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| Trivially, the Euclidean plane cannot be decomposed into disks of equal radii. Thus any partition requires either the sets to be not all disks (i.e. a hexagonal partition) or that they be of different size (Apollonian packings). We quantify the phenomenon and use it to improve Pleijel's bound on the number of nodal domains of a Laplacian eigenfunction (a numerically more explicit improvement has been given by Bourgain). |
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