$SPAD/src/algebra carten.spad

Stephen M. Watt

May 20, 1991

Abstract

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 minix to minix+dim−1.

GradedAlgebra(R,E) denotes “E-graded R-algebra”. A graded algebra is a graded module together with a degree preserving R-linear map, called the product.

The name “product” is written out in full so inner and outer products with the same mapping type can be distinguished by name.

GradedModule(R,E) denotes “E-graded R-module”, i.e. collection of R-modules indexed by an abelian monoid E. An element g of G[s] for some specific s in E is said to be an element of G with degree s. Sums are defined in each module G[s] so two elements of G have a sum if they have the same degree.

Morphisms can be defined and composed by degree to give the mathematical category of graded modules.

Contents

1 category GRMOD GradedModule 3 2 category GRALG GradedAlgebra 5 3 domain CARTEN CartesianTensor 6 4 package CARTEN2 CartesianTensorFunctions2 17 5 License 18

1 category GRMOD GradedModule

⟨category GRMOD GradedModule⟩≡

)abbrev category GRMOD GradedModule
++ Author: Stephen M. Watt
++ Date Created: May 20, 1991
++ Date Last Updated: May 20, 1991
++ Basic Operations: +, *, degree
++ Related Domains: CartesianTensor(n,dim,R)
++ Also See:
++ AMS Classifications:
++ Keywords: graded module, tensor, multi-linear algebra
++ Examples:
++ References: Algebra 2d Edition, MacLane and Birkhoff, MacMillan 1979
++ Description:
++  GradedModule(R,E) denotes ``E-graded R-module'', i.e. collection of
++  R-modules indexed by an abelian monoid E.
++  An element \spad{g} of \spad{G[s]} for some specific \spad{s} in \spad{E}
++  is said to be an element of \spad{G} with {\em degree} \spad{s}.
++  Sums are defined in each module \spad{G[s]} so two elements of \spad{G}
++  have a sum if they have the same degree.
++
++  Morphisms can be defined and composed by degree to give the
++  mathematical category of graded modules.
GradedModule(R: CommutativeRing, E: AbelianMonoid): Category ==
    SetCategory with
        degree: % -> E
            ++ degree(g) names the degree of g.  The set of all elements
            ++ of a given degree form an R-module.
        0: constant -> %
            ++ 0 denotes the zero of degree 0.
        _*: (R, %) -> %
            ++ r*g is left module multiplication.
        _*: (%, R) -> %
            ++ g*r is right module multiplication.
        _-: % -> %
            ++ -g is the additive inverse of g in the module of elements
            ++ of the same grade as g.
        _+: (%, %) -> %
            ++ g+h is the sum of g and h in the module of elements of
            ++ the same degree as g and h.  Error: if g and h
            ++ have different degrees.
        _-: (%, %) -> %
            ++ g-h is the difference of g and h in the module of elements of

++ the same degree as g and h.  Error: if g and h
++ have different degrees.
add
(x: %) - (y: %) == x+(-y)

2 category GRALG GradedAlgebra

⟨category GRALG GradedAlgebra⟩≡

)abbrev category GRALG GradedAlgebra
++ Author: Stephen M. Watt
++ Date Created: May 20, 1991
++ Date Last Updated: May 20, 1991
++ Basic Operations: +, *, degree
++ Related Domains: CartesianTensor(n,dim,R)
++ Also See:
++ AMS Classifications:
++ Keywords: graded module, tensor, multi-linear algebra
++ Examples:
++ References: Encyclopedic Dictionary of Mathematics, MIT Press, 1977
++ Description:
++  GradedAlgebra(R,E) denotes ``E-graded R-algebra''.
++  A graded algebra is a graded module together with a degree preserving
++  R-linear map, called the {\em product}.
++
++  The name ``product'' is written out in full so inner and outer products
++  with the same mapping type can be distinguished by name.
GradedAlgebra(R: CommutativeRing, E: AbelianMonoid): Category ==
    Join(GradedModule(R, E),RetractableTo(R)) with
        1: constant -> %
            ++ 1 is the identity for \spad{product}.
        product: (%, %) -> %
            ++ product(a,b) is the degree-preserving R-linear product:
            ++
            ++   \spad{degree product(a,b) = degree a + degree b}
            ++   \spad{product(a1+a2,b) = product(a1,b) + product(a2,b)}
            ++   \spad{product(a,b1+b2) = product(a,b1) + product(a,b2)}
            ++   \spad{product(r*a,b) = product(a,r*b) = r*product(a,b)}
            ++   \spad{product(a,product(b,c)) = product(product(a,b),c)}
  add
        if not (R is %) then
            0: % == (0$R)::%
            1: % == 1$R::%
            (r: R)*(x: %) == product(r::%, x)
            (x: %)*(r: R) == product(x, r::%)

3 domain CARTEN CartesianTensor

⟨domain CARTEN CartesianTensor⟩≡

)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{k1}-th index of
++ t and the \spad{k2}-th index of s.
++ For example, if \spad{r = contract(s,2,t,1)} for rank 3 tensors
++ 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)    -> %
++ 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

get   ==> elt$Rep
set_! ==> setelt$Rep
-- Use row-major order:
--   x[h,i,j] <-> x[(h-minix)*dim^2+(i-minix)*dim+(j-minix)]
Rep := IndexedVector(R,0)
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
    for j in 0..ny-1 repeat
        set_!(z, ioff + j, get(x,i) * 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)
            for k in 1..dim for xk in xl.. by xok repeat
                set_!(z, zl, get(z,zl) + get(x,xk))
z
contract(x, i, y, j) ==
rx := rank x
ry := rank y
i < 1 or i > rx or j < 1 or j > ry =>
    error "Improper index for contraction"
rly:= (ry-j) pretend NNI;  nly:= dim^rly;  oly:= 1;    zoly:= 1
rhy:= (j -1) pretend NNI; nhy:= dim^rhy
ohy:= nly*dim; zohy:= zoly*nly
rlx:= (rx-i) pretend NNI;  nlx:= dim^rlx
olx:= 1;        zolx:= zohy*nhy
rhx:= (i -1) pretend NNI;  nhx:= dim^rhx
ohx:= nlx*dim;  zohx:= zolx*nlx
z := new(nlx*nhx*nly*nhy, 0)
for dxh in 1..nhx _
for xh in 0.. by ohx for zhx in 0.. by zohx repeat
    for dxl in 1..nlx _
    for xl in xh.. by olx for zlx in zhx.. by zolx repeat
        for dyh in 1..nhy _
        for yh in 0.. by ohy for zhy in zlx.. by zohy repeat
            for dyl in 1..nly _
            for yl in yh.. by oly for zly in zhy.. by zoly repeat
                set_!(z, zly, 0)
                for k in 1..dim _
                for xk in xl.. by nlx for yk in yl.. by nly repeat
                    set_!(z, zly, get(z,zly)+get(x,xk)*get(y,yk))

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

4 package CARTEN2 CartesianTensorFunctions2

⟨package CARTEN2 CartesianTensorFunctions2 ⟩≡

)abbrev package CARTEN2 CartesianTensorFunctions2
++ Author: Stephen M. Watt
++ Date Created: December 1986
++ Date Last Updated: May 30, 1991
++ Basic Operations:  reshape, map
++ Related Domains: CartesianTensor
++ Also See:
++ AMS Classifications:
++ Keywords: tensor
++ Examples:
++ References:
++ Description:
++   This package provides functions to enable conversion of tensors
++   given conversion of the components.
CartesianTensorFunctions2(minix, dim, S, T): CTPcat == CTPdef where
    minix:  Integer
    dim:    NonNegativeInteger
    S, T:   CommutativeRing
    CS  ==> CartesianTensor(minix, dim, S)
    CT  ==> CartesianTensor(minix, dim, T)
    CTPcat == with
        reshape: (List T, CS) -> CT
            ++ reshape(lt,ts) organizes the list of components lt into
            ++ a tensor with the same shape as ts.
        map: (S->T,   CS) -> CT
            ++ map(f,ts) does a componentwise conversion of the tensor ts
            ++ to a tensor with components of type T.
    CTPdef == add
        reshape(l, s) == unravel l
        map(f, s)     == unravel [f e for e in ravel s]

5 License

⟨license⟩≡

--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

⟨* ⟩≡ ⟨license⟩

⟨category GRMOD GradedModule⟩ ⟨category GRALG GradedAlgebra⟩ ⟨domain CARTEN CartesianTensor⟩ ⟨package CARTEN2 CartesianTensorFunctions2 ⟩

References^{[1] nothing}

19