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spad
)abbrev domain CARTEN CartesianTensor
++ Author: Stephen M. Watt
++ Date Created: December 1986
++ Date Last Updated: May 15, 1991
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: tensor, graded algebra
++ Examples:
++ References:
++ Description:
++   CartesianTensor(minix,dim,R) provides Cartesian tensors with
++   components belonging to a commutative ring R.  These tensors
++   can have any number of indices.  Each index takes values from
++   \spad{minix} to \spad{minix + dim - 1}.
CartesianTensor(minix, dim, R): Exports == Implementation where
NNI ==> NonNegativeInteger
I   ==> Integer
DP  ==> DirectProduct
SM  ==> SquareMatrix
minix: Integer
dim: NNI
R: CommutativeRing
Exports ==> Join(GradedAlgebra(R, NNI), GradedModule(I, NNI)) with
coerce: DP(dim, R) -> %
++ coerce(v) views a vector as a rank 1 tensor.
coerce: SM(dim, R)  -> %
++ coerce(m) views a matrix as a rank 2 tensor.
coerce: List R -> %
++ coerce([r_1,...,r_dim]) allows tensors to be constructed
++ using lists.
coerce: List % -> %
++ coerce([t_1,...,t_dim]) allows tensors to be constructed
++ using lists.
rank: % -> NNI
++ rank(t) returns the tensorial rank of t (that is, the
++ number of indices).  This is the same as the graded module
++ degree.
elt: (%) -> R
++ elt(t) gives the component of a rank 0 tensor.
elt: (%, I) -> R
++ elt(t,i) gives a component of a rank 1 tensor.
elt: (%, I, I) -> R
++ elt(t,i,j) gives a component of a rank 2 tensor.
elt: (%, I, I, I) -> R
++ elt(t,i,j,k) gives a component of a rank 3 tensor.
elt: (%, I, I, I, I) -> R
++ elt(t,i,j,k,l) gives a component of a rank 4 tensor.
elt: (%, List I) -> R
++ elt(t,[i1,...,iN]) gives a component of a rank \spad{N} tensor.
-- This specializes the documentation from GradedAlgebra.
product: (%,%) -> %
++ product(s,t) is the outer product of the tensors s and t.
++ For example, if \spad{r = product(s,t)} for rank 2 tensors s and t,
++ then \spad{r} is a rank 4 tensor given by
++     \spad{r(i,j,k,l) = s(i,j)*t(k,l)}.
"*": (%, %) -> %
++ s*t is the inner product of the tensors s and t which contracts
++ the last index of s with the first index of t, i.e.
++     \spad{t*s = contract(t,rank t, s, 1)}
++     \spad{t*s = sum(k=1..N, t[i1,..,iN,k]*s[k,j1,..,jM])}
++ This is compatible with the use of \spad{M*v} to denote
++ the matrix-vector inner product.
contract:  (%, Integer, %, Integer) -> %
++ contract(t,i,s,j) is the inner product of tenors s and t
++ which sums along the \spad{i}-th index of
++ t and the \spad{j}-th index of s.
++ For example, if \spad{r = contract(s,2,t,1)} for
++ rank 3 tensors \spad{s} and \spad{t}, then \spad{r} is
++ the rank 4 \spad{(= 3 + 3 - 2)} tensor  given by
++     \spad{r(i,j,k,l) = sum(h=1..dim,s(i,h,j)*t(h,k,l))}.
contract:  (Integer, %, Integer, %, Integer) -> %
++ contract(n,t,i,s,j) is n-fold inner product of tenors
++ s and t which sums along n indices of t starting at
++ \spad{i} and n indices of s starting at \spad{j}.
++ For example, if \spad{r = contract(2,s,2,t,1)} for
++ rank 3 tensors \spad{s} and \spad{t}, then \spad{r} is
++ the rank 2 \spad{(= 3 + 3 - 2*2)} tensor  given by
++ \spad{r(i,l) = sum(h2=1..dim,sum(h1=1..dim,s(i,h1,h2)*t(h1,h2,l)))}.
contract:  (%, Integer, Integer)    -> %
++ contract(t,i,j) is the contraction of tensor t which
++ sums along the \spad{i}-th and \spad{j}-th indices.
++ For example,  if
++ \spad{r = contract(t,1,3)} for a rank 4 tensor t, then
++ \spad{r} is the rank 2 \spad{(= 4 - 2)} tensor given by
++     \spad{r(i,j) = sum(h=1..dim,t(h,i,h,j))}.
transpose: % -> %
++ transpose(t) exchanges the first and last indices of t.
++ For example, if \spad{r = transpose(t)} for a rank 4 tensor t, then
++ \spad{r} is the rank 4 tensor given by
++     \spad{r(i,j,k,l) = t(l,j,k,i)}.
transpose: (%, Integer, Integer) -> %
++ transpose(t,i,j) exchanges the \spad{i}-th and \spad{j}-th indices of t.
++ For example, if \spad{r = transpose(t,2,3)} for a rank 4 tensor t, then
++ \spad{r} is the rank 4 tensor given by
++     \spad{r(i,j,k,l) = t(i,k,j,l)}.
reindex: (%, List Integer) -> %
++ reindex(t,[i1,...,idim]) permutes the indices of t.
++ For example, if \spad{r = reindex(t, [4,1,2,3])}
++ for a rank 4 tensor t,
++ then \spad{r} is the rank for tensor given by
++     \spad{r(i,j,k,l) = t(l,i,j,k)}.
kroneckerDelta:  () -> %
++ kroneckerDelta() is the rank 2 tensor defined by
++    \spad{kroneckerDelta()(i,j)}
++       \spad{= 1  if i = j}
++       \spad{= 0 if  i \~= j}
leviCivitaSymbol: () -> %
++ leviCivitaSymbol() is the rank \spad{dim} tensor defined by
++ \spad{leviCivitaSymbol()(i1,...idim) = +1/0/-1}
++ if \spad{i1,...,idim} is an even/is nota /is an odd permutation
++ of \spad{minix,...,minix+dim-1}.
ravel:     % -> List R
++ ravel(t) produces a list of components from a tensor such that
++   \spad{unravel(ravel(t)) = t}.
unravel:   List R -> %
++ unravel(t) produces a tensor from a list of
++ components such that
++   \spad{unravel(ravel(t)) = t}.
sample:    () -> %
++ sample() returns an object of type %.
Implementation ==> add
PERM  ==> Vector Integer  -- 1-based entries from 1..n
INDEX ==> Vector Integer  -- 1-based entries from minix..minix+dim-1
-- Use row-major order:
--   x[h,i,j] <-> x[(h-minix)*dim^2+(i-minix)*dim+(j-minix)]
Rep := PrimitiveArray(R)
--get   ==> elt$Rep --set_! ==> setelt$Rep
get(x:Rep,i:Integer):R == QAREF1(x pretend SExpression,i)$Lisp set_!(x:Rep,i:Integer,s:R):R == QSETAREF1(x pretend SExpression,i,s)$Lisp
n:     Integer
r,s:   R
x,y,z: %
---- Local stuff
dim2: NNI := dim^2
dim3: NNI := dim^3
dim4: NNI := dim^4
sample()==kroneckerDelta()$% int2index(n: Integer, indv: INDEX): INDEX == n < 0 => error "Index error (too small)" rnk := #indv for i in 1..rnk repeat qr := divide(n, dim) n := qr.quotient indv.((rnk-i+1) pretend NNI) := qr.remainder + minix n ~= 0 => error "Index error (too big)" indv index2int(indv: INDEX): Integer == n: I := 0 for i in 1..#indv repeat ix := indv.i - minix ix<0 or ix>dim-1 => error "Index error (out of range)" n := dim*n + ix n lengthRankOrElse(v: Integer): NNI == v = 1 => 0 v = dim => 1 v = dim2 => 2 v = dim3 => 3 v = dim4 => 4 rx := 0 while v ~= 0 repeat qr := divide(v, dim) v := qr.quotient if v ~= 0 then qr.remainder ~= 0 => error "Rank is not a whole number" rx := rx + 1 rx -- l must be a list of the numbers 1..#l mkPerm(n: NNI, l: List Integer): PERM == #l ~= n => error "The list is not a permutation." p: PERM := new(n, 0) seen: Vector Boolean := new(n, false) for i in 1..n for e in l repeat e < 1 or e > n => error "The list is not a permutation." p.i := e seen.e := true for e in 1..n repeat not seen.e => error "The list is not a permutation." p -- permute s according to p into result t. permute_!(t: INDEX, s: INDEX, p: PERM): INDEX == for i in 1..#p repeat t.i := s.(p.i) t -- permsign!(v) = 1, 0, or -1 according as -- v is an even, is not, or is an odd permutation of minix..minix+#v-1. permsign_!(v: INDEX): Integer == -- sum minix..minix+#v-1. maxix := minix+#v-1 psum := (((maxix+1)*maxix - minix*(minix-1)) exquo 2)::Integer -- +/v ~= psum => 0 n := 0 for i in 1..#v repeat n := n + v.i n ~= psum => 0 -- Bubble sort! This is pretty grotesque. totTrans: Integer := 0 nTrans: Integer := 1 while nTrans ~= 0 repeat nTrans := 0 for i in 1..#v-1 for j in 2..#v repeat if v.i > v.j then nTrans := nTrans + 1 e := v.i; v.i := v.j; v.j := e totTrans := totTrans + nTrans for i in 1..dim repeat if v.i ~= minix+i-1 then return 0 odd? totTrans => -1 1 ---- Exported functions ravel x == [get(x,i) for i in 0..#x-1] unravel l == -- lengthRankOrElse #l gives sytnax error nz: NNI := # l lengthRankOrElse nz z := new(nz, 0) for i in 0..nz-1 for r in l repeat set_!(z, i, r) z kroneckerDelta() == z := new(dim2, 0) for i in 1..dim for zi in 0.. by (dim+1) repeat set_!(z, zi, 1) z leviCivitaSymbol() == nz := dim^dim z := new(nz, 0) indv: INDEX := new(dim, 0) for i in 0..nz-1 repeat set_!(z, i, permsign_!(int2index(i, indv))::R) z -- from GradedModule degree x == rank x rank x == n := #x lengthRankOrElse n elt(x) == #x ~= 1 => error "Index error (the rank is not 0)" get(x,0) elt(x, i: I) == #x ~= dim => error "Index error (the rank is not 1)" get(x,(i-minix)) elt(x, i: I, j: I) == #x ~= dim2 => error "Index error (the rank is not 2)" get(x,(dim*(i-minix) + (j-minix))) elt(x, i: I, j: I, k: I) == #x ~= dim3 => error "Index error (the rank is not 3)" get(x,(dim2*(i-minix) + dim*(j-minix) + (k-minix))) elt(x, i: I, j: I, k: I, l: I) == #x ~= dim4 => error "Index error (the rank is not 4)" get(x,(dim3*(i-minix) + dim2*(j-minix) + dim*(k-minix) + (l-minix))) elt(x, i: List I) == #i ~= rank x => error "Index error (wrong rank)" n: I := 0 for ii in i repeat ix := ii - minix ix<0 or ix>dim-1 => error "Index error (out of range)" n := dim*n + ix get(x,n) coerce(lr: List R): % == #lr ~= dim => error "Incorrect number of components" z := new(dim, 0) for r in lr for i in 0..dim-1 repeat set_!(z, i, r) z coerce(lx: List %): % == #lx ~= dim => error "Incorrect number of slices" rx := rank first lx for x in lx repeat rank x ~= rx => error "Inhomogeneous slice ranks" nx := # first lx z := new(dim * nx, 0) for x in lx for offz in 0.. by nx repeat for i in 0..nx-1 repeat set_!(z, offz + i, get(x,i)) z retractIfCan(x:%):Union(R,"failed") == zero? rank(x) => x() "failed" Outf ==> OutputForm mkOutf(x:%, i0:I, rnk:NNI): Outf == odd? rnk => rnk1 := (rnk-1) pretend NNI nskip := dim^rnk1 [mkOutf(x, i0+nskip*i, rnk1) for i in 0..dim-1]::Outf rnk = 0 => get(x,i0)::Outf rnk1 := (rnk-2) pretend NNI nskip := dim^rnk1 matrix [[mkOutf(x, i0+nskip*(dim*i + j), rnk1) for j in 0..dim-1] for i in 0..dim-1] coerce(x): Outf == mkOutf(x, 0, rank x) 0 == 0$R::Rep
1 == 1$R::Rep --coerce(n: I): % == new(1, n::R) coerce(r: R): % == new(1,r) coerce(v: DP(dim,R)): % == z := new(dim, 0) for i in 0..dim-1 for j in minIndex v .. maxIndex v repeat set_!(z, i, v.j) z coerce(m: SM(dim,R)): % == z := new(dim^2, 0) offz := 0 for i in 0..dim-1 repeat for j in 0..dim-1 repeat set_!(z, offz + j, m(i+1,j+1)) offz := offz + dim z x = y == #x ~= #y => false for i in 0..#x-1 repeat if get(x,i) ~= get(y,i) then return false true x + y == #x ~= #y => error "Rank mismatch" -- z := [xi + yi for xi in x for yi in y] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, get(x,i) + get(y,i)) z x - y == #x ~= #y => error "Rank mismatch" -- [xi - yi for xi in x for yi in y] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, get(x,i) - get(y,i)) z - x == -- [-xi for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, -get(x,i)) z n * x == -- [n * xi for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, n * get(x,i)) z x * n == -- [n * xi for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, n* get(x,i)) -- Commutative!! z r * x == -- [r * xi for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, r * get(x,i)) z x * r == -- [xi*r for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, r* get(x,i)) -- Commutative!! z product(x, y) == nx := #x; ny := #y z := new(nx * ny, 0) for i in 0..nx-1 for ioff in 0.. by ny repeat xi := get(x,i) if not zero? xi then for j in 0..ny-1 repeat set_!(z, ioff + j, xi * get(y,j)) z x * y == rx := rank x ry := rank y rx = 0 => get(x,0) * y ry = 0 => x * get(y,0) contract(x, rx, y, 1) contract(x, i, j) == rx := rank x i < 1 or i > rx or j < 1 or j > rx or i = j => error "Improper index for contraction" if i > j then (i,j) := (j,i) rl:= (rx- j) pretend NNI; nl:= dim^rl; zol:= 1; xol:= zol rm:= (j-i-1) pretend NNI; nm:= dim^rm; zom:= nl; xom:= zom*dim rh:= (i - 1) pretend NNI; nh:= dim^rh; zoh:= nl*nm xoh:= zoh*dim^2 xok := nl*(1 + nm*dim) z := new(nl*nm*nh, 0) for h in 1..nh _ for xh in 0.. by xoh for zh in 0.. by zoh repeat for m in 1..nm _ for xm in xh.. by xom for zm in zh.. by zom repeat for l in 1..nl _ for xl in xm.. by xol for zl in zm.. by zol repeat --set_!(z, zl, 0) zz:R:=0 for k in 1..dim for xk in xl.. by xok repeat --set_!(z, zl, get(z,zl) + get(x,xk)) zz := zz + get(x,xk) set_!(z, zl, zz) z contract(x, i, y, j) == contract(1, x, i, y, j) contract(n, x, i, y, j) == rx := rank x ry := rank y i < 1 or i+n-1 > rx or j < 1 or j+n-1 > ry => error "Improper index for contraction" -- width of trace nw:=dim^(n pretend NNI) -- rank of lower (right) part of y rly:= (ry-n-j+1) pretend NNI nly:= dim^rly -- spacing of lower y oly:= 1 zoly:= 1 -- rank of higher (left) part of y rhy:= (j-1) pretend NNI nhy:= dim^rhy -- spacing of higher y ohy:= nly*nw zohy:= zoly*nly -- rank of lower (right) part of x rlx:= (rx-n-i+1) pretend NNI nlx:= dim^rlx -- spacing of lower x olx:= 1 zolx:= zohy*nhy -- rank of higher (left) part of x rhx:= (i-1) pretend NNI nhx:= dim^rhx -- spacing of higher x ohx:= nlx*nw zohx:= zolx*nlx -- result z := new(nlx*nhx*nly*nhy, 0) -- higher x index for dxh in 1..nhx _ for xh in 0.. by ohx for zhx in 0.. by zohx repeat -- lower x index for dxl in 1..nlx _ for xl in xh.. by olx for zlx in zhx.. by zolx repeat -- higher y index for dyh in 1..nhy _ for yh in 0.. by ohy for zhy in zlx.. by zohy repeat -- lower y index for dyl in 1..nly _ for yl in yh.. by oly for zly in zhy.. by zoly repeat --trace r:R:=get(z,zly) for nk in 1..nw _ for xk in xl.. by nlx for yk in yl.. by nly repeat r:=r+get(x,xk)*get(y,yk) set_!(z, zly, r) z transpose x == transpose(x, 1, rank x) transpose(x, i, j) == rx := rank x i < 1 or i > rx or j < 1 or j > rx or i = j => error "Improper indicies for transposition" if i > j then (i,j) := (j,i) rl:= (rx- j) pretend NNI; nl:= dim^rl; zol:= 1; zoi := zol*nl rm:= (j-i-1) pretend NNI; nm:= dim^rm; zom:= nl*dim; zoj := zom*nm rh:= (i - 1) pretend NNI; nh:= dim^rh; zoh:= nl*nm*dim^2 z := new(#x, 0) for h in 1..nh for zh in 0.. by zoh repeat _ for m in 1..nm for zm in zh.. by zom repeat _ for l in 1..nl for zl in zm.. by zol repeat _ for p in 1..dim _ for zp in zl.. by zoi for xp in zl.. by zoj repeat for q in 1..dim _ for zq in zp.. by zoj for xq in xp.. by zoi repeat set_!(z, zq, get(x,xq)) z reindex(x, l) == nx := #x z: % := new(nx, 0) rx := rank x p := mkPerm(rx, l) xiv: INDEX := new(rx, 0) ziv: INDEX := new(rx, 0) -- Use permutation for i in 0..#x-1 repeat pi := index2int(permute_!(ziv, int2index(i,xiv),p)) set_!(z, pi, get(x,i)) z spad  Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6706617537521748663-25px001.spad using old system compiler. CARTEN abbreviates domain CartesianTensor ------------------------------------------------------------------------ initializing NRLIB CARTEN for CartesianTensor compiling into NRLIB CARTEN processing macro definition PERM ==> Vector Integer processing macro definition INDEX ==> Vector Integer compiling local get : (Rep,Integer) -> R CARTEN;get is replaced by QAREF1 Time: 0.02 SEC. compiling local set_! : (Rep,Integer,R) -> R CARTEN;set_! is replaced by QSETAREF1 Time: 0 SEC. compiling exported sample : () ->$
Time: 0 SEC.
compiling local int2index : (Integer,Vector Integer) -> Vector Integer
Time: 0.06 SEC.
compiling local index2int : Vector Integer -> Integer
Time: 0.01 SEC.
compiling local lengthRankOrElse : Integer -> NonNegativeInteger
Time: 0.01 SEC.
compiling local mkPerm : (NonNegativeInteger,List Integer) -> Vector Integer
Time: 0.01 SEC.
compiling local permute_! : (Vector Integer,Vector Integer,Vector Integer) -> Vector Integer
Time: 0 SEC.
compiling local permsign_! : Vector Integer -> Integer
Time: 0.04 SEC.
compiling exported ravel : $-> List R Time: 0 SEC. compiling exported unravel : List R ->$
Time: 0.01 SEC.
compiling exported kroneckerDelta : () -> $Time: 0 SEC. compiling exported leviCivitaSymbol : () ->$
Time: 0 SEC.
compiling exported degree : $-> NonNegativeInteger Time: 0 SEC. compiling exported rank :$ -> NonNegativeInteger
Time: 0 SEC.
compiling exported elt : $-> R Time: 0 SEC. compiling exported elt : ($,Integer) -> R
Time: 0 SEC.
compiling exported elt : ($,Integer,Integer) -> R Time: 0 SEC. compiling exported elt : ($,Integer,Integer,Integer) -> R
Time: 0.03 SEC.
compiling exported elt : ($,Integer,Integer,Integer,Integer) -> R Time: 0.04 SEC. compiling exported elt : ($,List Integer) -> R
Time: 0.01 SEC.
compiling exported coerce : List R -> $Time: 0.01 SEC. compiling exported coerce : List$ -> $Time: 0.02 SEC. compiling exported retractIfCan :$ -> Union(R,failed)
Time: 0 SEC.
processing macro definition Outf ==> OutputForm
compiling local mkOutf : ($,Integer,NonNegativeInteger) -> OutputForm Time: 0.02 SEC. compiling exported coerce :$ -> OutputForm
Time: 0 SEC.
compiling exported Zero : () -> $Time: 0.01 SEC. compiling exported One : () ->$
Time: 0 SEC.
compiling exported coerce : R -> $CARTEN;coerce;R$;29 is replaced by MAKEARR11r
Time: 0 SEC.
compiling exported coerce : DirectProduct(dim,R) -> $Time: 0.01 SEC. compiling exported coerce : SquareMatrix(dim,R) ->$
Time: 0.01 SEC.
compiling exported = : ($,$) -> Boolean
Time: 0.01 SEC.
compiling exported + : ($,$) -> $Time: 0 SEC. compiling exported - : ($,$) ->$
Time: 0 SEC.
compiling exported - : $->$
Time: 0 SEC.
compiling exported * : (Integer,$) ->$
Time: 0.01 SEC.
compiling exported * : ($,Integer) ->$
Time: 0 SEC.
compiling exported * : (R,$) ->$
Time: 0 SEC.
compiling exported * : ($,R) ->$
Time: 0.01 SEC.
compiling exported product : ($,$) -> $Time: 0.01 SEC. compiling exported * : ($,$) ->$
Time: 0 SEC.
compiling exported contract : ($,Integer,Integer) ->$
Time: 0 SEC.
compiling exported contract : ($,Integer,$,Integer) -> $Time: 0 SEC. compiling exported contract : (Integer,$,Integer,$,Integer) ->$
Time: 0.03 SEC.
compiling exported transpose : $->$
Time: 0 SEC.
compiling exported transpose : ($,Integer,Integer) ->$
Time: 0.01 SEC.
compiling exported reindex : ($,List Integer) ->$
Time: 0.01 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |CartesianTensor| REDEFINED
;;;     ***       |CartesianTensor| REDEFINED
Time: 0 SEC.
Warnings:
 int2index:  quotient has no value
 int2index:  remainder has no value
 index2int:  n has no value
 lengthRankOrElse:  quotient has no value
 lengthRankOrElse:  remainder has no value
 permsign_!:  nTrans has no value
 elt:  n has no value
Cumulative Statistics for Constructor CartesianTensor
Time: 0.41 seconds
finalizing NRLIB CARTEN
Processing CartesianTensor for Browser database:
--------constructor---------
--------(coerce (% (DirectProduct dim R)))---------
--------(coerce (% (SquareMatrix dim R)))---------
--------(coerce (% (List R)))---------
--------(coerce (% (List %)))---------
--------(rank ((NonNegativeInteger) %))---------
--------(elt (R %))---------
--------(elt (R % (Integer)))---------
--------(elt (R % (Integer) (Integer)))---------
--------(elt (R % (Integer) (Integer) (Integer)))---------
--------(elt (R % (Integer) (Integer) (Integer) (Integer)))---------
--------(elt (R % (List (Integer))))---------
--------(product (% % %))---------
--------(* (% % %))---------
--------(contract (% % (Integer) % (Integer)))---------
--------(contract (% (Integer) % (Integer) % (Integer)))---------
--------(contract (% % (Integer) (Integer)))---------
--------(transpose (% %))---------
--------(transpose (% % (Integer) (Integer)))---------
--------(reindex (% % (List (Integer))))---------
--------(kroneckerDelta (%))---------
--------(leviCivitaSymbol (%))---------
--------(ravel ((List R) %))---------
--------(unravel (% (List R)))---------
--------(sample (%))---------
; compiling file "/var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN.lsp" (written 25 APR 2017 08:36:33 PM):
; /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN.fasl written
; compilation finished in 0:00:00.421
------------------------------------------------------------------------
CartesianTensor is now explicitly exposed in frame initial
CartesianTensor will be automatically loaded when needed from
/var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN

fricas
X:=unravel([script(x,[[i]]) for i in 0..2^2-1])$CartesianTensor(1,2,EXPR INT) (1) Type: CartesianTensor?(1,2,Expression(Integer)) fricas Y:=unravel([script(y,[[i]]) for i in 0..2^2-1])$CartesianTensor(1,2,EXPR INT) (2)
Type: CartesianTensor?(1,2,Expression(Integer))
fricas
XY:=contract(contract(X,1,Y,1),1,2) (3)
Type: CartesianTensor?(1,2,Expression(Integer))
fricas
test(XY=contract(2,X,1,Y,1)) (4)
Type: Boolean
fricas
XY:=contract(X,2,Y,1) (5)
Type: CartesianTensor?(1,2,Expression(Integer))
fricas
test(XY=contract(1,X,2,Y,1)) (6)
Type: Boolean

fricas
X:=unravel([script(x,[[i]]) for i in 0..2^3-1])$CartesianTensor(1,2,EXPR INT) (7) Type: CartesianTensor?(1,2,Expression(Integer)) fricas Y:=unravel([script(y,[[i]]) for i in 0..2^3-1])$CartesianTensor(1,2,EXPR INT) (8)
Type: CartesianTensor?(1,2,Expression(Integer))
fricas
XY:=contract(contract(contract(X,1,Y,1),1,3),1,2) (9)
Type: CartesianTensor?(1,2,Expression(Integer))
fricas
test(XY=contract(3,X,1,Y,1)) (10)
Type: Boolean
fricas
test(product(X,Y)=contract(0,X,1,Y,1)) (11)
Type: Boolean
fricas
test(XY=contract(contract(contract(product(X,Y),1,4),1,3),1,2)) (12)
Type: Boolean

fricas
X:=unravel([script(x,[[i]]) for i in 0..2^4-1])$CartesianTensor(1,2,EXPR INT) (13) Type: CartesianTensor?(1,2,Expression(Integer)) fricas Y:=unravel([script(y,[[i]]) for i in 0..2^4-1])$CartesianTensor(1,2,EXPR INT) (14)
Type: CartesianTensor?(1,2,Expression(Integer))
fricas
XY:=contract(contract(contract(contract(X,1,Y,1),1,4),1,3),1,2) (15)
Type: CartesianTensor?(1,2,Expression(Integer))
fricas
test(XY=contract(4,X,1,Y,1)) (16)
Type: Boolean
fricas
XY:=contract(contract(X,3,Y,1),3,4) (17)
Type: CartesianTensor?(1,2,Expression(Integer))
fricas
test(product(X,Y)=contract(0,X,1,Y,1)) (18)
Type: Boolean
fricas
test(XY=contract(contract(product(X,Y),3,5),3,4)) (19)
Type: Boolean
fricas
test(XY=contract(2,X,3,Y,1)) (20)
Type: Boolean

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