S. P. Glasby

Enumerating the rationals from left to right

Abstract The rationals can be enumerated using Stern-Brocot sequences 풮ℬn, Calkin-Wilf sequences 풞풲n, or Farey sequences ℱn. We show that the rationals in 풮ℬn, 풞풲n, and ℱn can be computed using very similar second-order linear recurrence relations. (This, incidentally, obviates the need to compute previous sequences 풮ℬi, 풞풲i, ℱi with i < n.)

Worms by number

This paper investigates alternation patterns in length, shape and orientation of dorsal cirri (fleshy segmental appendages) of phyllodocidans, a large group of polychaete worms (Annelida). We document the alternation patterns in several families of Phyllodocida (Syllidae, Hesionidae, Sigalionidae, Polynoidae, Aphroditidae and Acoetidae) and identify the simple mathematical rule bases that describe the progression of these sequences. Two fundamentally different binary alternation patterns were found on the first four segments: 1011 for nereidiform families and 1010 for aphroditiform families. The alternation pattern in all aphroditiform families matches a simple one-dimensional cellular automaton and that for Syllidae (nereidiform) matches the Fibonacci string sequence. Hesionidae (nereidiform) showed the greatest variation in alternation patterns, but all corresponded to various known substitution rules. Comparison of binary patterns of the first 22 segments using a distance measure supports the current ideas on phylogeny within Phyllodocida. These results suggest that gene(s) involved in post-larval segmental growth employ a switching sequence that corresponds to simple mathematical substitution rules.

Primitive prime divisors and the nTH cyclotomic polynomial

Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called

Decomposing modular tensor products, and periodicity of ‘Jordan partitions’

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Twisted centralizer codes

, which is closely related to the cyclotomic polynomial

Decomposing modular tensor products: ‘Jordan partitions’, their parts and p-parts

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Generalized Quadrangles and Transitive Pseudo-Hyperovals

and to primitive prime divisors of

Point-primitive generalised hexagons and octagons

q^{n}-1

The number of composition factors of order p in completely reducible groups of characteristic p

. Our definition of

On the dimension of twisted centralizer codes

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