The singular chain complex of the little 2-cubes operad defines an operad in simplicial cocommutative coalgebras. This operad has a dual structure, formed by a cooperad in cosimplicial commutative algebras, which defines model of the prounipotent completion of the topological operad of little

If C is a cocommutative coalgebra, a bialgebra structure can be given to the symmetric algebra S(C). The symmetric product is twisted by a Laplace pairing and the twisted product of any number of elements of S(C) is calculated explicitly. This is used to recover important identities in the quantum field theory of interacting scalar bosons.

We define a new cyclic module, dual to the Connes-Moscovici cyclic module, for Hopf algebras, and give a characteristric map for coactions of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.

Suitable normalization of time{homogeneous rotation{invariant random walks on unit spheres S d R d+1 for d > 2 leads to a central limit theorem with a Gaussian limit measure. This paper is devoted to an associated Edgeworth expansion with respect to the total variation norm. This strong type of convergence is diierent from the classical case. The proof is performed in the more general setting o...