Localized Induction Hierarchy and Weingarten Systems

We describe a method of constructing Weingarten systems of triply orthogonal coordinates, related to the localized induction equation hierarchy of integrable geometric evolution equationsComment: AMSTeX file (10 pages) with one Postscript graphic; submitted to Physics Letters

Two Universality Properties Associated with the Monkey Model of Zipf's Law

The distribution of word probabilities in the monkey model of Zipf's law is associated with two universality properties: (1) the power law exponent converges strongly to $-1$ as the alphabet size increases and the letter probabilities are specified as the spacings from a random division of the unit interval for any distribution with a bounded density function on $[0,1]$; and (2), on a logarithmic scale the version of the model with a finite word length cutoff and unequal letter probabilities is approximately normally distributed in the part of the distribution away from the tails. The first property is proved using a remarkably general limit theorem for the logarithm of sample spacings from Shao and Hahn, and the second property follows from Anscombe's central limit theorem for a random number of i.i.d. random variables. The finite word length model leads to a hybrid Zipf-lognormal mixture distribution closely related to work in other areas.Comment: 14 pages, 3 figure

Tire tracks and integrable curve evolution

We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equation -- the bicycle equation. The same model, in dimension two, describes another mechanical device, the hatchet planimeter. Here is a sampler of our results. We express the linearized flow of the bicycle equation in terms of the geometry of the rear track; in dimension three, for closed front and rear tracks, this is a version of the Berry phase formula. We show that in all dimensions a sufficiently long bicycle also serves as a planimeter: it measures, approximately, the area bivector defined by the closed front track. We prove that the bicycle equation also describes rolling, without slipping and twisting, of hyperbolic space along Euclidean space. We relate the bicycle problem with two completely integrable systems: the AKNS (Ablowitz, Kaup, Newell and Segur) system and the vortex filament equation. We show that "bicycle correspondence" of space curves (front tracks sharing a common back track) is a special case of a Darboux transformation associated with the AKNS system. We show that the filament hierarchy, encoded as a single generating equation, describes a 3-dimensional bike of imaginary length. We show that a series of examples of "ambiguous" closed bicycle curves (front tracks admitting self bicycle correspondence), found recently F. Wegner, are buckled rings, or solitons of the planar filament equation. As a case study, we give a detailed analysis of such curves, arising from bicycle correspondence with multiply traversed circles.Comment: 61 pages, 22 figure

Distributions of Triplets in Genetic Sequences

Distributions of triplets in some genetic sequences are examined and found to be well described by a 2-parameter Markov process with a sparse transition matrix. The variances of all the relevant parameters are not large, indicating that most sequences gather in a small region in the parameter space. Different sequences have very near values of the entropy calculated directly from the data and the two parameters characterizing the Markov process fitting the sequence. No relevance with taxonomy or coding/noncoding is clearly observed.Comment: revtex, 17pages, 8 figures, submitted to Physica