Virtually Geometric Words and Whitehead’s Algorithm

A. N. Whitehead’s Geometric Algebra

Whitehead method and Genetic Algorithms

In this paper we discuss a genetic version (GWA) of the Whitehead’s algorithm, which is one of the basic algorithms in combinatorial group theory. It turns out that GWA is surprisingly fast and outperforms the standard Whitehead’s algorithm in free groups of rank ≥ 5. Experimenting with GWA we collected an interesting numerical data that clarifies the time-complexity of the Whitehead’s Problem ...

Counting Words of Minimum Length in an Automorphic Orbit

Abstract. Let u be a cyclic word in a free group Fn of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v : |v| = |u| and v = φ(u) for some φ ∈ AutFn}. In this paper, we prove that N(u) is bounded by a polynomial function with respect to |u| under the hypothesis that if two letters x, y with x 6= y occur in u, ...

1 2 Se p 20 06 Clusters , currents and Whitehead ’ s algorithm

Using geodesic currents, we provide a theoretical justification for some of the experimental results, obtained by Haralick, Miasnikov and Myasnikov via pattern recognition methods, regarding the behavior of Whitehead’s algorithm on non-minimal inputs. In particular we prove that the images of “random” elements of a free group F under the automorphisms of F form “clusters” that share similar nor...

Ecological Thought : Some New Implications Of Whitehead’s Philosophy

  This paper will discuss some new implications of Whitehead’s philosophy for environmental thought.[i] They fall under three headings: 1. The interrelations between three ways of thinking about the environment: bioregionalism, land ethics, and the philosophy of organism.  2.  The implications of Whitehead’s philosophy of organism for environmental aesthetics.  3.  The implications of Whit...