5,236 research outputs found
Analytic aspects of the shuffle product
There exist very lucid explanations of the combinatorial origins of rational
and algebraic functions, in particular with respect to regular and context free
languages. In the search to understand how to extend these natural
correspondences, we find that the shuffle product models many key aspects of
D-finite generating functions, a class which contains algebraic. We consider
several different takes on the shuffle product, shuffle closure, and shuffle
grammars, and give explicit generating function consequences. In the process,
we define a grammar class that models D-finite generating functions
Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology
In this review we discuss the relevance of the Hochschild cohomology of
renormalization Hopf algebras for local quantum field theories and their
equations of motion.Comment: 29 pages, eps figures; minor changes; final versio
Efficient implementation of the Hardy-Ramanujan-Rademacher formula
We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to
allow the partition function to be computed with softly optimal
complexity and very little overhead. A new implementation
based on these techniques achieves speedups in excess of a factor 500 over
previously published software and has been used by the author to calculate
, an exponent twice as large as in previously reported
computations.
We also investigate performance for multi-evaluation of , where our
implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to
power series methods on far denser sets of indices than previous
implementations. As an application, we determine over 22 billion new
congruences for the partition function, extending Weaver's tabulation of 76,065
congruences.Comment: updated version containing an unconditional complexity proof;
accepted for publication in LMS Journal of Computation and Mathematic
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