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 Topics FrontPage SandBox SandBoxGradedTensor <-- You are here. spad)abbrev domain GTEN GradedTensor GradedTensor(n:NonNegativeInteger, m:NonNegativeInteger, R:CommutativeRing,dim:NonNegativeInteger): Join(GradedAlgebra(R, NonNegativeInteger), GradedModule(Integer, NonNegativeInteger), Eltable(Integer,R)) with coerce: DirectProduct(dim, R) -> GradedTensor(1,0,R,dim) ++ coerce(v) views a vector as a (1,0)-tensor. coerce: SquareMatrix(dim, R) -> GradedTensor(0,2,R,dim) ++ coerce(m) views a matrix as a (0,2)-tensor. coerce: List R -> GradedTensor(0,1,R,dim) ++ coerce([r_1,...,r_dim]) allows tensors to be constructed ++ using lists. coerce: List % -> GradedTensor(n,m+1,R,dim) ++ coerce([t_1,...,t_dim]) allows tensors to be constructed ++ using lists. rank: % -> DirectProduct(2,NonNegativeInteger) ++ rank(t) returns the tensorial rank of (n,m)-tensor t ++ [n,m] (that is, the number of contravariant and covariant ++ indices). elt: (%) -> R ++ elt(t) gives the component of a rank 0 tensor. elt: (%, Integer, Integer) -> R ++ elt(t,i,j) gives a component of a rank (2,0) (1,1) or (0,2)-tensor. E.g. ++ T(1,1), T(1,-1), T(-1,-1) elt: (%, Integer, Integer, Integer) -> R ++ elt(t,i,j,k) gives a component of a rank (3,0),(2,1),(1,2) or (0,3)-tensor. ++ E.g. T(1,1,1), T(1,1,-1), etc. elt: (%, Integer, Integer, Integer, Integer) -> R ++ elt(t,i,j,k,l) gives a component of a rank (4,0), (3,1),(2,2),(1,3) or (0,4)-tensor. ++ E.g. T(1,1,1,1), T(1,1,1,-1), etc. elt: (%, List Integer) -> R ++ elt(t,[i1,...,iN]) gives a component of a rank (n,m)-tensor when n+m=N. ++ E.g. T[1,1,1,1,1], T[1,1,1,1,-1], etc. -- 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..dim, 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 ++ \spad{r(i1,i2,...,in,j1,j2,...jm) = sum(h=1..dim,s(i1,i2,ii=-h,...,in)*t(j1,j2,jj=h,...,jm))}. contract: (%, Integer, Integer) -> % ++ contract(t,i,j) is the contraction of tensor t which ++ \spad{r(i1,i2,...,in) = sum(h=1..dim,s(i1,i2,ii=-h,...,ij=h,...,in))}. transpose: % -> % ++ transpose(t) exchanges the first and last indices of t. ++ \spad{r(i,...,l) = t(l,...,i)}. transpose: (%, Integer, Integer) -> % ++ transpose(t,i,j) exchanges the \spad{i}-th and \spad{j}-th indices of t. ++ \spad{r(...,i,...,j,...) = t(...,j,...,i,...)}. reindex: (%, List Integer) -> % ++ reindex(t,[i1,...,in]) permutes the indices of t. ++ \spad{r(j1,j2,...,jn) = t(ji1,ji2,...,jin)}. kroneckerDelta: () -> GradedTensor(1,1,R,dim) ++ kroneckerDelta() is the rank (1,1)-tensor defined by ++ \spad{kroneckerDelta()(i,j)} ++ \spad{= 1 if i = j} ++ \spad{= 0 if i \~= j} leviCivitaSymbol: () -> GradedTensor(0,dim,R,dim) ++ leviCivitaSymbol() is the rank (0,\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 %. == add Rep == CartesianTensor(1,dim,R) -- exports rank(t:%):DirectProduct(2,NonNegativeInteger)==directProduct [n,m] spad Compiling OpenAxiom source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1176878541039946027-25px001.spad using Spad compiler. GTEN abbreviates domain GradedTensor Adding Integer modemaps Adding NonNegativeInteger modemaps ------------------------------------------------------------------------ initializing NRLIB GTEN for GradedTensor compiling into NRLIB GTEN Adding $modemaps Adding R modemaps Adding Rep modemaps Adding DirectProduct(2,NonNegativeInteger) modemaps compiling exported rank : % -> DirectProduct(2,NonNegativeInteger) Adding Vector NonNegativeInteger modemaps Time: 0.09 SEC. (time taken in buildFunctor: 0) ;;; *** |GradedTensor| REDEFINED ;;; *** |GradedTensor| REDEFINED Time: 0 SEC. Cumulative Statistics for Constructor GradedTensor Time: 0.09 seconds finalizing NRLIB GTEN Processing GradedTensor for Browser database: -- coerce : DirectProduct(dim,R) -> GradedTensor(1,0,R,dim) -- coerce : SquareMatrix(dim,R) -> GradedTensor(0,2,R,dim) -- coerce : List R -> GradedTensor(0,1,R,dim) -- coerce : List % -> GradedTensor(n,+mOne(),R,dim) -- rank : % -> DirectProduct(2,NonNegativeInteger) -- elt : % -> R -- elt : (%,Integer,Integer) -> R -- elt : (%,Integer,Integer,Integer) -> R -- elt : (%,Integer,Integer,Integer,Integer) -> R -- elt : (%,List Integer) -> R -- product : (%,%) -> % -- ?*? : (%,%) -> % -- contract : (%,Integer,%,Integer) -> % -- contract : (%,Integer,Integer) -> % -- transpose : % -> % -- transpose : (%,Integer,Integer) -> % -- reindex : (%,List Integer) -> % -- kroneckerDelta : () -> GradedTensor(1,1,R,dim) -- leviCivitaSymbol : () -> GradedTensor(0,dim,R,dim) -- ravel : % -> List R -- unravel : List R -> % -- sample : () -> % --->-->GradedTensor(constructor): Not documented!!!! --->-->GradedTensor: Missing Description ; compiling file "/var/aw/var/LatexWiki/GTEN.NRLIB/code.lsp" (written 31 JUL 2013 03:29:02 PM): ; compiling (/VERSIONCHECK 2) ; compiling (DEFUN |GTEN;rank;$Dp;1| ...) ; compiling (DEFUN |GradedTensor| ...) ; compiling (DEFUN |GradedTensor;| ...) ; compiling (MAKEPROP (QUOTE |GradedTensor|) ...) ; /var/aw/var/LatexWiki/GTEN.NRLIB/code.fasl written ; compilation finished in 0:00:00.068 ------------------------------------------------------------------------ GradedTensor is now explicitly exposed in frame initial GradedTensor will be automatically loaded when needed from /var/aw/var/LatexWiki/GTEN.NRLIB/code.fasl

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