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.

Age-at-first-registration and heterogeneity in affective psychoses

Background: Previous research into age of onset in affective disorders has produced conflicting results. This paper examines the influence of heterogeneity on the age-at-first-registration distribution for the ICD-9 diagnostic group 'affective psychosis'. Method: For 1979-1991, data for age-at-first-registration for 4985 individuals diagnosed with affective psychosis (ICD-9 296.x) were extracted from a name-linked mental health register. These data were divided into (i) '296.1 only', a category used to code unipolar depression (males = 700; females = 1321); and (ii) '296 other', all 296 cases other than 296.1 (males = 1280; females = 1684). Inception rates for each 5-year age division were adjusted for the background population age-structure as a rate per 100 000 population. Results: The age-at-first-registration distribution for affective psychosis has a wide age range, with women outnumbering men. There is a near-linear increase in inception rates for both men and women with 296.1 only, while the bulk of those with affective psychoses (296 other) have an inverted U-shaped age distribution. Males have an earlier modal age-at-first-registration for 296 other compared to females. Conclusion: The heterogeneity in terms of subtypes and sex in affective psychosis clouds the interpretation of age-at-first-registration. Separating those with unipolar psychotic depression from other subclassifications and differentiating by sex may provide clues to factors that precipitate the onset of affective psychosis.