Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters

International audienceThis article provides a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method presented here is based on an analytic expression for the integral, derived by use of Laplace transform and D-finite functions properties. The formula has the form of a product between an exponential term and a convergent power series with positive coefficients. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm. Another contribution is the derivation of analytic bounds on the probability of collision itself, allowing for a very fast and - in most cases - very precise evaluation of the risk. The only other analytical method of the literature - based on an approximation - is shown to be a special case of the new formula. A numerical study illustrates the efficiency of the proposed algorithms on a broad variety of examples and favorably compares the approach to the other methods of the literature

Decay of the Sinai Well in D dimensions

We study the decay law of the Sinai Well in $D$ dimensions and relate the behavior of the decay law to internal distributions that characterize the dynamics of the system. We show that the long time tail of the decay is algebraic ($1/t$), irrespective of the dimension $D$.Comment: 14 pages, Figures available under request. Revtex. Submitted to Phys. Rev. E.,e-mail: [email protected]

Heat kernel and number theory on NC-torus

The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the number-theoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit on a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing phenomenon for a scalar theory. Although we find non-local counterterms in the NC $\phi^4$ theory on \T^4, we show that this theory can be made renormalizable at least at one loop, and may be even beyond

Degeneracy in infinite horizon optimization

We consider sequential decision problems over an infinite horizon. The forecast or solution horizon approach to solving such problems requires that the optimal initial decision be unique. We show that multiple optimal initial decisions can exist in general and refer to their existence as degeneracy. We then present a conceptual cost perturbation algorithm for resolving degeneracy and identifying a forecast horizon. We also present a general near-optimal forecast horizon.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47919/1/10107_2005_Article_BF01582295.pd