383 research outputs found
Hopf algebras: motivations and examples
This paper provides motivation as well as a method of construction for Hopf
algebras, starting from an associative algebra. The dualization technique
involved relies heavily on the use of Sweedler's dual
The mechanics of shuffle products and their siblings
We carry on the investigation initiated in [15] : we describe new shuffle
products coming from some special functions and group them, along with other
products encountered in the literature, in a class of products, which we name
-shuffle products. Our paper is dedicated to a study of the latter
class, from a combinatorial standpoint. We consider first how to extend
Radford's theorem to the products in that class, then how to construct their
bi-algebras. As some conditions are necessary do carry that out, we study them
closely and simplify them so that they can be seen directly from the definition
of the product. We eventually test these conditions on the products mentioned
above
Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach
Using a quantum field theory renormalization group-like differential
equation, we give a new proof of the recipe theorem for the Tutte polynomial
for matroids. The solution of such an equation is in fact given by some
appropriate characters of the Hopf algebra of isomorphic classes of matroids,
characters which are then related to the Tutte polynomial for matroids. This
Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte
polynomial convolution formula appearing in W. Kook {\it et. al., J. Comb.
Series} {\bf B 76} (1999).Comment: 14 pages, 3 figure
Moderate Growth Time Series for Dynamic Combinatorics Modelisation
Here, we present a family of time series with a simple growth constraint.
This family can be the basis of a model to apply to emerging computation in
business and micro-economy where global functions can be expressed from local
rules. We explicit a double statistics on these series which allows to
establish a one-to-one correspondence between three other ballot-like
strunctures
- …