284 research outputs found
Parametrization of Pythagorean triples by a single triple of polynomials
It is well known that Pythagorean triples can be parametrized by two triples
of polynomials with integer coefficients. We show that no single triple of
polynomials with integer coefficients in any number of variables is sufficient,
but that there exists a parametrization of Pythagorean triples by a single
triple of integer-valued polynomials.Comment: to appear in J. Pure Appl. Algebr
Improved sphere packing lower bounds from Hurwitz lattices
In this paper we prove an asymptotic lower bound for the sphere packing
density in dimensions divisible by four. This asymptotic lower bound improves
on previous asymptotic bounds by a constant factor and improves not just lower
bounds for the sphere packing density, but also for the lattice sphere packing
density and, in fact, the Hurwitz lattice sphere packing density.Comment: 12 page
Discrepancy-based error estimates for Quasi-Monte Carlo. I: General formalism
We show how information on the uniformity properties of a point set employed
in numerical multidimensional integration can be used to improve the error
estimate over the usual Monte Carlo one. We introduce a new measure of
(non-)uniformity for point sets, and derive explicit expressions for the
various entities that enter in such an improved error estimate. The use of
Feynman diagrams provides a transparent and straightforward way to compute this
improved error estimate.Comment: 23 pages, uses axodraw.sty, available at
ftp://nikhefh.nikhef.nl/pub/form/axodraw Fixed some typos, tidied up section
3.
Critical points and supersymmetric vacua, III: String/M models
A fundamental problem in contemporary string/M theory is to count the number
of inequivalent vacua satisfying constraints in a string theory model. This
article contains the first rigorous results on the number and distribution of
supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau
3-fold with flux. In particular, complete proofs of the counting formulas
in Ashok-Douglas and Denef-Douglas are given, together with van der Corput
style remainder estimates. We also give evidence that the number of vacua
satisfying the tadpole constraint in regions of bounded curvature in moduli
space is of exponential growth in .Comment: Final revision for publication in Commun. Math. Phys. Minor
corrections and editorial change
Sequential Quasi-Monte Carlo
We derive and study SQMC (Sequential Quasi-Monte Carlo), a class of
algorithms obtained by introducing QMC point sets in particle filtering. SQMC
is related to, and may be seen as an extension of, the array-RQMC algorithm of
L'Ecuyer et al. (2006). The complexity of SQMC is , where is
the number of simulations at each iteration, and its error rate is smaller than
the Monte Carlo rate . The only requirement to implement SQMC is
the ability to write the simulation of particle given as a
deterministic function of and a fixed number of uniform variates.
We show that SQMC is amenable to the same extensions as standard SMC, such as
forward smoothing, backward smoothing, unbiased likelihood evaluation, and so
on. In particular, SQMC may replace SMC within a PMCMC (particle Markov chain
Monte Carlo) algorithm. We establish several convergence results. We provide
numerical evidence that SQMC may significantly outperform SMC in practical
scenarios.Comment: 55 pages, 10 figures (final version
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