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

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

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

Efficient generation of energetic ions in multi-ion plasmas by radio-frequency heating

We describe a new technique for the efficient generation of high-energy ions with electromagnetic ion cyclotron waves in multi-ion plasmas. The discussed ‘three-ion’ scenarios are especially suited for strong wave absorption by a very low number of resonant ions. To observe this effect, the plasma composition has to be properly adjusted, as prescribed by theory. We demonstrate the potential of the method on the world-largest plasma magnetic confinement device, JET (Joint European Torus, Culham, UK), and the high-magnetic-field tokamak Alcator C-Mod (Cambridge, USA). The obtained results demonstrate efficient acceleration of 3He ions to high energies in dedicated hydrogen–deuterium mixtures. Simultaneously, effective plasma heating is observed, as a result of the slowing-down of the fast 3He ions. The developed technique is not only limited to laboratory plasmas, but can also be applied to explain observations of energetic ions in space-plasma environments, in particular, 3He-rich solar flares.This paper is dedicated to the late P. E. M. Vandenplas, founder and first director of LPP-ERM/KMS, in recognition of his lifelong outstanding commitment to fusion research, in particular to ICRH. The support from the JET and Alcator C-Mod Teams is warmly acknowledged. We are grateful to A. Cardinali, C. Castaldo, R. Dumont, J. Eriksson, T. Fülöp, C. Giroud, C. Hellesen, S. Menmuir and M. Schneider for fruitful discussions. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 under grant agreement no. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work was also supported by the US DoE, Office of Science, Office of Fusion Energy Sciences, SciDAC Center for Simulation of Wave Plasma Interactions under DE-FC02-01ER54648 and the User Facility Alcator C-Mod under DE-FC02-99ER54512. The Alcator C-Mod Team author list is reproduced from ref. 12. The JET Contributors author list is reproduced from ref. 33.Peer ReviewedPostprint (author's final draft