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# 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