Growth Rates and Explosions in Sandpiles

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in ? ^d . Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height h≤2d?2, the diameter of the set of sites that topple has order n ^1/d . This was previously known only for h<d. Our proof uses a strong form of the least action principle for sandpiles, and a novel method of background modification. We can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d?1. On the other hand, we show that if the background height 2d?2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random, then adding finitely many particles creates an explosion (a sandpile that never stabilizes).     

Numerical simulation of flow-induced fiber orientation using normalization of second moment

A normalization scheme for the numerical solution of the moment approximation equation in fiber suspension flows is presented. Here, normalization refers to rescaling the trace of the second moment tensor to unity at each time step. The equivalence between the normalization scheme and the quadratic closure model is analytically proved. The performance of the scheme is investigated in simple shear flow with respect to the quadratic and hybrid closures, and a stochastic Monte-Carlo simulator that provides exact solution. The proposed scheme is a computationally efficient alternative to the quadratic closure: it performs equally well and is more efficient regarding computational time.     

Novel Copolymers of 4-Fluorostyrene. 9. Some Ring-Disubstituted 2-Phenyl-1,1-dicyanoethylenes

Novel copolymers of trisubstituted ethylene monomers, ring-disubstituted 2-phenyl-1,1-dicyanoethylenes, RC₆H₃CH=C(CN)₂ (where R = 3-Br-4-CH₃O, 5-Br-2-CH₃O, 2-F-5-CH₃, 2-F-6-CH₃, 3-F-2-CH₃, 3-F-4-CH₃, 4-F-2-CH₃, 4-F-3-CH₃) and 4-fluorostyrene were prepared at equimolar monomer feed composition by solution copolymerization in the presence of a radical initiator (ABCN) at 70°C. The composition of the copolymers was calculated from nitrogen analysis, and the structures were analyzed by IR, ¹H and ¹³C-NMR. The order of relative reactivity (1/r ₁) for the monomers is 3-F-4-CH₃(1.64) > 5-Br-2-CH₃O (1.62) > 3-Br-4-CH₃O (1.36) > 4-F-2-CH₃(1.3) > 4-F-3-CH₃(1.26) > 3-F-2-CH₃(1.11) > 2-F-5-CH₃ (0.98) > 2-F-6-CH₃ (0.97). High T_g of the copolymers, in comparison with that of poly(4-fluorostyrene) indicates a substantial decrease in chain mobility of the copolymer due to the high dipolar character of the trisubstituted ethylene monomer unit. Decomposition of the copolymers in nitrogen occurred in two steps, first in the 290–400°C range with residue, which then decomposed in 400–800°C range.