PathSignatures : Table of Contents

• PathSignatures -- a package for working with signatures of algebraic paths
• adjointWord -- image of a word through the shuffle algebra homomorphism induced by a polynomial map
• ambient -- exported from the NCALgebra package.
• antipode(NCRingElement) -- The antipode of a tensor
• Array _ NCPolynomialRing -- create a word from an array
• CAxisTensor -- the signature tensor of the canonical axis path at a given level
• CMonTensor -- the signature tensor of the canonical monomial path at a given level
• Computing Path Varieties
  • Polynomial paths of degree m
  • A family of paths on a cone
  • The universal variety and toric coordinates
• concatPath -- concatenation of paths
• halfshuffle -- compute the half-shuffle of an ordered pair of words
• inner -- compute the inner product of two tensors.
• lie -- lie bracket of two elements
• lieBasis -- basis element corresponding to a Lyndon word in a Lie algebra
• linPath -- constructor of single piece polynomial path
• lyndonShuffle -- compute the representation of a tensor as a shuffle polynomial in Lyndon words
• lyndonWords -- compute all Lyndon words of at most a given length on a given number of letters
• matrixAction -- Diagonal matrix action on a tensor.
• NCPolynomialRing -- exported from the NCAlgebra package, used to encode tensor algebras
• NCRingElement -- exported from the NCAlgebra package, used to encode tensors
• Path
• polyPath -- constructor of single piece polynomial path
• pwLinPath -- constructor of a piecewise linear path from a matrix
• sgnVolTensor -- The signed volume tensor in a free associative algebra.
• shuffle -- shuffle product of two words
• sig -- compute the signature of a piecewise polynomial path.
• substitute(Path,Ring) -- changes the coefficient ring of a path
• tensorArray -- k-th level component of a tensor.
• tensorExp -- Compute a component of the exponential of a tensor.
• tensorLog -- Compute a component of the logarithm of a tensor.
• tensorParametrization -- Constructs the morphism that maps a word to its coefficient in the given tensor.
• wordAlgebra -- create a free algebra over a given alphabet
• wordFormat -- display a tensor in word notation
• wordString -- a string representing a word in wordFormat