path: root/src/algebra/carten.spad.pamphlet

\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra carten.spad}
\author{Stephen M. Watt}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{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)

@
\section{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::%)

@
\section{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),_
                  Eltable(I,R),CoercibleFrom DP(dim,R),_
                  CoercibleFrom SM(dim,R),CoercibleFrom List R,_
                  CoercibleFrom List %) with

        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, 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

        ---- Local stuff
        dim2: NNI := dim**2
        dim3: NNI := dim**3
        dim4: NNI := dim**4

	sample()==kroneckerDelta()$%
        int2index(n: Integer, indv: INDEX): INDEX ==
            negative? n => 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
                negative? ix 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: %) ==
            not one?(#x)    => 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
                negative? ix 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: Integer * 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: Integer ==
            -- [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: 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: 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

@
\section{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]

@
\section{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>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}