Analysis and Assessment of the Effects of Corrosive Hydrogen Media on the Stress-Strain State of a Spherical Titanium Alloy Shell

We consider the creation of a mathematical model describing the effect of corrosive hydrogen environment on the stress state of a hollow spherical shell made of titanium alloy grade VT1-0, the load is evenly distributed throughout the shell. The solution of the problem in practice was carried out by two-step method of sequential perturbation of parameters using MatLab and Maple programs. To solve the system of solving differential equations the finite difference method was applied. The solution of the diffusion equation of the aggressive hydrogen medium has been considered and the obtained solution has been compared with the results of the classical theory which does not take into account the aggressive effect of the corrosive medium.

The Goldilocks Zone: Towards Better Understanding of Neural Network Loss Landscapes

We explore the loss landscape of fully-connected and convolutional neural networks using random, low-dimensional hyperplanes and hyperspheres. Evaluating the Hessian, H, of the loss function on these hypersurfaces, we observe 1) an unusual excess of the number of positive eigenvalues of H, and 2) a large value of Tr(H)/||H|| at a well defined range of configuration space radii, corresponding to a thick, hollow, spherical shell we refer to as the Goldilocks zone. We observe this effect for fully-connected neural networks over a range of network widths and depths on MNIST and CIFAR-10 datasets with the ReLU and tanh non-linearities, and a similar effect for convolutional networks. Using our observations, we demonstrate a close connection between the Goldilocks zone, measures of local convexity/prevalence of positive curvature, and the suitability of a network initialization. We show that the high and stable accuracy reached when optimizing on random, low-dimensional hypersurfaces is directly related to the overlap between the hypersurface and the Goldilocks zone, and as a corollary demonstrate that the notion of intrinsic dimension is initialization-dependent. We note that common initialization techniques initialize neural networks in this particular region of unusually high convexity/prevalence of positive curvature, and offer a geometric intuition for their success. Furthermore, we demonstrate that initializing a neural network at a number of points and selecting for high measures of local convexity such as Tr(H)/||H||, number of positive eigenvalues of H, or low initial loss, leads to statistically significantly faster training on MNIST. Based on our observations, we hypothesize that the Goldilocks zone contains an unusually high density of suitable initialization configurations.

Three-dimensional carbon boron nitrides with a broken, hollow, spherical shell for water treatment

Broken hollow spherical shell like 3D C-BNs with a very fast dye adsorption rate for water purification.