In this article, we discuss the question "What Level Set Methods can do for image science". We examine the scope of these techniques in image science, in particular in image segmentation, and introduce some relevant level set techinquies that are potnetially useful for this class of applications. We will show that image science demands multi-disciplinary knowledge and flexible but still robust methods. That is why the Level Set Method has become a thriving technique in this field.

In this work, we are interested in a complex fluid-kinetic model that aims to take into account the compressibility of the droplets of the spray. The ambient fluid is described by Euler-like equations, in which the transfer of momentum and energy form the droplets is taken into account, while the spray is represented by a probability density function satisfying a Vlasov-like equation. Implicit terms crop up because of the compressibility of the droplets. After having derived the model, we prove that golbal conservations are satisfied. Then we present two numerical tests. The first one enables us to validate the numerical code, while the second one is performed in a physically realistic situation.

Using a rather simple model of coupled, time-dependent Ginzburg-Landau equations with two order parameters, we demonstrate that the total Hamilitonian energy of the system contains at least three levels describing point vortices, domain walls and configurations. The global in time dynamics contain then also at least three different time scales for nontrivial motions of domain walls, boundaries of domain walls (frational degree vortices) and paired vortices. In particular, we rigorously show, after an intial time period of adjusting, the domain walls start to move according to motion by the mean-curvature that straighten out the domain walls while the boundaries of such domain walls are essentially fixed. After this motion is completed, the fractional degree vortices begin to move at the next time scale. The motion is relatively simple as it is of constant speed and toward each other to form vortex pairs. Finally, these vortex pairs may move in the final time scale very much like the ordinary vortices in a single time-dependent Ginzburg-Laudau equation.

This paper deals with the Cauchy problem for the quasilinear parabolic equations with arbitrary degeneracy. We give a new and natural definition of BV entropy solution. The uniqueness of the natural BV entropy solutions is obtained. In order to prove the uniqueness, the discontinuity conditions for these solutions are established. Also, we construct an example of the natural BVentropy solution.

We present and discuss a framework for computer-aided multiscale analysis, which enables models at a fine (microscopic/ stochastic) level of description to perform modeling tasks at a coarse (macroscopic/systems) level. These macroscopic modeling tasks, yeilding infomration over long time and large scales, are accomplished through approximately initialized calls to the microscopic simular for only short times and small spatial domains. Traditionally modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive realtions). An arsenal of analytical and numerical techniques for the efficent solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in [1], when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathmatics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales, "coarse" bifurcation analysis, optimization, and control) directly. In effect, the procedure constitutes a system identification based, "closure-on-demand" computational toolkit, bridging microscopic/ stochastic simulation with traditional continuum scientific computation amd numerical analysis. We will breifly survey the application of these "numerical enabling technology" ideas through examples including the computation of coarsely self-similar solutions, and discuss various features, limitations and potential extensions of the approach.

The system of balance laws describing a compressible fluid flow in a nozzle forms a non-strictly hyperbolic system of partial differential equations which, also, is not fully conservative due to the effect of the geometry. First, we investigate the general properties of the system and determine all possible wave combinations. Second, we construct analytically the solutions of the Riemann problem for any values of the left-and right-handed states. For certain values we obtain up to three solution whose structure is carefully described here. In some range of Riemann data, no solutions exists. When three solutions are avialable, then exactly one of them contains two stationary waves which are superimposed in the physical space. We include also numerical plots of these solutions.

A kind of strongly coupled cross diffusion parabolic system, which can be used as the energy transport model in semiconductor science, is studied in this paper. The existence and uniqueness of strong soltution to the initial boundary value problem are obtained, under the condition that the initial data are a small perturbation of an isothermal stationary soltuion. Based on this result, an application of Newton iteration on solving this problem is also given.

The Schrodinger-Poisson-X_alpha (S-P-X_alpha) model is a "local one particle approximation" of the time dependent Hartree-Fock equations. It describes the time evolution of electrons in a quantum model respecting the Pauli principle in an approximate fashion which yields an effective potential that is the difference of the nonlocal Coulomb potential and the third root of the local density. We sketch the formal derivation, existence and uniqueness analysis of the S-P-X_alpha model with/without an external potential. In this paper we deal with numerical simulations based on a time-splitting spectral method, which was used and studied recently for the nonlinear Schrodinger (NLS) equation in the semi-classical regime and shows much better spatial and temporal resolution than finite difference methods. Extensive numerical results of position density an Winger measures in 1d, 2d, and 3d for the S-P-X_alpha model with/without an external potential are presented. These results give an insight to understand the interplay between the nonlocal ("weak") and the local ("strong") nonlinearity.

Human ear has the masking property that certain audible sound becomes inaudible in the presence of another sound. Masking is quantified by the raised threshold from the absolute hearing threshold in quiet. It is of scientific and practical importance to compute masking thresholds. Empirical models on masking have applications in low bit rate digital music compression. A first principle based two level model is developed with partial differential equation (PDE) at the peripheral level and a similarity transform to represent all higher levels of auditory perception. Modeled masking thresholds of banded noise by tonal signals agree well with hearing data.