92 research outputs found
Beating the Generator-Enumeration Bound for -Group Isomorphism
We consider the group isomorphism problem: given two finite groups G and H
specified by their multiplication tables, decide if G cong H. For several
decades, the n^(log_p n + O(1)) generator-enumeration bound (where p is the
smallest prime dividing the order of the group) has been the best worst-case
result for general groups. In this work, we show the first improvement over the
generator-enumeration bound for p-groups, which are believed to be the hard
case of the group isomorphism problem. We start by giving a Turing reduction
from group isomorphism to n^((1 / 2) log_p n + O(1)) instances of p-group
composition-series isomorphism. By showing a Karp reduction from p-group
composition-series isomorphism to testing isomorphism of graphs of degree at
most p + O(1) and applying algorithms for testing isomorphism of graphs of
bounded degree, we obtain an n^(O(p)) time algorithm for p-group
composition-series isomorphism. Combining these two results yields an algorithm
for p-group isomorphism that takes at most n^((1 / 2) log_p n + O(p)) time.
This algorithm is faster than generator-enumeration when p is small and slower
when p is large. Choosing the faster algorithm based on p and n yields an upper
bound of n^((1 / 2 + o(1)) log n) for p-group isomorphism.Comment: 15 pages. This is an updated and improved version of the results for
p-groups in arXiv:1205.0642 and TR11-052 in ECC
Real Separated Algebraic Curves, Quadrature Domains, Ahlfors Type Functions and Operator Theory
The aim of this paper is to inter-relate several algebraic and analytic
objects, such as real-type algebraic curves, quadrature domains, functions on
them and rational matrix functions with special properties, and some objects
from Operator Theory, such as vector Toeplitz operators and subnormal
operators. Our tools come from operator theory, but some of our results have
purely algebraic formulation. We make use of Xia's theory of subnormal
operators and of the previous results by the author in this direction. We also
correct (in Section 5) some inaccuracies in two papers by the author in Revista
Matematica Iberoamericana (1998).Comment: 43 pages, 2 figures; zip archiv
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