In this paper, we present a novel pairwise-force smoothed particle hydrodynamics (PF-SPH) model to allow modeling of various interactions at interfaces in real time. Realistic capture of interactions at interfaces is a challenging problem for SPH-based simulations, especially for scenarios involving multiple interactions at different interfaces. Our PF-SPH model can readily handle multiple kinds of interactions simultaneously in a single simulation; its basis is to use a larger support radius than that used in standard SPH. We adopt a novel anisotropic filtering term to further improve the performance of interaction forces. The proposed model is stable; furthermore, it avoids the particle clustering problem which commonly occurs at the free surface. We show how our model can be used to capture various interactions. We also consider the close connection between droplets and bubbles, and show how to animate bubbles rising in liquid as well as bubbles in air. Our method is versatile, physically plausible and easy-to-implement. Examples are provided to demonstrate the capabilities and effectiveness of our approach.

We find the shock wave solutions in a class of cosmological backgrounds with a null singularity, each of these backgrounds admits a matrix description. A shock wave solution breaks all supersymmetry meanwhile indicates that the interaction between two static D0-branes cancel, thus provides basic evidence for the matrix description. The probe action of a D0-brane in the background of another suggests that the usual perturbative expansion of matrix model breaks down.

When two topologically identical shapes are blended, various possible transformation paths exist from the source shape to the target shape. Which one is the most plausible? Here we propose that the transformation process should obey a quasi-physical law. This paper combines morphing with deformation theory from continuum mechanics. By using strain energy, which reflects the magnitude of deformation, as an objective function, we convert the problem of path interpolation into an unconstrained optimization problem. To reduce the number of variables in the optimization we adopt shape functions, as used in the finite element method (FEM). A point-to-point correspondence between the source and target shapes is naturally established using these polynomial functions plus a distance map.

Based on earlier work of the latter two named authors on the higher super-Teichmueller space with N=1, a component of the flat OSp(1\2) connections on a punctured surface, here we extend to the case N=2 of flat OSp(2\2) connections. Indeed, we construct here coordinates on the higher super-Teichmueller space of a surface F with at least one puncture associated to the supergroup OSp(2\2), which in particular specializes to give another treatment for N=1 simpler than the earlier work. The Minkowski space in the current case, where the corresponding super Fuchsian groups act, is replaced by the superspace R2,2\4, and the familiar lambda lengths are extended by odd invariants of triples of special isotropic vectors in R2,2\4 as well as extra bosonic parameters, which we call ratios, defining a flat R+-connection on F. As in the pure bosonic or N=1 cases, we derive the analogue of Ptolemy transformations for all these new variables.

We derive a lower bound on the ground state energy of the Hubbard model for given value of the total spin. In combination with the upper bound derived previously by Giuliani, our result proves that in the low density limit, the leading order correction compared to the ground state energy of a non-interacting lattice Fermi gas is given by $8\pi a \rho_u \rho_d$, where $\rho_{u(d)}$ denotes the density of the spin-up (down) particles, and $a$ is the scattering length of the contact interaction potential. This result extends previous work on the corresponding continuum model to the lattice case.