(Non)Commutative Hopf algebras of trees and (quasi)symmetric functions

The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes-Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson-Schwinger equations.Comment: For March 2006 CIRM conference "Renormalization and Galois theories

Combinatorics of Rooted Trees and Hopf Algebras

We begin by considering the graded vector space with a basis consisting of rooted trees, graded by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices. We define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the multiplicities associated with each operator. The symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this gives the Hopf algebra of Grossman and Larson. We show the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.Comment: 19 pages; final revision has minor corrections, slightly expanded sect. 4 and additional reference

Correlation Statistics of Irregular and Spiral Galaxies Mapped in HI

Several measures of galaxy size and mass obtained from the neutral hydrogen mapping of 70 dwarf irregular galaxies presented in Paper I (Hoffman et al. 1996) are compared statistically to those for the set of all available HI-mapped dwarfs and HI-mapped spirals distributed within the same spatial volume to investigate variations in Tully-Fisher relations and in surface densities as functions of galaxy size and luminosity or mass. Some ambiguities due to the ``non-commutativity'' of the correlations among the variables are addressed and linear regressions of logarithms of blue luminosity, HI and optical radii, velocity profile half-width incorporating rotation and random motions, HI mass, and indicative dynamical mass are presented and analyzed. The surface density of HI is almost constant along the sequence of size/mass/luminosity while surface density of blue luminosity increases with galaxy size. For quantities not involving HI we find no evidence for a ``break'' between dwarfs and spirals, but we do find some curvature in velocity vs. radius and in the Tully-Fisher relation. There is an indication for a difference in the correlations involving HI mass or radius between dwarfs alone and spirals alone, in the sense that irregulars have somewhat more HI mass or slightly larger HI radii than spirals at a given blue luminosity, optical radius, or velocity profile width.Comment: AASTeX, to appear in ApJ, 26 pages + 3 tables + 12 figure

Political and Public Finance Motives for Tariffs

Governments face many constraints when making taxation decisions, including revenue needs, political objectives, and administrative capacities. Tariffs have an appealing combination of features for politicians: they provide a stream of revenue that is easy to collect, as well as satisfying political objectives in import-competing industries. This paper describes the tax structure governments choose when they are not purely benevolent. In the model the government must finance a stream of public expenditures while simultaneously seeking campaign contributions to maximize political support. The predictions of the model are consistent with observed taxation decisions in developing and industrialized countries.tariffs, political economy, development, tax regimes