Commutators and squares in free groups

Let F_2 be the free group generated by x and y. In this article, we prove that the commutator of x^m and y^n is a product of two squares if and only if mn is even. We also show using topological methods that there are infinitely many obstructions for an element in F_2 to be a product of two squares.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-27.abs.htm

A New Type of Massive Spin-One Boson: and its Relation with Maxwell Equations

First, we show that in the $(1,0)\oplus(0,1)$ representation space there exist not one but two theories for charged particles. In the Weinberg construct, the boson and its antiboson carry {\it same} relative intrinsic parity, whereas in our construct the relative intrinsic parities of the boson and its antiboson are {\it opposite}. These results originate from the commutativity of the operations of Charge conjugation and Parity in Weinberg's theory, and from the anti-commutativity of the operations of Charge conjugation and Parity in our theory. We thus claim that we have constructed a first non-trivial quantum theory of fields for the Wigner-type particles. Second, the massless limit of both theories seems formally identical and suggests a fundamental modification of Maxwell equations. At its simplest level, the modification to Maxwell equations enters via additional boundary condition(s).Comment: Invited talk presented at ``The Present Status of Quantum Theory of Light: A Symposium to Honour Jean-Pierre Vigier,'' held at York University, Toronto, August 27-30, 1995. 15 pages, latex (crckapb.sty needed

Transforming AdaPT to Ada9x

How the concepts of AdaPT can be transformed into programs using the object oriented features proposed in the preliminary mapping for Ada9x are described. Emphasizing, as they do, the importance of data types as units of program, these features match well with the development of partitions as translations into Abstract Data Types which was exploited in the Ada83 translation covered in report R3. By providing a form of polymorphic type, the Ada83 version also gives support for the conformant partition idea which could be achieved in Ada83 only by using UNCHECKED CONVERSIONS. It is assumed that the reader understands AdaPT itself, but the translation into Ada83 is briefly reviewed, by applying it to a small example. This is then used to show how the same translation would be achieved in the 9x version. It is important to appreciate that the distribution features which are proposed in current mapping are not used or discussed in any detail, as those are not well matched to the AdaPT approach. Critical evaluation and comparison of these approaches is given in a separate report