A mapping approach to synchronization in the "Zajfman trap". II: the observed bunch

We extend a recently introduced mapping model, which explains the bunching phenomenon in an ion beam resonator for two ions [Geyer, Tannor, J. Phys. B 37 (2004) 73], to describe the dynamics of the whole ion bunch. We calculate the time delay of the ions from a model of the bunch geometry and find that the bunch takes on a spherical form at the turning points in the electrostatic mirrors. From this condition we derive how the observed bunch length depends on the experimental parameters. We give an interpretation of the criteria for the existence of the bunch, which were derived from the experimental observations by Pedersen et al [Pedersen etal, Phys. Rev. A 65 042704].Comment: 25 pages, 6 figures; added new section 5 and clarified text; submitted to J. Phys.

Fedosov supermanifolds: II. Normal coordinates

The study of recently introduced Fedosov supermanifolds is continued. Using normal coordinates, properties of even and odd symplectic supermanifolds endowed with a symmetric connection respecting given sympletic structure are studied.Comment: 12 pages, Late

Monte Carlo likelihood inference for missing data models

We describe a Monte Carlo method to approximate the maximum likelihood estimate (MLE), when there are missing data and the observed data likelihood is not available in closed form. This method uses simulated missing data that are independent and identically distributed and independent of the observed data. Our Monte Carlo approximation to the MLE is a consistent and asymptotically normal estimate of the minimizer $\theta^*$ of the Kullback--Leibler information, as both Monte Carlo and observed data sample sizes go to infinity simultaneously. Plug-in estimates of the asymptotic variance are provided for constructing confidence regions for $\theta^*$. We give Logit--Normal generalized linear mixed model examples, calculated using an R package.Comment: Published at http://dx.doi.org/10.1214/009053606000001389 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Surreal Time and Ultratasks

This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible