Universality of Low-Energy Scattering in 2+1 Dimensions: The Non Symmetric Case

For a very large class of potentials, $V(\vec{x})$, $\vec{x}\in R^2$, we prove the universality of the low energy scattering amplitude, $f(\vec{k}', \vec{k})$. The result is $f=\sqrt{\frac{\pi}{2}}\{1/log k)+O(1/(log k)^2)$. The only exceptions occur if $V$ happens to have a zero energy bound state. Our new result includes as a special subclass the case of rotationally symmetric potentials, $V(|\vec{x}|)$.Comment: 65 pages, Latex, significant changes, new sections and appendice

On the Nonlocal Equations and Nonlocal Charges Associated with the Harry Dym Hierarchy

A large class of nonlocal equations and nonlocal charges for the Harry Dym hierarchy is exhibited. They are obtained from nonlocal Casimirs associated with its bi-Hamiltonian structure. The Lax representation for some of these equations is also given.Comment: to appear in Journal of Mathematical Physics, 17 pages, Late

Deformed Harry Dym and Hunter-Zheng Equations

We study the deformed Harry Dym and Hunter-Zheng equations with two arbitrary deformation parameters. These reduce to various other known models in appropriate limits. We show that both these systems are bi-Hamiltonian with the same Hamiltonian structures. They are integrable and belong to the same hierarchy corresponding to positive and negative flows. We present the Lax pair description for both the systems and construct the conserved charges of negative order from the Lax operator. For the deformed Harry Dym equation, we construct the non-standard Lax representation for two special classes of values of the deformation parameters. In general, we argue that a non-standard description will involve a pseudo-differential operator of infinite order.Comment: Latex file, 15 page

Living in Plurilingual Spaces: Self-Study, Critical Friendship, and the Plurality of Publics

In this article we engage in collaborative self-study through a critical friendship that is specifically designed to evoke our personal histories in relation to how we approach our practices as teacher educators. In particular, we focus on understanding the conceptual origins of our pedagogies of teacher education and how our identities as teacher educators were shaped, and continue to be shaped, by colleagues, teacher candidates, and by the process of self-study itself. We argue that the multiple studies that are available to any self-study research program creates a plurality of publics in which we identify, and are identified, in particular ways by particular members of the teacher education community. One significant avenue for our developing line of work concerns the ways in which our shared bilingualism plays out during our critical friendship, and how the use of multiple languages helps us to reframe our identities as teacher educators, particularly whilst engaged in translanguaging practices