The purpose of this paper is to derive modeling equations for debris flows on real terrain. Thus, we use curvilinear coordinates adapted to the topography as introduced, e.g., by Bouchut and Westdickenberg, and develop depth-averaged models of gravity-driven saturated mixtures of solid grains and pore fluid on an arbitrary rigid basal surface. First, by only specifying the interaction force and ordering approximations in terms of an aspect ratio between a typical length perpendicular to the topography, and a typical length parallel to the topography, we derive the governing equations for the shallow flow of a binary mixture, driven by gravitational force. In doing so, the non-uniformity through the avalanche depth of the constituent velocities and of the solid volume fraction is accounted for by coefficients of Boussinesq type. Then, the material behaviour peculiarities of both constituents properly enter the theory. One constituent is a granular solid. For its stresses we propose three models, one of them of Mohr-Coulomb type. The other constituent is a Newtonian/non-Newtonian fluid with small viscosity, obeying a viscous bottom friction condition. The final governing equations for the shallow flow of the mixture, incorporating the constitutive assumptions, are deduced, and the limiting equilibrium is then investigated.

By using a new bilinear estimate, a pointwise estimate of the generalized Oseen kernel, and an idea of a fractional bootstrap, we establish optimal local smoothing and decay estimates of solutions to the Navier-Stokes equations with fractional dissipation. We also show that solutions are analytic in space variables.

We extend previous work and present a general approach for solving partial differential equations in complex, stationary, or moving geometries with Dirichlet, Neumann, and Robin boundary conditions. Using an implicit representation of the geometry through an auxilliary phase field function, which replaces the sharp boundary of the domain with a diffuse layer (e.g. diffuse domain), the equation is reformulated on a larger regular domain. The resulting partial differential equation is of the same order as the original equation, with additional lower order terms to approximate the boundary conditions. The reformulated equation can be solved by standard numerical techniques. We use the method of matched asymptotic expansions to show that solutions of the reformulated equations converge to those of the original equations. We provide numerical simulations which confim this analysis. We also present applications of the method to growing domains and complex three-dimensional structures and we discuss applications to cell biology and heteroepitaxy.

A time domain blind source separation algorithm of convolutive sound mixtures is studied based on a compact partial inversion formula in closed form. An L1-constrained minimization problem is formulated to find demixing filter coefficients for source separation while capturing scaling invariance and sparseness of solutions. The minimization aims to reduce (lagged) cross correlations of the mixture signals, which are modeled stochastically. The problem is non-convex, however it is put in a nonlinear least squares form where the robust and convergent Levenberg-Marquardt iterative method is applicable to compute local minimizers. Efficiency is achieved in recovering lower dimensional demixing filter solutions than the physical ones. Computations on recorded and synthetic mixtures show satisfactory performance, and are compared with other iterative methods.

The dispersion relation and shock structure of a gas mixture undergoing a bimolecular chemical reaction are studied by means of a hydrodynamic model deduced from the relevant kinetic equations. Qualitative changes in the solution, in particular loss of smoothness, for varying parameters (including Mach number and strength of the chemical reaction rate) are investigated, and numerical results are presented. In the limits of vanishing or diverging reactive relaxation times the “equilibrium” and “frozen” thermodynamical situations are recovered.

This paper discusses the Lagrange interpolation problem in continuous bivariate spline spaces over regular triangulations. By using the so-called Lagrange interpolation set along piecewise algebraic curves, we develop a new approach of constructing the interpolation set for continuous spline spaces. We show the property of this set on star region, and construct the interpo- lation set for continuous bivariate spline spaces over arbitrary triangulations. The construction only depends on the number of points on the piecewise algebraic curve in each cell.

A generalization of the classical Erdös and Rényi (ER) random graph is introduced and investigated. A generalized random graph (GRG) admits different values of probabilities for its edges rather than a single probability uniformly for all edges as in the ER model. In probabilistic terms, the vertices of a GRG are no longer statistically identical in general, giving rise to the pos- sibility of complex network topology. Depending on their surrounding edge probabilities, vertices of a GRG can be either “homogeneous” or “heterogeneous”. We study the statistical properties of the degree of a single vertex, as well as the degree distribution over the whole GRG. We distinguish the degree distribution for the entire random graph ensemble and the degree frequency for a particular graph realization, and study the mathematical relationship between them. Finally, the connectivity of a GRG, a property which is highly related to the degree distribution, is briefly discussed and some useful results are derived.

We study explicit dispersion and uniform L1-stability estimates to the Vlasov-Poisson system for a collisionless plasma in a half space, when the initial data is sufficiently small and decays fast enough in phase space. This extends the previous results on the dispersion and stability estimates for the whole space case.

We describe an algorithm for solving steady one-dimensional convex-like Hamilton-Jacobi equations using a L1-minimization technique on piecewise linear approximations. For a large class of convex Hamiltonians, the algorithm is proven to be convergent and of optimal complexity whenever the viscosity solution is q-semiconcave. Numerical results are presented to illustrate the performance of the method.

We study the reliability of large networks of pulse-coupled oscillators in response to fluctuating stimuli. Reliability means that a stimulus elicits essentially identical responses upon repeated presentations. We view the problem on two scales: neuronal reliability, which concerns the repeatability of spike times of individual oscillators embedded within a network, and pooled-response reliability, which addresses the repeatability of the total output from the network. We find that individual embedded oscillators can be reliable or unreliable depending on network conditions, whereas pooled responses of sufficiently large networks are mostly reliable.