Breadth-first search and the Andrews-Curtis conjecture

Andrews and Curtis conjectured in 1965 that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. Recent computational work by Miasnikov and Myasnikov on this problem has been based on genetic algorithms. We show that a computational attack based on a breadth-first search of the tree of equivalent presentations is also viable, and seems to outperform that based on genetic algorithms. It allows us to extract shorter proofs (in some cases, provably shortest) and to consider the length thirteen case for two generators. We prove that, up to equivalence, there is a unique minimum potential counterexample.

Irreducible cyclic presentations of the trivial group

We produce families of irreducible cyclic presentations of the trivial group. These families comprehensively answer questions about such presentations asked by Dunwoody and by Edjvet, Hammond, and Thomas. Our theorems are purely theoretical, but their derivation is based on practical computations. We explain how we chose the computations and how we deduced the theorems.

A report of a national mutation testing service for the MEN1 gene: clinical presentations and implications for mutation testing

Introduction: Mutation testing for the MEN1 gene is a useful method to diagnose and predict individuals who either have or will develop multiple endocrine neoplasia type 1 ( MEN 1). Clinical selection criteria to identify patients who should be tested are needed, as mutation analysis is costly and time consuming. This study is a report of an Australian national mutation testing service for the MEN1 gene from referred patients with classical MEN 1 and various MEN 1- like conditions. Results: All 55 MEN1 mutation positive patients had a family history of hyperparathyroidism, had hyperparathyroidism with one other MEN1 related tumour, or had hyperparathyroidism with multiglandular hyperplasia at a young age. We found 42 separate mutations and six recurring mutations from unrelated families, and evidence for a founder effect in five families with the same mutation. Discussion: Our results indicate that mutations in genes other than MEN1 may cause familial isolated hyperparathyroidism and familial isolated pituitary tumours. Conclusions: We therefore suggest that routine germline MEN1 mutation testing of all cases of classical'' MEN1, familial hyperparathyroidism, and sporadic hyperparathyroidism with one other MEN1 related condition is justified by national testing services. We do not recommend routine sequencing of the promoter region between nucleotides 1234 and 1758 ( Genbank accession no. U93237) as we could not detect any sequence variations within this region in any familial or sporadic cases of MEN1 related conditions lacking a MEN1 mutation. We also suggest that testing be considered for patients < 30 years old with sporadic hyperparathyroidism and multigland hyperplasia