On the zeros of the Abelian integrals for a class of Liénard systems

A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this paper, we investigated a class of Liénard systems of the form x'=y, y'=f(x)+y g(x) with deg f=5 and deg g=4. We proved that the related elliptic integrals of the Liénard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3.

A multiliteracies project in the middle school : Parents as coteachers

This paper examines the experiences of one middle years’ English and Studies of Society and Environment (SoSE) teacher who adopted a multiliteracies project-based orientation to a unit on War and Refugees. It details the multiliteracies teaching and learning cycle, which is based on four non-hierarchical, pedagogical orientations: situated practice, overt instruction, critical framing and transformed practice (New London Group, 2000; Kalantzis & Cope, 2005a). Following the work of Kalantzis and Cope (2005a), it draws out the knowledge processes exacted in each of these four phases: experiencing the known and the new; conceptualising by naming and theorising; analysing functionally and critically; and, applying appropriately and creatively. Two parents were invited to enter the study as coteachers with the teacher and researcher. Using Bourdieu’s (1992) construct of capital, the findings report on how the multiliteracies approach enabled them to engage in school-based literacy practices differently than they had done previously in classrooms. An unexpected finding concerns the teacher’s altered view about how his role and status were perceived by the parents.

The middle years learner

This guide offers a systematic overview of the philosophy, principles, and issues in middle schooling, including contributions from academics and school-based practitioners on intellectual and emotional development in early adolescence, pedagogy, curricula, and the assessment of middle years students.

Middle years teacher education : new programs and research directions

Teacher education programs focussing on the development of specialist teachers for 'the middle years' have proliferated in Australian universities in recent years. This paper provides some insights into middle years' teacher education programs at the University of Queensland, Edith Cowan and Flinders Universities with regard to their: philosophical underpinnings; specific educational context; scope and nature of the program. In addition, some of the research directions and efficacy strategies utilised in conjunction with the programs will be shared, along with some early findings from a longitudinal study in one of the programs. We propose that the pattern of programmatic growth heralds a new time for teacher education, and we speculate about the production of new kinds of teacher identities as graduates take their place in the profession.