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A self-affine tile in R n is a set T of positive measure with, where A is an expanding n_n real matrix with |det(A)| =m an integer, andn is a set of m digits. It is known that self-affine tiles always give tilings of R n by translation. This paper extends known characterizations of digit sets D yielding self-affine tiles. It proves several results about the structure of tilings of R n possible using such tiles, and gives examples showing the possible relations between self-replicating tilings and general tilings, which clarify results of Kenyon on self-replicating tilings.

The empirical mode decomposition (EMD) was a method pioneered by (N. Huang et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear nonstationary time series analysis, Proc. Roy. Soc. Lond. A 454 (1998) 903-995) as an alternative technique to the traditional Fourier and wavelet techniques for studying signals. It decomposes a signal into several components called intrinsic mode functions (IMFs), which have shown to admit better behaved instantaneous frequencies via Hilbert transforms. In this paper, we propose an alternative algorithm for EMD based on iterating certain filters, such as Toeplitz filters. This approach yields similar results as the more traditional sifting algorithm for EMD. In many cases the convergence can be rigorously proved.

The notion of ‘finite type’ iterated function systems of contractive similitudes is introduced, and a scheme for computing the exact Hausdorff dimension of their attractors in the absence of the open set condition is described. This method extends a previous one by Lalley, and applies not only to the classes of self-similar sets studied by Edgar, Lalley, Rao and Wen, and others, but also to some new classes that are not covered by the previous ones.