We describe the dynamic response of a two-dimensional hexagonal packing of uncompressed stainless steel spheres excited by localized impulsive loadings. After the initial impact strikes the system, a characteristic wave structure emerges and continuo
usly decays as it propagates through the lattice. Using an extension of the binary collision approximation (BCA) for one-dimensional chains, we predict its decay rate, which compares well with numerical simulations and experimental data. While the hexagonal lattice does not support constant speed traveling waves, we provide scaling relations that characterize the power law decay of the wave velocity. Lastly, we discuss the effects of weak disorder on the directional amplitude decay rates.
The Qth-power algorithm produces a useful canonical P-module presentation for the integral closures of certain integral extensions of $P:=mathbf{F}[x_n,...,x_1]$, a polyonomial ring over the finite field $mathbf{F}:=mathbf{Z}_q$ of $q$ elements. Here
it is shown how to use this for several small primes $q$ to reconstruct similar integral closures over the rationals $mathbf{Q}$ using the Chinese remainder theorem to piece together presentations in different positive characteristics, and the extended Euclidean algorithm to reconstruct rational fractions to lift these to presentations over $mathbf{Q}$.
The Qth-power algorithm for computing structured global presentations of integral closures of affine domains over finite fields is modified to compute structured presentations of integral closures of ideals in affine domains over finite fields relati
ve to a local monomial ordering. A non-homogeneous version of the standard (homogeneous) Rees algebra is introduced as well.
