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We construct two area preserving maps from the square to the p-ball (the domain defined by |x|p + |y|p ≤ |a|p ) and use them to obtain low distorted uniform grids on the p-ball.

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First a new mathematical approach is presented to evaluate the electric and magnetic field of the lightning, via engineering model with variable shape return stroke channel; next, an inverse procedure is exposed for the reconstruction of both spatial and temporal waveforms of the lightning return stroke current, throughout a numerical field synthesis procedure, based on regularization of ill-posed problems. The approach uses as input data the acquisition of time domain recordings of electric and/or magnetic field generated by the lightning current, at various locations on the ground and transforms these signals into harmonics, by Fourier decomposition. This combination, between the proposed solving procedures and harmonic filtering, yields numerical results that are in good agreement with the testing functions

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