Dependence of volume FEL (VFEL) threshold conditions on undulator parameters

The idea and principles of volume free electron lasers were proposed in [1-4]. It was shown there that volume distributed feedback (VDFB) can essentially reduce the threshold current of generation and provide the possibility of smooth frequency tuning. The present work considers an undulator VFEL with multiwave VDFB. It is shown that dependence of threshold current on the interaction length changes in the point of roots degeneration. This leads to the sharp decrease of start current if condition of dynamical diffraction is fulfilled. The dependence of amplification coefficient changes, too. So the interaction length for generation appears shorter. The proposed scheme can be used for generation in wide spectral range from microwaves to X-rays. The operating features of undulator VFEL is considered.Comment: Latex, 6 pages with 2 Postscript figure

The rise and stability of the Earth's atmosphere

History and stability of earth atmospher

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of an m-component system of hydrodynamic type, u_t=V(u)u_x, by the generalized hodograph method requires the diagonalizability of the mxm matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains -- infinite-component systems of hydrodynamic type for which the infinite matrix V(u) is `sufficiently sparse'. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.Comment: 36 pages, the classification results and proofs are refined. A section on generating functions is adde

Primary School Teachers’ Constructions of Mathematics Attainment Differences: A Critical and Bioecological Exploration

There is a persistent gap between the mathematical attainment of children from vulnerable groups and their peers. This has a significant effect upon the access of children from disadvantaged backgrounds to educational and social opportunities both in childhood and into their adult lives. It also impacts upon their perceptions of their mathematics abilities. It is therefore important that educational psychologists seek to equalise opportunities for mathematical success, regardless of a child’s circumstances. Teachers’ perspectives surrounding the mathematics attainment gap not only impact upon how they interact with students; they can also directly affect students’ mathematics performance. Despite this, little research has been undertaken to explore what factors influence teachers’ constructions of attainment differences. While some studies have considered teachers’ mathematics attainment views as part of intervention evaluations or quantitative studies, there is little in-depth research considering the breadth and origin of their views. This is of importance to educational psychologists as teachers’ perspectives will affect their responsiveness to psychological approaches and interventions designed to reduce the mathematics attainment gap. In this research I present four case studies that explore the ways in which primary teachers conceptualise mathematics attainment differences and how this is influenced by their personal characteristics, contexts and experiences (bioecology). Completing four semi-structured interviews with each participant, I analysed these interviews to identify each teacher’s bioecological influences. I then critically examined their views around mathematics attainment differences to identify themes in their perspectives. Finally, these analyses were combined to consider how each teacher’s bioecology influenced their conceptualisations of mathematics attainment differences. Each of the teachers in this study presented different views surrounding the origins of mathematics attainment differences and how these differences should be approached. Exploration of their bioecology in relation to these views suggested there were multiple interconnected influences upon their perspectives. Teachers’ own experiences of learning mathematics at school and the impact of universal attainment expectations were consistently related to teachers’ views, although the type of influence they conferred was highly variable. As teachers’ views and influences were so varied, different psychological approaches and knowledge would be required when working with each teacher to reduce the mathematics attainment gap within their classes most effectively: one approach would be unlikely to fit all. The findings of this research suggest that deeper exploration of teacher perspectives can be supportive to understanding their views around mathematics attainment differences. Greater knowledge of teachers’ perspectives and influences may support educational psychologists to tailor their training and casework to address mathematical attainment differences more effectively. In addition, exploration of views and influences upon them allows both teachers and educational psychologists the time and space to critically reflect upon their own assumptions and practice. Future research with different teacher groups and demographics is suggested to broaden our understanding of how teachers form their mathematics attainment views. Further exploration of the importance of the wider educational context and teachers’ school experiences on their views and practices is also suggested