31,636 research outputs found
The importance of the Selberg integral
It has been remarked that a fair measure of the impact of Atle Selberg's work
is the number of mathematical terms which bear his name. One of these is the
Selberg integral, an n-dimensional generalization of the Euler beta integral.
We trace its sudden rise to prominence, initiated by a question to Selberg from
Enrico Bombieri, more than thirty years after publication. In quick succession
the Selberg integral was used to prove an outstanding conjecture in random
matrix theory, and cases of the Macdonald conjectures. It further initiated the
study of q-analogues, which in turn enriched the Macdonald conjectures. We
review these developments and proceed to exhibit the sustained prominence of
the Selberg integral, evidenced by its central role in random matrix theory,
Calogero-Sutherland quantum many body systems, Knizhnik-Zamolodchikov
equations, and multivariable orthogonal polynomial theory.Comment: 43 page
Generalizing Boolean Satisfiability II: Theory
This is the second of three planned papers describing ZAP, a satisfiability
engine that substantially generalizes existing tools while retaining the
performance characteristics of modern high performance solvers. The fundamental
idea underlying ZAP is that many problems passed to such engines contain rich
internal structure that is obscured by the Boolean representation used; our
goal is to define a representation in which this structure is apparent and can
easily be exploited to improve computational performance. This paper presents
the theoretical basis for the ideas underlying ZAP, arguing that existing ideas
in this area exploit a single, recurring structure in that multiple database
axioms can be obtained by operating on a single axiom using a subgroup of the
group of permutations on the literals in the problem. We argue that the group
structure precisely captures the general structure at which earlier approaches
hinted, and give numerous examples of its use. We go on to extend the
Davis-Putnam-Logemann-Loveland inference procedure to this broader setting, and
show that earlier computational improvements are either subsumed or left intact
by the new method. The third paper in this series discusses ZAPs implementation
and presents experimental performance results
One-Loop Gauge Theory Amplitudes in N=4 Super Yang-Mills from MHV Vertices
We propose a new, twistor string theory inspired formalism to calculate loop
amplitudes in N=4 super Yang-Mills theory. In this approach, maximal helicity
violating (MHV) tree amplitudes of N=4 super Yang-Mills are used as vertices,
using an off-shell prescription introduced by Cachazo, Svrcek and Witten, and
combined into effective diagrams that incorporate large numbers of conventional
Feynman diagrams. As an example, we apply this formalism to the particular
class of MHV one-loop scattering amplitudes with an arbitrary number of
external legs in N=4 super Yang-Mills. Remarkably, our approach naturally leads
to a representation of the amplitudes as dispersion integrals, which we
evaluate exactly. This yields a new, simplified form for the MHV amplitudes,
which is equivalent to the expressions obtained previously by Bern, Dixon,
Dunbar and Kosower using the cut-constructibility approach.Comment: Latex, 35 pages, 3 figures. v2: remarks on gauge invariance added.
Published version to appear in Nuclear Physics
Short expressions of permutations as products and cryptanalysis of the Algebraic Eraser
On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the
\emph{Algebraic Eraser} scheme for key agreement over an insecure channel,
using a novel hybrid of infinite and finite noncommutative groups. They also
introduced the \emph{Colored Burau Key Agreement Protocol (CBKAP)}, a concrete
realization of this scheme.
We present general, efficient heuristic algorithms, which extract the shared
key out of the public information provided by CBKAP. These algorithms are,
according to heuristic reasoning and according to massive experiments,
successful for all sizes of the security parameters, assuming that the keys are
chosen with standard distributions.
Our methods come from probabilistic group theory (permutation group actions
and expander graphs). In particular, we provide a simple algorithm for finding
short expressions of permutations in , as products of given random
permutations. Heuristically, our algorithm gives expressions of length
, in time and space . Moreover, this is provable from
\emph{the Minimal Cycle Conjecture}, a simply stated hypothesis concerning the
uniform distribution on . Experiments show that the constants in these
estimations are small. This is the first practical algorithm for this problem
for .
Remark: \emph{Algebraic Eraser} is a trademark of SecureRF. The variant of
CBKAP actually implemented by SecureRF uses proprietary distributions, and thus
our results do not imply its vulnerability. See also arXiv:abs/12020598Comment: Final version, accepted to Advances in Applied Mathematics. Title
slightly change
Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices
In a recent work Killip and Nenciu gave random recurrences for the
characteristic polynomials of certain unitary and real orthogonal upper
Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are
beta-generalizations of the classical groups. Left open was the direct
calculation of certain Jacobians. We provide the sought direct calculation.
Furthermore, we show how a multiplicative rank 1 perturbation of the unitary
Hessenberg matrices provides a joint eigenvalue p.d.f generalizing the circular
beta-ensemble, and we show how this joint density is related to known
inter-relations between circular ensembles. Projecting the joint density onto
the real line leads to the derivation of a random three-term recurrence for
polynomials with zeros distributed according to the circular Jacobi
beta-ensemble.Comment: 23 page
The orbifold transform and its applications
We discuss the notion of the orbifold transform, and illustrate it on simple
examples. The basic properties of the transform are presented, including
transitivity and the exponential formula for symmetric products. The connection
with the theory of permutation orbifolds is addressed, and the general results
illustrated on the example of torus partition functions
Finite Form of the Quintuple Product Identity
The celebrated quintuple product identity follows surprisingly from an
almost-trivial algebraic identity, which is the limiting case of the
terminating q-Dixon formula.Comment: 1 pag
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