Complexity of terms, composition, and hypersubstitution

We consider four useful measures of the complexity of a term: the maximum depth (usually called the depth), the minimum depth, the variable count, and the operation count. For each of these, we produce a formula for the complexity of the compositionSmn(s,t1,…,tn)in terms of the complexity of the inputss,t1,…,tn. As a corollary, we also obtain formulas for the complexity ofσˆ[t]in terms of the complexity oftwhentis a compound term andσis a hypersubstitution. We then apply these formulas to the theory ofM-solid varieties, examining thek-normalization chains of a variety with respect to the four complexity measures.

Computing the Fourier transform in geophysics with the transform decomposition DFT

The Fourier transform and its computationally efficient discrete implementation, the fast Fourier transform (FFT), are omnipresent in geophysical processing. While a general implementation of the discrete Fourier transform (DFT) will take on the order [Formula: see text] operations to compute the transform of an N point sequence, the FFT algorithm accomplishes the DFT with an operation count proportional to [Formula: see text] When a large percentage of the output coefficients of the transform are not desired, or a majority of the inputs to the transform are zero, it is possible to further reduce the computation required to perform the DFT. Here, we review one possible approach to accomplishing this reduction and indicate its application to phase‐shift migration.