Abstract. We describe the Multiple Solutions Method, a one-dimensional sampling of the six-dimensional orbital confidence region that is widely applicable in the field of asteroid orbit determination. In many situations there is one predominant direction of uncertainty in an orbit determination or orbital prediction, i.e., a "weak" direction. The idea is to record Multiple Solutions by following this, typically curved, weak direction, or Line Of Variations (LOV). In this paper we describe the method and give new insights into the mathematics behind this tool. We pay particular attention to the problem of how to ensure that the coordinate systems are properly scaled so that the weak direction really reflects the intrinsic direction of greatest uncertainty. We also describe how the multiple solutions can be used even in the absence of a nominal orbit solution, which substantially broadens the realm of applications. There are numerous applications for multiple solutions; we discuss a few problems in asteroid orbit determination and prediction where we have had good success with the method. In particular, we show that multiple solutions can be used effectively for potential impact monitoring, preliminary orbit determination, asteroid identification, and for the recovery of lost asteroids.

No abstract

No abstract

We study the Newton-Hooke groups in (2 + 1)-dimensions. A complete classification of both classical and quantum elementary systems is achieved by explicit computation of coadjoint orbits and unitary irreducible representations of the extended (by central extensions) Newton-Hooke groups. In addition, we show the quantization à la Moyal of a classical system using the Stratonovich-Weyl correspondence. An analysis of the origin of the "exotic" central extension, which only appears in (2 + 1)-dimensions, is presented.

We develop a systematic method to construct induced representations of quantum algebras. The procedure makes use of two Hopf algebras with a nondegenerate pairing and a pair of dual bases for them. We apply the method on three different quantum deformations of the Galilei algebra in (1+1) dimensions. We obtain several families of induced representations including some results already known.