4,517 research outputs found
An Efficient Quantum Algorithm for some Instances of the Group Isomorphism Problem
In this paper we consider the problem of testing whether two finite groups
are isomorphic. Whereas the case where both groups are abelian is well
understood and can be solved efficiently, very little is known about the
complexity of isomorphism testing for nonabelian groups. Le Gall has
constructed an efficient classical algorithm for a class of groups
corresponding to one of the most natural ways of constructing nonabelian groups
from abelian groups: the groups that are extensions of an abelian group by
a cyclic group with the order of coprime with . More precisely,
the running time of that algorithm is almost linear in the order of the input
groups. In this paper we present a quantum algorithm solving the same problem
in time polynomial in the logarithm of the order of the input groups. This
algorithm works in the black-box setting and is the first quantum algorithm
solving instances of the nonabelian group isomorphism problem exponentially
faster than the best known classical algorithms.Comment: 20 pages; this is the full version of a paper that will appear in the
Proceedings of the 27th International Symposium on Theoretical Aspects of
Computer Science (STACS 2010
Stable Flags and the Riemann-Hilbert Problem
We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise
logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We
show that the solutions of the Riemann-Hilbert problem are in bijection with
some families of local filtrations which are stable under the prescribed
monodromy maps. We introduce the notion of Birkhoff-Grothendieck
trivialisation, and show that its computation corresponds to geodesic paths in
some local affine Bruhat-Tits building. We use this to compute how the type of
a bundle changes under stalk modifications, and give several corresponding
algorithmic procedures.Comment: 39 page
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
Effective inverse spectral problem for rational Lax matrices and applications
We reconstruct a rational Lax matrix of size R+1 from its spectral curve (the
desingularization of the characteristic polynomial) and some additional data.
Using a twisted Cauchy--like kernel (a bi-differential of bi-weight (1-nu,nu))
we provide a residue-formula for the entries of the Lax matrix in terms of
bases of dual differentials of weights nu and 1-nu respectively. All objects
are described in the most explicit terms using Theta functions. Via a sequence
of ``elementary twists'', we construct sequences of Lax matrices sharing the
same spectral curve and polar structure and related by conjugations by rational
matrices. Particular choices of elementary twists lead to construction of
sequences of Lax matrices related to finite--band recurrence relations (i.e.
difference operators) sharing the same shape. Recurrences of this kind are
satisfied by several types of orthogonal and biorthogonal polynomials. The
relevance of formulae obtained to the study of the large degree asymptotics for
these polynomials is indicated.Comment: 33 pages. Version 2 with added references suggested by the refere
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