For an integer n, we wish to dispose the n-1 non-zero elements of

Z_n in a circular arrangement [a_1, a_2, ... , a_{n-1}] (a_n = a_1)

so that the set of differences a_{i+1} - a_i (i = 1, 2, ... , n-1)

is itself Z_n \ {0} . A basic recipe for doing this for any odd

n was given in 1978 by Friedlander, Gordon and Miller. Variants

of this recipe are available for n \equiv 1 (mod 2i), with

i = 2, 3, 4, ... . Entirely different procedures have been

published by the speaker for prime values of n . Now procedures

have been found specifically for values n = pq where p and q

are distinct odd primes. Many of these new procedures produce

arrangements in which any element x and its negative -x (mod n)

are (n-1)/2 positions apart (in either direction).

The talk will pay little heed to possible applications. However,

for small values of n , the arrangements give us neighbour designs,

as in statistical design theory.

# Cycling with mirrors, modulo n

Speaker:

Donald Preece

Date/Time:

Mon, 15/03/2010 - 16:30

Room:

M103

Seminar series: