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A New Kind of Permutation Entropy Used to Classify Sleep Stages from Invisible EEG Microstructure
(2017)
Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic
(2020)
Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.