2,333 research outputs found
(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
Quasi-shuffle products revisited
Quasi-shuffle products, introduced by the first author, have been useful in
studying multiple zeta values and some of their analogues and generalizations.
The second author, together with Kajikawa, Ohno, and Okuda, significantly
extended the definition of quasi-shuffle algebras so it could be applied to
multiple zeta q-values. This article extends some of the algebraic machinery of
the first author's original paper to the more general definition, and uses this
extension to obtain various algebraic formulas in the quasi-shuffle algebra in
a transparent way. Some applications to multiple zeta values, interpolated
multiple zeta values, multiple q-zeta values, and multiple polylogarithms are
given.Comment: This is an extensively revised and expanded version of the Max Planck
Institute preprint (MPIM 2012-16) with the same title. 27 Oct 16: minor
revision and corrections 3 Apr 17: additional revision and correction
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
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