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We present a two-parameter family (G m,k) m,k∈N ≥2 , of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of G m,k is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erd˝ os– Rényi random graph. As well(More)
The classical multi-agent rendezvous problem asks for a deterministic algorithm by which n points scattered in a plane can move about at constant speed and merge at a single point, assuming each point can use only the locations of the others it sees when making decisions and that the visibility graph as a whole is connected. In time complexity analyses of(More)
In multi-agent rendezvous it is naturally assumed that agents have a maximum speed of movement. In the absence of any distributed control issues, this imposes a lower bound on the time to rendezvous, for idealised point agents, proportional to the diameter of a configuration. Assuming bounded visibility, we consider &#937;(<i>n</i><sup>2</sup> log <i>n</i>)(More)
We prove several expanders with exponent strictly greater than 2. For any finite set A ⊂ R, we prove the following six-variable expander results: |(A−A)(A−A)(A−A)| |A| 2+ 1 8 log 17 16 |A| , ∣∣∣∣A+A A+A + AA ∣∣∣∣ |A| 2 17 log 16 17 |A| , ∣∣∣∣AA+AA A+A ∣∣∣∣ |A| 18 log |A| , ∣∣∣∣AA+A AA+A ∣∣∣∣ |A| 18 log |A| .
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