Symmetry Properties of a Generalized Korteweg-de Vires Equation and some Explicit Solutions

The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invarint solution for it are obtained by means of this technique. Polynomial, trigonometric and elliptic function solutions can be calculated. It is shown that this generalized equation can be reduced to a first-order equation under a particular second-order differential constraint which resembles a Schrodinger equation. For a particular instance in which the constraint is satisfied, the generalized equation is reduced to a quadrature. A condition which ensures that the reciprocal of a solution is also a solution is given, and a first integral to this constraint is found

Region Operators of Wigner Function: Transformations, Realizations and Bounds

An integral of the Wigner function of a wavefunction |psi >, over some region S in classical phase space is identified as a (quasi) probability measure (QPM) of S, and it can be expressed by the |psi > average of an operator referred to as the region operator (RO). Transformation theory is developed which provides the RO for various phase space regions such as point, line, segment, disk and rectangle, and where all those ROs are shown to be interconnected by completely positive trace increasing maps. The latter are realized by means of unitary operators in Fock space extended by 2D vector spaces, physically identified with finite dimensional systems. Bounds on QPMs for regions obtained by tiling with discs and rectangles are obtained by means of majorization theory.Comment: 16 pages, 4 figures. Hurst Bracken Festschrift, Reports of Mathematical Physics, Feb 2006, to appea

Discussing the Importance of Ontology and Epistemology Awareness in Practitioner Research.

This paper uses the focus of identity and acculturation within schools as the basis for a reflection on the ways in which researchers ground their investigations. It identifies the necessity for researchers to ensure that their own ontological perceptions, epistemological stances and methods for data gathering and interpretation are closely aligned. By investigating the ways in which a diversity of methodological approaches are used to address the issue of identity formation, as reflected in three school-based studies (Houlette et al, 2004; illbourn, 2006; Nasir et al, 2009), the paper facilitates a teasing out of ontological and epistemological issues. The practical implications are that, through a deeper awareness of the ontological substructures informing their studies, researchers will be more clearly positioned to iteratively reflect upon, and define how best to engage with, their research projects