We examine the signatures of internal structure emerged from quasistatic shear responses of granular materials based on three-dimensional discrete element simulations. Granular assemblies consisting of spheres or nonspherical particles of different polydispersity are sheared from different initial densities and under different loading conditions (drained or undrained) steadily to reach the critical state (a state featured by constant stress and constant volume). The radial distribution function used to measure the packing structure is found to remain almost unchanged during the shearing process, regardless of the initial states or loading conditions of an assembly. Its specific form, however, varies with polydispersities in both grain size and grain shape. Set Voronoi tessellation is employed to examine the characteristics of local volume and anisotropy, and deformation. The local inverse solid fraction and anisotropy index are found following inverse Weibull and log-normal distributions, respectively, which are unique at the critical states. With further normalization, an invariant distribution for local volume and anisotropy is observed whose function can be determined by the polydispersities in both particle size and grain shape but bears no relevance to initial densities or loading conditions (or paths). An invariant Gaussian distribution is found for the local deformation for spherical packings, but no invariant distribution can be found for nonspherical packings where the distribution of normalized local volumetric strain is dependent on initial states.This corrects the article DOI 10.1103/PhysRevE.100.043203.Whenever a dynamical process unfolds on static networks, the dynamical state of any focal individual will be exclusively influenced by directly connected neighbors, rather than by those unconnected ones, hence the arising of the dynamical correlation problem, where mean-field-based methods fail to capture the scenario. The dynamic correlation coupling problem has always been an important and difficult problem in the theoretical field of physics. The explicit analytical expressions and the decoupling methods often play a key role in the development of corresponding field. Cariprazine mw In this paper, we study the cyclic three-state dynamics on static networks, which include a wide class of dynamical processes, for example, the cyclic Lotka-Volterra model, the directed migration model, the susceptible-infected-recovered-susceptible epidemic model, and the predator-prey with empty sites model. We derive the explicit analytical solutions of the propagating size and the threshold curve surface for the four different dynamics. We compare the results on static networks with those on annealed networks and made an interesting discovery for the symmetrical dynamical model (the cyclic Lotka-Volterra model and the directed migration model, where the three states are of rotational symmetry), the macroscopic behaviors of the dynamical processes on static networks are the same as those on annealed networks; while the outcomes of the dynamical processes on static networks are different with, and more complicated than, those on annealed networks for asymmetric dynamical model (the susceptible-infected-recovered-susceptible epidemic model and the predator-prey with empty sites model). We also compare the results forecasted by our theoretical method with those by Monte Carlo simulations and find good agreement between the results obtained by the two methods.Absorbing boundary conditions are presented for three-dimensional time-dependent Schrödinger-type of equations as a means to reduce the cost of the quantum-mechanical calculations. The boundary condition is first derived from a semidiscrete approximation of the Schrödinger equation with the advantage that the resulting formulas are automatically compatible with the finite-difference scheme and no further discretization is needed in space. The absorbing boundary condition is expressed as a discrete Dirichlet-to-Neumann map, which can be further approximated in time by using rational approximations of the Laplace transform to enable a more efficient implementation. This approach can be applied to domains with arbitrary geometry. The stability of the zeroth-order and first-order absorbing boundary conditions is proved. We tested the boundary conditions on benchmark problems. The effectiveness is further verified by a time-dependent Hartree-Fock model with Skyrme interactions. The accuracy in terms of energy and nucleon density is examined as well.Slow dynamic nonlinearity describes a poorly understood, creeplike phenomena that occurs in brittle composite materials such as rocks and cement. It is characterized by a drop in stiffness induced by a mechanical conditioning, followed by a log(time) recovery. A consensus theoretical understanding of the behavior has not been developed. Here we introduce an alternative experimental venue with which to inform theory. Unconsolidated glass bead packs are studied rather than rocks or cement because the structure and internal contacts of bead packs are less complex and better understood. Slow dynamics has been observed in such systems previously. However, the measurements to date tend to be irregular. Particular care is used here in the experimental design to overcome the difficulties inherent in bead pack studies. This includes the design of the bead pack support, the use of low-frequency conditioning, and the use of ultrasonic waves as a probe with coda wave interferometry to assess changes. Slow dynamics is observed in our system after three different methods for low-frequency conditioning, one of which has not been reported in the literature previously.An angular momentum conservative pure bulk viscosity term for smoothed particle hydrodynamics (SPH) is proposed in the present paper. This formulation permits independent modeling of shear and bulk viscosities, which is of paramount importance for fluids with large bulk viscosity in situations where sound waves or large Mach numbers are expected. With this aim a dissipative term proportional to the rate of change of the volume is considered at the particle level. The equations of motion are derived from the minimization of a Lagrangian combined with an appropriate dissipation function that depends on this rate of change of particle volume, in analogy with the corresponding entropy production contribution in fluids. Due to the Galilean invariance of the formulation, the new term is shown to exactly conserve linear momentum. Moreover, its invariance under solid-body rotations also ensures the conservation of angular momentum. Two verification cases are proposed the one-dimensional propagation of a sound pulse and a two-dimensional case, modeling the time decay of an accelerating-decelerating pipe flow.