6,864 research outputs found
Regions of linearity, Lusztig cones and canonical basis elements for the quantized enveloping algebra of type A_4
Let U_q be the quantum group associated to a Lie algebra g of rank n. The
negative part U^- of U has a canonical basis B with favourable properties,
introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig
lead to a set of alternative parametrizations of the canonical basis, one for
each reduced expression for the longest word in the Weyl group of g. We show
that if g is of type A_4 there are close relationships between the Lusztig
cones, canonical basis elements and the regions of linearity of
reparametrization functions arising from the above parametrizations. A graph
can be defined on the set of simplicial regions of linearity with respect to
adjacency, and we further show that this graph is isomorphic to the graph with
vertices given by the reduced expressions of the longest word of the Weyl group
modulo commutation and edges given by long braid relations.
Keywords: Quantum group, Lie algebra, Canonical basis, Tight monomials, Weyl
group, Piecewise-linear functions.Comment: 61 pages, 17 figures, uses picte
How Good are Genetic Algorithms at Finding Large Cliques: An Experimental Study
This paper investigates the power of genetic algorithms at solving the MAX-CLIQUE problem. We measure the performance of a standard genetic algorithm on an elementary set of problem instances consisting of embedded cliques in random graphs. We indicate the need for improvement, and introduce a new genetic algorithm, the multi-phase annealed GA, which exhibits superior performance on the same problem set.
As we scale up the problem size and test on \hard" benchmark instances, we notice a
degraded performance in the algorithm caused by premature convergence to local minima. To alleviate this problem, a sequence of modi cations are implemented ranging from changes in input representation to systematic local search. The most recent version, called union GA, incorporates the features of union cross-over, greedy replacement, and diversity enhancement. It shows a marked speed-up in the number of iterations required to find a given solution, as well as some improvement in the clique size found.
We discuss issues related to the SIMD implementation of the genetic algorithms on a Thinking Machines CM-5, which was necessitated by the intrinsically high time complexity (O(n3)) of the serial algorithm for computing one iteration.
Our preliminary conclusions are: (1) a genetic algorithm needs to be heavily customized to work "well" for the clique problem; (2) a GA is computationally very expensive, and its use is only recommended if it is known to find larger cliques than other algorithms; (3) although our customization e ort is bringing forth continued improvements, there is no clear evidence, at this time, that a GA will have better success in circumventing local minima.NSF (CCR-9204284
On Scalable Particle Markov Chain Monte Carlo
Particle Markov Chain Monte Carlo (PMCMC) is a general approach to carry out
Bayesian inference in non-linear and non-Gaussian state space models. Our
article shows how to scale up PMCMC in terms of the number of observations and
parameters by expressing the target density of the PMCMC in terms of the basic
uniform or standard normal random numbers, instead of the particles, used in
the sequential Monte Carlo algorithm. Parameters that can be drawn efficiently
conditional on the particles are generated by particle Gibbs. All the other
parameters are drawn by conditioning on the basic uniform or standard normal
random variables; e.g. parameters that are highly correlated with the states,
or parameters whose generation is expensive when conditioning on the states.
The performance of this hybrid sampler is investigated empirically by applying
it to univariate and multivariate stochastic volatility models having both a
large number of parameters and a large number of latent states and shows that
it is much more efficient than competing PMCMC methods. We also show that the
proposed hybrid sampler is ergodic
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