Rota-Baxter Algebras and Dendriform Algebras

In this paper we study the adjoint functors between the category of Rota-Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota-Baxter algebras and then apply it to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota-Baxter algebras.Comment: Typos corrected and the last section on analog of Poincare-Birkhoff-Witt theorem deleted for a gap in the proo

Algebraic Birkhoff decomposition and its applications

Central in the Hopf algebra approach to the renormalization of perturbative quantum field theory of Connes and Kreimer is their Algebraic Birkhoff Decomposition. In this tutorial article, we introduce their decomposition and prove it by the Atkinson Factorization in Rota-Baxter algebra. We then give some applications of this decomposition in the study of divergent integrals and multiple zeta values.Comment: 39 pages. To appear in "Automorphic Forms and Langlands Program

Structure theorems of mixable shuffle algebras and free commutative Rota-Baxter algebras

We study the ring theoretical structures of mixable shuffle algebras and their associated free commutative Rota-Baxter algebras. For this study we utilize the connection of the mixable shuffle algebras with the overlapping shuffle algebra of Hazewinkel, quasi-shuffle algebras of Hoffman and quasi-symmetric functions. This connection allows us to apply methods and results on shuffle products and Lyndon words on ordered sets. As a result, we obtain structure theorems for a large class of mixable shuffle algebras and free commutative Rota-Baxter algebras with various coefficient rings.Comment: 30 pages. Corrected typos and improved presentatio

Efficient Representation and Encoding of Distributive Lattices

This thesis presents two new representations of distributive lattices with an eye towards efficiency in both time and space. Distributive lattices are a well-known class of partially-ordered sets having two natural operations called meet and join. Improving on all previous results, we develop an efficient data structure for distributive lattices that supports meet and join operations in O(log n) time, where n is the size of the lattice. The structure occupies O(n log n) bits of space, which is as compact as any known data structure and within a logarithmic factor of the information-theoretic lower bound by enumeration. The second representation is a bitstring encoding of a distributive lattice that uses approximately 1.26n bits. This is within a small constant factor of the best known upper and lower bounds for this problem. A lattice can be encoded or decoded in O(n log n) time

Succinct and Compact Data Structures for Intersection Graphs

This thesis designs space efficient data structures for several classes of intersection graphs, including interval graphs, path graphs and chordal graphs. Our goal is to support navigational operations such as adjacent and neighbourhood and distance operations such as distance efficiently while occupying optimal space, or a constant factor of the optimal space. Using our techniques, we first resolve an open problem with regards to succinctly representing ordinal trees that is able to convert between the index of a node in a depth-first traversal (i.e. pre-order) and in a breadth-first traversal (i.e. level-order) of the tree. Using this, we are able to augment previous succinct data structures for interval graphs with the \GDistance operation. We also study several variations of the data structure problem in interval graphs: beer interval graphs and dynamic interval graphs. In beer interval graphs, we are given that some vertices of the graph are beer nodes (representing beer stores) and we consider only those paths that pass through at least one of these beer nodes. We give data structure results and prove space lower bounds for these graphs. We study dynamic interval graphs under several well known dynamic models such as incremental or offline, and we give data structures for each of these models. Finally we consider path graphs where we improve on previous works by exploiting orthogonal range reporting data structures. For optimal space representations, we improve the run time of the queries, while for non-optimal space representations (but optimal query times), we reduce the space needed