We present preliminary results from an observational campaign aimed at the study of the binary fraction and binary radial distribution in Galactic globular clusters. In particular, we concentrate on the ongoing observational campaign for the search of spectroscopic binaries.

The orbital fits of multi-planetary systems from radial velocity data has proved to be a complex task. In some cases, different orbital solutions provide similarly good fits, especially when two planets are near mean-motion resonances. Ferraz-Mello et al (2005) and Goździewski et al (2005) showed that the published best fits of systems HD82932 and HD160691 are dynamically unstable, and re-determined their orbital parameters with Monte Carlo and genetic algorithms. In both cases dynamically stable orbits were found with RMS similar to the published orbits. It was also shown that uncertainties in the stellar mass Ferraz Mello et al (2005) and the stellar jitter Gozdziewski et al (2005) can significantly affect the orbital determination. Ford (2005) used a Markov chain Monte Carlo technique to quantify the orbit uncertainties. For some planetary systems he found a strong correlation between the orbital elements and/or significant non-Gaussian error distribution in the parameter space. As a consequence, the actual uncertainties in the orbital fits can be much larger (or smaller) than those published.

The electronic structure of dislocations in GaN is controversial. Several experimental techniques such as carrier mobility studies and cathodoluminescence experiments have indicated that dislocations are charged while theoretical studies point to intrinsic states and/or point defect accumulation along the core as a source of electrical activity. Electron Energy Loss Spectroscopy (EELS) studies have the ability to probe the electronic structure of extended defects. Here we report rst principles calculations of the EELS spectrum applied to edge dislocations in GaN. It is found that the electrostatic potential at N atoms in the vicinity of the dislocation varies by the order of a volt and casts doubt on any simple interpretation of core loss spectroscopy. On the other hand, low loss spectroscopy leads directly to detailed information about any gap states. The low loss spectrum obtained by the theory is in good agreement with recent experimental work and indicates that threading dislocations in p-type GaN possess acceptor levels in the upper half of the gap.

Research in Celestial Mechanics, for the past three years, has mainly focused on the understanding of Chaos on all its aspects. The always larger number of potential applications (meteors, KBO, NEA, asteroids of the main belt but also exoplanets or galactic motions) and the development of new efficient tools, like the symplectic integrators, have allowed the passage from QUALITATIVE models (for example the transfer mechanisms) to real QUANTITATIVE results (like the calculation of lifetimes). This important step has contributed to (re)create collaborations between theoreticians and observers (for example, in the prediction of catastrophic impacts) and to situate the Celestial Mechanics in a wider scientific context.

We present density–functional theory studies for a variety of surfaces and extended defects in GaN. According to previous theoretical studies1 {100} type surfaces are electrically inactive. They play an important role in GaN since similar configurations occur at open–core screw dislocations and nanopipes as well as at the core of threading edge dislocations. Domain boundaries are found to consist of four–fold coordinated atoms and are also found to be electrically inactive. Thus, except for full–core screw dislocations which possess heavily strained bonds all investigated extended defects do not induce deep states into the band–gap. However, electrically active impurities in particular gallium vacancies and oxygen related defect complexes are found to be trapped at the stress field of the extended defects.

We present density-functional theory studies for a variety of surfaces and extended defects in GaN. According to previous theoretical studies {1010} type surfaces are electrically inactive. They play an important role in GaN since similar configurations occur at open-core screw dislocations and nanopipes as well as at the core of threading edge dislocations. Domain boundaries are found to consist of four-fold coordinated atoms and are also found to be electrically inactive. Thus, except for full-core screw dislocations which possess heavily strained bonds all investigated extended defects do not induce deep states into the band-gap. However, electrically active impurities in particular gallium vacancies and oxygen related defect complexes are found to be trapped at the stress field of the extended defects.

The generalized binomial function can be obtained as the solution of the equation y = 1 +zyα which satisfies y(0) = 1 where α ≠ 1 is assumed to be real and positive. The technique of Lagrange inversion can be used to express as a series which converges for |z| < α-α|a — l|α-1. We obtain a representation of the function as a contour integral and show that if α > 1 it is an analytic function in the complex z plane cut along the nonnegative real axis. For 0 < α < 1 the region of analyticity is the sector |arg(—z)| < απ. In either case defined by the series can be continued beyond the circle of convergenece of the series through a functional equation which can be derived from the integral representation.

In this paper we consider the groups G = G(α, n) defined by the presentations . We derive a formula for [G′: ″] and determine the order of G whenever n ≦ 7. We show that G is a finite soluble group if n is odd, but that G can be infinite when n is even, n ≧ 8. We also show that G(6, 10) is a finite insoluble group involving PSU(3, 4), and that the group H with presentation is a finite group of deficiency zero of order at least 114,967,210,176,000.