For three particles P,Q,R travelling on a straight line, let vPQ be the (relative) velocity of P as measured by Q, and define vQR, vPR similarly. According to classical mechanics, the velocity v of a particle moving on a line can be any real number, and relative velocities add by the simple formula vPR = vPQ + vQR. On the other hand, the special theory of relativity says velocities are restrict...

While we were graduate students, Marty Isaacs and I worked together on the character theory of finite groups, studying in particular the character degrees of finite p-groups. Somewhat later, my interests turned to ring theory and infinite group theory. On the other hand, Marty continued with character theory and soon became a leader in the field. Indeed, he has had a superb career as a research...

The subgroup R of the group G has been termed a retract of the group G whenever there exists an idempotent endomorphism of G which maps G upon R. This definition is in strict analogy to the topological concept of retract. Thus one may be tempted to define absolute retracts in like similarity to topological usage. However, we shall prove in the course of the present note that the identity is the...

We present a brief overview of methods and results in geometric group theory, with the goal of introducing the reader to both topological and metric perspectives. Prerequisites are kept to a minimum: we require only basic algebra, graph theory, and metric space topology.

There is currently much interest in bringing together the tradition of categorial grammar, and especially the Lambek calculus, with the recent paradigm of linear logic to which it has strong ties. One active research area is designing non-commutative versions of linear logic (Abrusci 1995, Retor e 1993) which can be sensitive to word order while retaining the hypothetical reasoning capabilities...

This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research.

2 Preface Group theory is a central part of modern mathematics. Its origins lie in geometry (where groups describe in a very detailed way the symmetries of geometric objects) and in the theory of polynomial equations (developed by Galois, who showed how to associate a finite group with any polynomial equation in such a way that the structure of the group encodes information about the process of...

where xi is the the Cartesian coordinate of a point particle of mass m, Fi is the external force applied to the particle, and the force is assumed to be conservative, the negative gradient of a potential energy function V (x). Newtonian mechanics for rigid bodies can be formulated in a similar way, and this approach can even be extended to relativistic point mechanics. The calculus of variation...

1. The importance of the concept group-matrix in the theory of finite groups was recognized by Dedekind as early as 1880. The development of a general theory of group-matrices is due, however, to Frobenius, (Berliner Sitzungsberichte, from 1896 to the present). In particular, Frobenius has applied the theory to the representation of a finite group as a (non-modular) linear group. Since linear c...

Cooperative learning (CL) is more than having students work in groups: it is a fundamental shift from teacher as information provider and sole source of truth, to teacher as facilitator [2]. It involves the use of tasks whose completion requires the combined efforts and skills of the individual group members. Group investigation (GI) is one form of CL, and the focus of this paper. The following...