A method maps PDEs from the unit interval to self-similar fractals via isometry and nonlocal approximations, yielding self-similar particle systems for transport, Burgers, and heat equations.
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For diagonally aligned self-affine carpets with weak separation on axis projections, Hausdorff dimension equals the limit of the Barański formula and box-counting dimension equals the limit of the Feng-Wang formula over n-fold IFS compositions.
Ahlfors regular sets are shown to correspond to tree-like structures, enabling the study of asymptotic limits for their counting functions.
citing papers explorer
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Weakly separated self-affine carpets
For diagonally aligned self-affine carpets with weak separation on axis projections, Hausdorff dimension equals the limit of the Barański formula and box-counting dimension equals the limit of the Feng-Wang formula over n-fold IFS compositions.
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On the asymptotics of counting functions for Ahlfors regular sets
Ahlfors regular sets are shown to correspond to tree-like structures, enabling the study of asymptotic limits for their counting functions.