770 research outputs found
A GAP package for braid orbit computation, and applications
Let G be a finite group. By Riemann's Existence Theorem, braid orbits of
generating systems of G with product 1 correspond to irreducible families of
covers of the Riemann sphere with monodromy group G. Thus many problems on
algebraic curves require the computation of braid orbits. In this paper we
describe an implementation of this computation. We discuss several
applications, including the classification of irreducible families of
indecomposable rational functions with exceptional monodromy group
Solving equations in the relational algebra
Enumerating all solutions of a relational algebra equation is a natural and
powerful operation which, when added as a query language primitive to the
nested relational algebra, yields a query language for nested relational
databases, equivalent to the well-known powerset algebra. We study
\emph{sparse} equations, which are equations with at most polynomially many
solutions. We look at their complexity, and compare their expressive power with
that of similar notions in the powerset algebra.Comment: Minor revision, accepted for publication in SIAM Journal on Computin
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
Voicing Transformations and a Linear Representation of Uniform Triadic Transformations (Preprint name)
Motivated by analytical methods in mathematical music theory, we determine the structure of the subgroup of generated by the three voicing reflections. We determine the centralizer of in both and the monoid of affine transformations, and recover a Lewinian duality for trichords containing a generator of . We present a variety of musical examples, including Wagner's hexatonic Grail motive and the diatonic falling fifths as cyclic orbits, an elaboration of our earlier work with Satyendra on Schoenberg, String Quartet in minor, op. 7, and an affine musical map of Joseph Schillinger. Finally, we observe, perhaps unexpectedly, that the retrograde inversion enchaining operation RICH (for arbitrary 3-tuples) belongs to the setwise stabilizer in of root position triads. This allows a more economical description of a passage in Webern, Concerto for Nine Instruments, op. 24 in terms of a morphism of group actions. Some of the proofs are located in the Supplementary Material file, so that this main article can focus on the applications
Coordination and Learning with a Partial Language
This paper explores how efficiency promotes the use of structure in language. It starts from the premise that one of language’s central characteristics is to provide a means for saying novel things about novel circumstances, its creativity. It is reasonable to expect that in a rich and changing environment, language will be incomplete. This encourages reliance on structure. It is shown how creative language use emerges form common knowledge structures, even if those structures are consistent with an a priori absence of a common language. ZUSAMMENFASSUNG - (Koordination und Lernen mit einer Partialsprache) In diesem Beitrag wird die Anwendung von Strukturen in einer Sprache aus Effizienzsicht begründet. Der Artikel geht davon aus, daß eines der wichtigsten Merkmale der Sprache in ihrer Kreativität zu sehen ist, d. h. als Mittel, um Neues über neue Sachverhalte auszusagen. Es ist deshalb zu erwarten, daß in einer vielfältigen und sich verändernden Umwelt die Sprache unvollständig bleiben wird. Dies fördert die Anwendung von Strukturen. Es wird gezeigt, wie die kreative Sprachanwendung aus allgemeinen Wissensstrukturen entsteht, auch dann, wenn diese Strukturen a priori noch keine gemeinsame Sprache bilden.Language; Coordination; Optimal Learning; Common Knowledge
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