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We derive a rigorous scaling law for minimizers in a natural version of the regularized Cross-Newell model for pattern formation far from threshold. These energy-minimizing solutions support defects having the same character as what is seen in experimental studies of the corresponding physical systems and in numerical simulations of the microscopic equations that describe these systems., Comment: 26 pages, 5 figures, submitted to the Journal of Nonlinear Science

We study the linearized Fopl - von Karman theory of a long, thin rectangular elastic membrane that is bent through an angle $2 \alpha$. We prove rigorous bounds for the minimum energy of this configuration in terms of the plate thickness $\sigma$ and the bending angle. We show that the minimum energy scales as $\sigma^{5/3} \alpha^{7/3}$. This scaling is in sharp contrast with previously obtained results for the linearized theory of thin sheets with isotropic compression boundary conditions, where the energy scales as $\sigma$., Comment: 15 pages, 2 figures, iopart style files; Changes - 10/25 -- Changes to the introduction, fixed the references

We investigate the propagation of transverse elastic waves in crumpled media. We set up the wave equation for transverse waves on a generic curved, strained surface via a Langrangian formalism and use this to study the scaling behaviour of the dispersion curves near the ridges and on the flat facets. This analysis suggests that ridges act as barriers to wave propagation and that modes in a certain frequency regime could be trapped in the facets. A simulation study of the wave propagation qualitatively supported our analysis and showed interesting effects of the ridges on wave propagation., Comment: RevTex 12 pages, 7 figures, Submitted to PRE

We study the dynamics of the front separating a spatio-temporally chaotic region from a stable steady region using a simple model applicable to periodically forced systems. In particular, we investigate both the coarsening of the front induced by the inherent `noise' of the chaotic region, and the long wavelength dynamics causing the front to develop cusps.

The crumpling of a thin sheet can be understood as the condensation of elastic energy into a network of ridges which meet in vertices. Elastic energy condensation should occur in response to compressive strain in elastic objects of any dimension greater than 1. We study elastic energy condensation numerically in 2-dimensional elastic sheets embedded in spatial dimensions 3 or 4 and 3-dimensional elastic sheets embedded in spatial dimensions 4 and higher. We represent a sheet as a lattice of nodes with an appropriate energy functional to impart stretching and bending rigidity. Minimum energy configurations are found for several different sets of boundary conditions. We observe two distinct behaviors of local energy density fall-off away from singular points, which we identify as cone scaling or ridge scaling. Using this analysis we demonstrate that there are marked differences in the forms of energy condensation depending on the embedding dimension., Comment: 31 pages, 19 figures, RevTeX4, submitted to PRE

We prove that an m-dimensional unit ball D^m in the Euclidean space {\mathbb R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B_r^d \subset {\mathbb R}^d of radius r < 1/2 unless one of two conditions is met -- (1)The embedding manifold has dimension d >= 2m. (2) The embedding is not smooth. The proof uses differential geometry to show that if d<2m and the embedding is smooth and isometric, we can construct a line from the center of D^m to the boundary that is geodesic in both D^m and in the embedding manifold {\mathbb R}^d. Since such a line has length 1, the diameter of the embedding ball must exceed 1., Comment: 20 Pages, 3 Figures

We investigate a nonlinear dynamical system which ``remembers'' preselected values of a system parameter. The deterministic version of the system can encode many parameter values during a transient period, but in the limit of long times, almost all of them are forgotten. Here we show that a certain type of stochastic noise can stabilize multiple memories, enabling many parameter values to be encoded permanently. We present analytic results that provide insight both into the memory formation and into the noise-induced memory stabilization. The relevance of our results to experiments on the charge-density wave material $NbSe_3$ is discussed., Comment: 29 pages, 6 figures, submitted to Physical Review E

We report short-term memory formation in a nonlinear dynamical system with many degrees of freedom. The system ``remembers'' a sequence of impulses for a transient period, but it coarsens and eventually ``forgets'' nearly all of them. The memory duration increases as the number of degrees of freedom in the system increases. We demonstrate the existence of these transient memories in a laboratory experiment., Comment: Significantly revised version with additional authors, accepted by PRL. 13 pages, 4 figures. Postscript version with higher resolution figures available at ftp://haber.uchicago.edu/pub/memories.ps