Many practical problems necessitate faster and faster computations. Simple physical estimates show that eventually, to move beyond a limit caused by the speed of light restrictions on communication speed, we will need to use quantum-or, more generally, reversible-computing. Thus, we need to be able to transform the existing algorithms into a reversible form. Such transformation schemes exist. However, such schemes are not very efficient. Indeed, in general, when we write an algorithm, we composed it of several pre-existing modules. It would be nice to be able to similarly compose a reversible version of our algorithm from reversible versions of these moduli-but the existing transformation schemes cannot do it, they require that we, in effect, program everything "from scratch". It is therefore desirable to come up with alternative transformations, transformation that transform compositions into compositions and thus, transform a modular program in an efficient way-by utilizing transformed moduli. Such transformations are proposed in this paper.

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In many applications, including analysis of seismic signals, Daubechies wavelets perform much better than other families of wavelets. In this paper, we provide a possible theoretical explanation for the empirical success of Daubechies wavelets. Specifically, we show that these wavelets are optimal with respect to any optimality criterion that satisfies the natural properties of scale- and shift-invariance.