Rational approximation and arithmetic progressions

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed

The motion of stars near the Galactic center: A comparison of the black hole and fermion ball scenarios

After a discussion of the properties of degenerate fermion balls, we analyze the orbits of the stars S0-1 and S0-2, which have the smallest projected distances to Sgr A*, in the supermassive black hole as well as in the fermion ball scenarios of the Galactic center. It is shown that both scenarios are consistent with the data, as measured during the last six years by Genzel et al. and Ghez et al. The free parameters of the projected orbit of a star are the unknown components of its velocity v_z and distance z to Sgr A* in 1995.4, with the z-axis being in the line of sight. We show, in the case of S0-1 and S0-2, that the z-v_z phase-space which fits the data, is much larger for the fermion ball than for the black hole scenario. Future measurements of the positions or radial velocities of S0-1 and S0-2 could reduce this allowed phase-space and eventually rule out one of the currently acceptable scenarios. This may shed some light into the nature of the supermassive compact dark object, or dark matter in general at the center of our Galaxy.Comment: 30 pages, 12 figures, Latex, aasms4 styl

Modelling of advanced three-ion ICRF heating and fast ion generation scheme for tokamaks and stellarators

Absorption of ion-cyclotron range of frequencies waves at the fundamental resonance is an efficient source of plasma heating and fast ion generation in tokamaks and stellarators. This heating method is planned to be exploited as a fast ion source in the Wendelstein 7-X stellarator. The work presented here assesses the possibility of using the newly developed three-ion species scheme (Kazakov et al (2015) Nucl. Fusion 55 032001) in tokamak and stellarator plasmas, which could offer the capability of generating more energetic ions than the traditional minority heating scheme with moderate input power. Using the SCENIC code, it is found that fast ions in the MeV range of energy can be produced in JET-like plasmas. The RF-induced particle pinch is seen to strongly impact the fast ion pressure profile in particular. Our results show that in typical high-density W7-X plasmas, the three-ion species scheme generates more energetic ions than the more traditional minority heating scheme, which makes three-ion scenario promising for fast-ion confinement studies in W7-X

Estimation of gestational age from fundal height: a solution for resource-poor settings

Many women in resource-poor settings lack access to reliable gestational age assessment because they do not know their last menstrual period; there is no ultrasound (US) and methods of newborn gestational age dating are not practised by birth attendants. A bespoke multiple-measures model was developed to predict the expected date of delivery determined by US. The results are compared with both a linear and a nonlinear model. Prospectively collected early US and serial symphysis-pubis fundal height (SFH) data were used in the models. The data were collected from Karen and Burmese women attending antenatal care on the Thai–Burmese border. The multiple-measures model performed best, resulting in a range of accuracy depending on the number of SFH measures recorded per mother (for example six SFH measurements resulted in a prediction accuracy of ±2 weeks). SFH remains the proxy for gestational age in much of the resource-poor world. While more accurate measures should be encouraged, we demonstrate that a formula that incorporates at least three SFH measures from an individual mother and the slopes between them provide a significant increase in the accuracy of prediction compared with the linear and nonlinear formulae also using multiple SFH measures