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- Peter Hegarty, Dmitry Zhelezov
- Combinatorics, Probability & Computing
- 2014

We present a two-parameter family (Gm,k)m,k∈N≥2 , of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of Gm,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 Erdős– Rényi random graph. As well as… (More)

- Peter Hegarty, Anders Martinsson, Dmitry Zhelezov
- ArXiv
- 2013

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)

- Peter Hegarty, Dmitry Zhelezov
- Geometry, Structure and Randomness in…
- 2014

- Antal Balog, Oliver Roche-Newton, Dmitry Zhelezov
- Electr. J. Comb.
- 2017

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| .

- Ljudmila A. Bordag, Ivan P. Yamshchikov, Dmitry Zhelezov
- Int. J. Comput. Math.
- 2016

- Peter Hegarty, Anders Martinsson, Dmitry Zhelezov
- ICDCN
- 2016

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 Ω(<i>n</i><sup>2</sup> log <i>n</i>)… (More)

- Dmitry Zhelezov
- Electronic Notes in Discrete Mathematics
- 2013

Let B be a set of real numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = {bibj | bi, bj ∈ B} cannot be greater than O(n1+1/ √ log logn) and present an example of a product set containing an arithmetic progression of length Ω(n log n), so the obtained upper bound is close to the optimal.

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