We present a simple proof on the formation of flocking to the Cucker-Smale system based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the initial states in which the exponential convergence to flocking state will occur. For large particle systems, we give a rigorous justification for the mean-field limit from the many particle Cucker-Smale system to the Vlasov equation with flocking dissipation as the number of particles goes to infinity.

This paper is concerned with fast solution of high frequency acoustic scattering problems in two dimensions. We introduce a directional multiscale algorithm for the $N$-body problem of the two dimensional Helmholtz kernel. The algorithm follows the approach developed in Engquist and Ying, SIAM J. Sci. Comput., 29 (4), 2007, where the three dimensional case was studied. The main observation is that, for two regions that follow a directional parabolic geometric conguration, the interaction between these two regions through the 2D Helmholtz kernel is approximately low rank. We propose an improved randomized procedure for generating the low rank separated representation for the interaction between these regions. Based on this representation, the computation of the far field interaction is organized in a multidirectional and multiscale way to achieve maximum efficiency. The proposed algorithm is accurate and has the optimal $O(NlogN)$ complexity for problems from two dimensional scattering applications. Finally, we combine this fast directional algorithm with standard boundary integral formulations to solve acoustic scattering problems that are of thousands of wavelengths in size.

Periodic arrays are structures consisting of geometrically identical subdomains, usually named periodic cells. In this paper, by taking the Helmholtz equation as a model, we consider the definition and evaluation of scattering operators for general semi-infinite periodic arrays. The well-posedness of the Helmholtz equation is established via the limiting absorption principle. A method based on the doubling procedure and extrapolation technique is first proposed to compute the scattering operators of Sommerfeld-to-Sommerfeld type. The advantages of this method are the robustness and simplicity of implementation. However, it suffers from the heavy computational cost and the resonance wavenumbers. To overcome these shortcomings, we propose another more efficient method based on a conjecture about the asymptotic behavior of limiting absorption principle solutions. Numerical evidences suggest that this method presents the same results as the first one.

We study the global properties of the solutions for the initial value problem for the Einstein-Boltzmann system with positive cosmological constant and arbitrarily large initial data, in the spatially homogeneous case, in a Robertson-Walker space-time.

In this paper we prove the uniqueness of weak solutions to a model of electro-kinetic fluid which consists of a momentum equation together with transport equations of charges. Our result is new in that it holds even when the momentum vanishes. The existence of weak solutions has been proved in P.Biler, W. Hebisch and T. Nadzieja, Nonlinear Anal. TMA, 23, 1189-1209, 1994, R. Ryham, C. Liu and Z-Q. Wang, Preprint, 2005, R. Ryham, arXiv: 0810.2064 v1 (math.AP) 12, Oct 2008.

In this paper, we investigate a new Gradient-Vector-Flow (GVF)-inspired static external force field for active contour models, deriving from the edge map of a given image and allowing to increase the capture range. Contrary to prior related works, we reduce the number of unknowns to a single one v by assuming that the expected vector field is the gradient field of a scalar function. The model is phrased in terms of a functional minimization problem comprising a data fidelity term and a regularizer based on the supremum norm of $Dv$.

The minimization is achieved by solving a second order singular degenerate parabolic equation. A comparison principle as well as the existence/uniqueness of a viscosity solution together with regularity results are established. Experimental results for image segmentation with details of the algorithm are also presented.

We study a stochastic Cucker-Smale flocking system in which particles interact with the environment through white noise. We provide the definition of flocking for the stochastic system, and show that when the communication rate is constant, the system exhibits a flocking behavior independent of the initial configurations. For the case of a radially symmetric communication rate with a positive lower bound, we show that the relative fluctuations of the particle velocity around the mean velocity have a uniformly bounded variance in time. We conclude with numerical simulations that validate our analytical results.

We study the link which exists between microscopic (detailed) models for the evolution of the electronic configurations in a population of ions and the macroscopic (average ion) models. Rigorous asymptotics are presented in situations where they exist (large temperature; almost empty or almost full shells), and numerical simulations are presented.

We study the evolution of the probability density of an asexual, one locus population under natural selection and random evolution. This evolution is governed by a Fokker-Planck equation with degenerate coefficients on the boundaries, supplemented by a pair of conservation laws. It is readily shown that no classical or standard weak solution definition yields solvability of the problem. We provide an appropriate definition of weak solution for the problem, for which we show existence and uniqueness. The solution displays a very distinctive structure and, for large time, we show convergence to a unique stationary solution that turns out to be a singular measure supported at the endpoints. An exponential rate of convergence to this steady state is also proved.

We consider the linear growth-fragmentation equation arising in the modelling of cell division or polymerisation processes. For constant coefficients, we prove that the dynamics converges to the steady state with an exponential rate. The control on the initial data uses an elaborate $L1$-norm that seems to be necessary. It also reflects the main idea of the proof, which is to use an anti-derivative of the solution. The main technical difficulty is related to the entropy dissipation rate, which is too weak to produce a Poincaré inequality.

The MacPherson-Srolovitz formula has been recently established as a generalization of the two dimensional von Neumann relation for microstructure coarsening. In this paper, we present an extension of the MacPherson-Srolovitz formula under more general junction conditions.