Expansion formulas. II. Variations on a theme

AbstractIn “Expansion Formulas, I” [S. A. Joni, J. Math. Anal. Appl. 81 (1981)], it was shown that the Steffensen formula or polynomial sequences of binomial type gives rise to a method for generating a certain class of expansion identities. Special cases of this class of identities were studied by Carlitz [SIAM J. Appl. Math. 26 (1974), 431–436; 8 (1977), 320–336]. Since the Umbral calculus for polynomial sequences of binomial type has been generalized to encompass the theories of composition sequences [A. M. Garsia and S. A. Joni, Comm. Algebra, in press] and factor sequences [S. Roman and G.-C. Rota, Adv. in Math. 27 (1978), 95–188], we herein extend the results of part I to these two more general settings

Exponential renormalization

Moving beyond the classical additive and multiplicative approaches, we present an "exponential" method for perturbative renormalization. Using Dyson's identity for Green's functions as well as the link between the Faa di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure for renormalization scheme maps with the Rota-Baxter property. To our best knowledge, although very natural from group-theoretical and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method, let us mention the notions of counterfactors and of order n bare coupling constants).Comment: revised version; accepted for publication in Annales Henri Poincar

From sets to functions: Three elementary examples

A sequence of binomial type is a basis for [x] satisfying a binomial-like identity, e.g. powers, rising and falling factorials. Given two sequences of binomial type, the authors describe a totally combinatorial way of finding the change of basis matrix: to each pair of sequences is associated a poset whose Whitney numbers of the 1st and 2nd kind give the entries of the matrix and its inverse.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24525/1/0000804.pd

Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.