------------------------------------ initializing NRLIB CMAP for CellMap compiling into NRLIB CMAP ****** Domain: R already in scope compiling exported = : ($,$) -> Boolean Time: 0.09 SEC. compiling exported cellMap : (List Segment Expression R,List Expression R -> List Expression R) -> $Time: 0.02 SEC. compiling local faceLoHi : ($,NonNegativeInteger,Boolean) -> $****** comp fails at level 9 with expression: ****** error in function faceLoHi (SEQ (|:=| (|:| |l| (|NonNegativeInteger|)) (|#| (|x| |d|))) (|:| |v| (|List| (|Expression| R))) (REPEAT (IN |j| (SEGMENT 1 |l|)) (SEQ (IF (= |j| |i|) (IF |lo| (|:=| (|:| |s| (|Expression| R)) (|lo| ((|x| |d|) |i|))) (|:=| (|:| |s| (|Expression| R)) (|hi| ((|x| |d|) |i|)))) (IF (> |j| |i|) (|:=| (|:| |s| (|Expression| R)) (|::| (|subscript| '$ (|construct| (|::| (- |j| 1) (|OutputForm|)))) (|Expression| R))) (|:=| (|:| |s| (|Expression| R)) (|::| (|subscript| '$(|construct| (|::| |j| (|OutputForm|)))) (|Expression| R))))) (|exit| 1 (|:=| |v| (|concat| |v| (|::| |s| (|Expression| R))))))) (|:=| |vv| (|delete| |v| (SEGMENT |i| |i|))) (|:=| (|:| |dd| (|List| (|Segment| (|Expression| R)))) (|delete| (|x| |d|) (SEGMENT |i| |i|))) (|:=| (|:| |ff| (|Mapping| (|List| (|Expression| R)) (|List| (|Expression| R)))) (+-> |vv| ((|x| |f|) |v|))) (|exit| 1 (|cellMap| |dd| |ff|))) ****** level 9 ******$x:= lo $m:=$ $f:= ((((|s| #) (|j| # #) (|v| #) (|l| # #) ...))) >> Apparent user error: Cannot coerce lo of mode (Boolean) to mode (Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))) fricas)clear all All user variables and function definitions have been cleared. R ==> EXPR INT Type: Void fricasOF ==> OutputForm Type: Void fricas-- Cell map R2 ==> CellMap(INT,2) Type: Void fricasR3 ==> CellMap(INT,3) Type: Void fricasR4 ==> CellMap(INT,4) Type: Void fricasQ2 ==> [0..1,0..1::R] Type: Void fricasQ3 ==> concat(Q2,[0..1::R]) Type: Void fricas--xs:List Symbol:=coordSymbols('x,4)$R4 ---------------------------------------------------------------- -- https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant ---------------------------------------------------------------- -- Example 1 F1:=cellMap(Q2,X+->[X.1^2*X.2,5*X.1+sin(X.2)])\$R2 CellMap is an unknown constructor and so is unavailable. Did you mean to use -> but type something different instead?

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