Oscar Arratia

Long term impact risk for (101955) 1999 RQ36

Abstract The potentially hazardous Asteroid (101955) 1999 RQ 36 has a possibility of colliding with the Earth in the latter half of the 22nd century, well beyond the traditional 100-year time horizon for routine impact monitoring. The probabilities accumulate to a total impact probability of approximately 10 - 3 , with a pair of closely related routes to impact in 2182 comprising more than half of the total. The analysis of impact possibilities so far in the future is strongly dependent on the action of the Yarkovsky effect, which raises new challenges in the careful assessment of longer term impact hazards. Even for asteroids with very precisely determined orbits, a future close approach to Earth can scatter the possible trajectories to the point that the problem becomes like that of a newly discovered asteroid with a weakly determined orbit. If the scattering takes place late enough so that the target plane uncertainty is dominated by Yarkovsky accelerations then the thermal properties of the asteroid, which are typically unknown, play a major role in the impact assessment. In contrast, if the strong planetary interaction takes place sooner, while the Yarkovsky dispersion is still relatively small compared to that derived from the measurements, then precise modeling of the nongravitational acceleration may be unnecessary.

BICROSSPRODUCT STRUCTURE OF THE NULL-PLANE QUANTUM POINCARE ALGEBRA

A nonlinear change of basis allows us to show that the non-standard quantum deformation of the (3 + 1) Poincare algebra has a bicrossproduct structure. Quantum universal R-matrix, Pauli - Lubanski and mass operators are presented in the new basis.

Contraction of Representations of 1+1 Kinematical Groups and Quantization

We study the contraction from the Poincare to the Galilei group in (1+1) dimensions. We apply these results to quantize, in the sense of Moyal, galilean and relativistic elementary systems using the method of the Stratonovich–Weyl Correspondence.

Representations of quantum bicrossproduct algebras

We present a method to construct induced representations of quantum algebras which have a bicrossproduct structure. We apply this procedure to some quantum kinematical algebras in (1 + 1) dimensions with this kind of structure: null-plane quantum Poincare algebra, non-standard quantum Galilei algebra and quantum κ-Galilei algebra.

Induced representations of quantum kinematical algebras and quantum mechanics

Unitary representations of kinematical symmetry groups of quantum systems are fundamental in quantum theory. In this paper we propose their generalization to quantum kinematical groups. Using the method, proposed by us in a recent paper (Arratia O and del Olmo M A 2001 Preprint math.QA/0110275) to induce representations of quantum bicrossproduct algebras, we construct the representations of the family of standard quantum inhomogeneous algebras Uλ(isoω(2)). This family contains the quantum Euclidean, Galilei and Poincare algebras, all of them in (1+1) dimensions. As byproducts we obtain the actions of these quantum algebras on regular co-spaces that are an algebraic generalization of the homogeneous spaces and q-Casimir equations which play the role of q-Schrodinger equations.

Induced representations of quantum (1+1) Galilei algebras

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.

Dynamical systems and quantum bicrossproduct algebras

We present a unified study of some aspects of quantum bicrossproduct algebras of inhomogeneous Lie algebras, such as Poincare, Galilei and Euclidean in N dimensions. The action associated with the bicrossproduct structure allows us to obtain a nonlinear action over a new group linked to the translations. This new nonlinear action associates a dynamical system with each generator which is the object of our study.

Moyal quantization on the cylinder

Abstract The Stratonovich-Weyl kernels for a wide class of phase spaces are known. In this paper we enlarge this class by solving the problem for the cylinder. No Ansatz is made at the beginning, rather a constructive point of view is adopted along the work.

Group Contractions and Stratonovich-Weyl Kernels

We study the Inonu-Wigner contractions of one dimensional groups and how kernels of Stratonovich-Weyl type, defined on coadjoint orbits of these groups, are transformed under contraction.

Single and multiple solution algorithms to scan asteroid databases for identifications

The process of cataloguing the minor planet population of the Solar System has experienced a great advance in the last decades with the start-up of several surveys. The large volume of data generated by them has increased with time and given rise to huge databases of asteroids with uneven qualities. In fact, a significant fraction of these objects have not been enough observed, thus leading to the computation of very poor quality orbits as to carry out useful predictions of the positions of such asteroids. As a result, some objects can get lost, which is particularly embarrassing for those with Earth crossing orbits. When this situation persists for a long time, the aforementioned databases end up contaminated in the sense that they contain more than one discovery for the same physical object and some kind of action must be taken. The algorithms for asteroid identifications are thought precisely to mitigate this problem and their design will depend upon the quality of the available data for the objects to be identified. In this paper we will distinguish two cases: when both objects have a nominal orbit and when one of them lacks it. In addition, when the available data poorly constrain the solution, other orbits in the neighbourhood of the nominal one are also compatible with the observations. Using these alternative orbits allows us to find many identifications that otherwise would be missed. Finally, we will show the efficiency of all these algorithms when applied to the datasets distributed by the Minor Planet Center.