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Proof of Murphy-Cohen Conjecture on One-Dimensional Hard Ball Systems*

Abstract

We prove the Murphy and Cohen's conjecture that the maximum number of collisions of n + 1 elastic particles moving freely on a line is \( \frac{{n{\left( {n + 1} \right)}}} {2} \) if no interior particle has mass less than the arithmetic mean of the masses of its immediate neighbors. In fact, we prove the stronger result that, for the same conclusion, the condition that no interior particle has mass less than the geometric mean, rather than the arithmetic mean, of the masses of its immediate neighbors suffices.

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Author information

Correspondence to Lizhou Chen.

Additional information

*Project supported by the Special Funds for Chinese Major State Basic Research Projects "Nonlinear Science".

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Chen, L. Proof of Murphy-Cohen Conjecture on One-Dimensional Hard Ball Systems*. Chin. Ann. Math. Ser. B 28, 293–298 (2007). https://doi.org/10.1007/s11401-006-0135-2

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Keywords

  • Hard ball
  • Elastic collision
  • Billiard
  • Reflection group
  • Numbers game

2000 MR Subject Classification

  • 70F99
  • 51F15
  • 20F55