13,932 research outputs found
Modular symbols and Hecke operators
We survey techniques to compute the action of the Hecke operators on the
cohomology of arithmetic groups. These techniques can be seen as
generalizations in different directions of the classical modular symbol
algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in
papers of the author and the author with Mark McConnell. Some results are
unpublished work of Mark McConnell and Robert MacPherson.Comment: 11 pp, 2 figures, uses psfrag.st
Below All Subsets for Some Permutational Counting Problems
We show that the two problems of computing the permanent of an
matrix of -bit integers and counting the number of
Hamiltonian cycles in a directed -vertex multigraph with
edges can be reduced to relatively
few smaller instances of themselves. In effect we derive the first
deterministic algorithms for these two problems that run in time in
the worst case. Classic time algorithms for the two
problems have been known since the early 1960's. Our algorithms run in
time.Comment: Corrected several technical errors, added comment on how to use the
algorithm for ATSP, and changed title slightly to a more adequate on
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
A linear time algorithm for the orbit problem over cyclic groups
The orbit problem is at the heart of symmetry reduction methods for model
checking concurrent systems. It asks whether two given configurations in a
concurrent system (represented as finite strings over some finite alphabet) are
in the same orbit with respect to a given finite permutation group (represented
by their generators) acting on this set of configurations by permuting indices.
It is known that the problem is in general as hard as the graph isomorphism
problem, whose precise complexity (whether it is solvable in polynomial-time)
is a long-standing open problem. In this paper, we consider the restriction of
the orbit problem when the permutation group is cyclic (i.e. generated by a
single permutation), an important restriction of the problem. It is known that
this subproblem is solvable in polynomial-time. Our main result is a
linear-time algorithm for this subproblem.Comment: Accepted in Acta Informatica in Nov 201
- …