Nonhomogeneous subalgebras of Lie and special Jordan superalgebras

We consider polynomial identities satisfied by nonhomogeneous subalgebras of Lie and special Jordan superalgebras: we ignore the grading and regard the superalgebra as an ordinary algebra. The Lie case has been studied by Volichenko and Baranov: they found identities in degrees 3, 4 and 5 which imply all the identities in degrees <= 6. We simplify their identities in degree 5, and show that there are no new identities in degree 7. The Jordan case has not previously been studied: we find identities in degrees 3, 4, 5 and 6 which imply all the identities in degrees <= 6, and demonstrate the existence of further new identities in degree 7. our proofs depend on computer algebra: we use the representation theory of the symmetric group, the Hermite normal form of an integer matrix, the LLL algorithm for lattice basis reduction, and the Chinese remainder theorem. (C) 2009 Elsevier Inc. All rights reserved.

The Development of Dalton`s Atomic Theory as a Case Study in the History of Science: Reflections for Educators in Chemistry

The inclusion of the history of science in science curricula-and specially, in the curricula of science teachers-is a trend that has been followed in several countries. The reasons advanced for the study of the history of science are manifold. This paper presents a case study in the history of chemistry, on the early developments of John Dalton`s atomic theory. Based on the case study, several questions that are worth discussing in educational contexts are pointed out. It is argued that the kind of history of science that was made in the first decades of the twentieth century (encyclopaedic, continuist, essentially anachronistic) is not appropriate for the development of the competences that are expected from the students of sciences in the present. Science teaching for current days will benefit from the approach that may be termed the ""new historiography of science"".