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In this paper we investigate the dimensional structure of probability distributions on Euclidean space and characterize a class of regular distributions. We obtain a consistent estimator of dimension based on a nearest neighbor statistic and in addition obtain asymptotic confidence intervals for dimension in the case of regular distributions. Although many examples of point estimation of dimension have recently appeared in the literature on chaotic attractors in dynamical systems, questions of consistency and interval estimation Blazer Nike Femme Bleu have not previously been addressed systematically.

A simplified model of plane Couette flow is derived by means of a cross-stream (y) Galerkin expansion in terms of trigonometric functions appropriate for idealized stress-free boundary conditions at the plates. A set of partial differential equations is obtained, governing the in-plane (x–z) space-dependence of a velocity field taken in the form: u=U0(x,z,t)+[1+U1(x,z,t)]sin(πy/2), v=V1(x,z,t)cos(πy/2), w=W0(x,z,t)+W1(x,z,t)sin(πy/2). Beyond Lorenz-like Waleffe's modeling (Waleffe 1997), this Swift–Hohenberg type of approach is expected to give an access to the microscopic mechanism of spatiotemporal intermittency typical of the transition to turbulence in plane Couette flow (Pomeau 1986, Bergé et al. 1998).

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