We model the evolution of the concentration field of macromolecules in a symmetric field-flow fractionation (FFF) channel by a onedimensional advection-diffusion equation. The coefficients are precisely determined from the fluid dynamics. This model gives quantitative predictions of the time of elution of the molecules and the width in time of the concentration pulse. The model is rigorously su...

The long term aim is to use modern dynamical systems theory to derive discretisations of noisy, dissipative partial differential equations. As a first step we here consider a small domain and apply stochastic centre manifold techniques to derive a model. The approach automatically parametrises subgrid scale processes induced by spatially distributed stochastic noise. It is important to discreti...

Giuseppe Gaeta Centre de Physique Th eorique, Ecole Polytechnique F 91128 Palaiseau (France) and Departamento de Fisica Teorica II, Universidad Complutense E 28040 Madrid (Spain) Summary. We prove that any dynamical system on a G-manifold M which is equivariant under the G action, can be decomposed into the semidirect product of an autonomous dynamics in the G-orbit space = M=G, and a dynamics ...

In shallow turbulent flows such as floods and tsunami vertical mixing tends to smooth out the flow characteristics in cross-sectional direction. The evolution of the average cross-flow characteristics presents considerable interest. We model such flows using the k-ω model of turbulence in the framework of the centre manifold theory. We tested the approach on an artificial diffusion problem for ...

A general nonparametric technique is proposed for the description of geometric manifold energy of unorganized data. Minimizing the energy leads to an optimal cycle, from which underlying manifolds are easily distinguished. We design a new framework for manifold clustering based on energy minimization. In addition, we propose the active tabu search method to approximately solve for the optimal s...

We study the existence of grain boundaries in the Swift-Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ODEs in normal form. We show persistence of the leading-order approximation using transversality induced by wavenum...

In this article we describe recent (within the past 4 or 5 years) results in minimal surface theory, with particular emphasis on the existence and regularity theory. For a more general survey, including some technical background discussion and a more comprehensive bibliography, the reader is referred to [30]. Throughout the article «?f will denote ÄJ-dimensional Hausdorff measure (in Euclidean ...

In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, w...

Consider the dynamics of turbulent flow in rivers, estuaries and floods. Based on the widely used k-ǫ model for turbulence, we use the techniques of centre manifold theory to derive dynamical models for the evolution of the water depth and of vertically averaged flow velocity and turbulent parameters. This new model for the shallow water dynamics of turbulent flow: resolves the vertical structu...

We consider pointed Lorentzian manifolds and construct “canonical” foliations by constant mean curvature (CMC) hypersurfaces. Our result assumes a uniform bound on the local sup-norm of the curvature of the manifold and on its local injectivity radius, only. The prescribed curvature problem under consideration is a nonlinear elliptic equation whose coefficients have limited regularity. The CMC ...