Nonlinear morphoelastic plates II: exodus to buckled states

Morphoelasticity is the theory of growing elastic materials. This theory is based on the multiple decomposition of the deformation gradient and provides a formulation of the deformation and stresses induced by growth. Following a companion paper, a general theory of growing nonlinear elastic Kirchhoff plate is described. First, a complete geometric description of incompatibility with simple examples is given. Second, the stability of growing Kirchhoff plates is analyzed

Dynamical Isometry is Achieved in Residual Networks in a Universal Way for any Activation Function

We demonstrate that in residual neural networks (ResNets) dynamical isometry is achievable irrespectively of the activation function used. We do that by deriving, with the help of Free Probability and Random Matrix Theories, a universal formula for the spectral density of the input-output Jacobian at initialization, in the large network width and depth limit. The resulting singular value spectrum depends on a single parameter, which we calculate for a variety of popular activation functions, by analyzing the signal propagation in the artificial neural network. We corroborate our results with numerical simulations of both random matrices and ResNets applied to the CIFAR-10 classification problem. Moreover, we study the consequence of this universal behavior for the initial and late phases of the learning processes. We conclude by drawing attention to the simple fact, that initialization acts as a confounding factor between the choice of activation function and the rate of learning. We propose that in ResNets this can be resolved based on our results, by ensuring the same level of dynamical isometry at initialization

Models of Galaxy Clusters with Thermal Conduction

We present a simple model of hot gas in galaxy clusters, assuming hydrostatic equilibrium and energy balance between radiative cooling and thermal conduction. For five clusters, A1795, A1835, A2199, A2390 and RXJ1347.5-1145, the model gives a good description of the observed radial profiles of electron density and temperature, provided we take the thermal conductivity $\kappa$ to be about 30% of the Spitzer conductivity. Since the required $\kappa$ is consistent with the recent theoretical estimate of Narayan & Medvedev (2001) for a turbulent magnetized plasma, we consider a conduction-based equilibrium model to be viable for these clusters. We further show that the hot gas is thermally stable because of the presence of conduction. For five other clusters, A2052, A2597, Hydra A, Ser 159-03 and 3C295, the model requires unphysically large values of $\kappa$ to fit the data. These clusters must have some additional source of heat, most likely an active galactic nucleus since all the clusters have strong radio galaxies at their centers. We suggest that thermal conduction, though not dominant in these clusters, may nevertheless play a significant role by preventing the gas from becoming thermally unstable.Comment: Published in ApJ; 22 pages, including 2 tables, 4 figures; typos corrected to match the published versio

Lessons learned from professional development workshops on using GIS to teach geography and history in the K-12 classroom

A GIS oriented professional development activity engaged social studies teachers with the importance of maps and graphics in teaching geography and history. With an introduction to ArcGIS Online and National Geographic Map Maker, the activity provided teachers with the ability to make their own maps and identify GIS materials for their classrooms. Conducting the workshop reinforced our belief in the need for considerable hands-on activity with participants allowed to work at their own pace. Pre- and post-event surveys showed positive gains regarding the teachers’ likelihood to include GIS based maps and graphics in teaching. The activity provided teachers with enough knowledge of GIS that they were ready to use the technology immediately

Transport in Almost Integrable Models: Perturbed Heisenberg Chains

The heat conductivity kappa(T) of integrable models, like the one-dimensional spin-1/2 nearest-neighbor Heisenberg model, is infinite even at finite temperatures as a consequence of the conservation laws associated with integrability. Small perturbations lead to finite but large transport coefficients which we calculate perturbatively using exact diagonalization and moment expansions. We show that there are two different classes of perturbations. While an interchain coupling of strength J_perp leads to kappa(T) propto 1/J_perp^2 as expected from simple golden-rule arguments, we obtain a much larger kappa(T) propto 1/J'^4 for a weak next-nearest neighbor interaction J'. This can be explained by a new approximate conservation law of the J-J' Heisenberg chain.Comment: 4 pages, several minor modifications, title change