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Symbolic dynamics of unimodal mappings: kneading sequences. The combinatorial approach to the dynamics of mappings, which this paper develops, starts with kneading sequences, as developed in [MT] and [CE]. Choose a < c < b E R., and set I = [a, b] and consider first unimodal maps, which we will take to mean continuous mappings f:(2) f is monotone decreasing (or increasing) on [a, c];(3) f is monotone increasing (or decreasing, respectively) on [c, b] .We apologize to readers used to unimodal maps with maxima; monic polynomials are best adapted to our purposes. Thus typical unimodal mappings are elements of the quadratic family pc(x) = X2 + c with the intervaland c = O.In this case, the kneading sequence of a point x E

This paper begins by presenting a simple explanation of the main ideas in fractal image compression. It then presents a brief discussion of the current state of the art along with some results comparing fractal encoding, JPEG, and a wavelet scheme. The conclusion contains references to many of the latest theoretical and implementation results.

The tritone paradox occurs when an ordered pair of tones is presented, with each tone consisting of a set of octave-related components, and the pitch classes of the tones separated by a half-octave. Such a pattern is heard as ascending in one key, but as descending in a different key. Further, the pattern in any one key is heard as ascending by some listeners but as descending by others. It was here found that this phenomenon occurs to a highly significant extent in a general population, and that it is distributed within the population in an orderly fashion. The findings also reveal a surprising ability within the general population to utilize absolute pitch.