(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

Flexible foam erectable space structures Patent

Self-erectable space structures of flexible foam for application in planetary orbit

Faith Forming Faith, Faith Shaping Ministry

Here are these three different gifts of our Lutheran theological and liturgical tradition that help us bear baptismal fruit that will last, once we’re out of the font: Luther’s baptismal theology of a daily dying and rising. The weekly rhythm of the assembly gathering around Word and Sacrament. The wonder and mystery that is the liturgical year. Each of these gifts in their own way urge and equip us to get beyond the waters of an individualized baptismal security tank and into the world to serve, the very place to which the tide of our baptismal waters is meant to carry us