148,498 research outputs found
The Hopf algebra of (q)multiple polylogarithms with non-positive arguments
We consider multiple polylogarithms in a single variable at non-positive
integers. Defining a connected graded Hopf algebra, we apply Connes' and
Kreimer's algebraic Birkhoff decomposition to renormalize multiple
polylogarithms at non-positive integer arguments, which satisfy the shuffle
relation. The q-analogue of this result is as well presented, and compared to
the classical case.Comment: some typos fixed, references update
The Shimura curve of discriminant 15 and topological automorphic forms
We find defining equations for the Shimura curve of discriminant 15 over
Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic
integers, as well as the higher cohomology. We apply this to calculate the
homotopy groups of a spectrum of "topological automorphic forms" associated to
this curve, as well as one associated to a quotient by an Atkin-Lehner
involution.Comment: 36 pages, 5 figures, updated with corrections and new introduction
(this version corrects image issues in the previous
Prime Labeling of Small Trees with Gaussian Integers
A graph on n vertices is said to admit a prime labeling if we can label its vertices with the first n natural numbers such that any two adjacent vertices have relatively prime labels. Here we extend the idea of prime labeling to the Gaussian integers, which are the complex numbers whose real and imaginary parts are both integers. We begin by defining an order on the Gaussian integers that lie in the first quadrant. Using this ordering, we show that all trees of order at most 72 admit a prime labeling with the Gaussian integers
- …