Tracking bifurcating solutions of a model biological pattern generator

We study heterogeneous steady-state solutions of a cell-chemotaxis model for generating biological spatial patterns in two-dimensional domains with zero flux boundary conditions. We use the finite-element package ENTWIFE to investigate bifurcation from the uniform solution as the chemotactic parameter varies and as the domain scale and geometry change. We show that this simple cell-chemotaxis model can produce a remarkably wide and surprising range of complex spatial patterns

Electron Spin-Lattice Relaxation of doped Yb3+ ions in YBa2Cu3Ox

The electron spin-lattice relaxation (SLR) times T1 of Yb3+‡ ions were measured from the temperature dependence of electron spin resonance linewidth in Y0.99Yb0.01Ba2Cu3Ox with different oxygen contents. Raman relaxation processes dominate the electron SLR. Derived from the temperature dependence of the SLR rate, the Debye temperature (Td) increases with the critical temperature Tc and oxygen content x. Keywords: EPR; ESR; Electron spin-lattice relaxation; Debye temperature; Critical temperatureComment: 5 Pages 4 Figure

Electron Spin-Lattice Relaxation of Er3+ ions in Er0.01Y0.99Ba2Cu3Ox

The temperature dependence of the electron spin-lattice relaxation SLR was studied in Er0.01Y0.99Ba2Cu3Ox compounds. The data derived from the electron spin resonance ESR and SLR measurements were compared to those from inelastic neutron scattering studies. SLR of Er3+ ions in the temperature range from 20 K to 65 K can be explained by the resonant phonon relaxation process with the involvement of the lowest excited crystalline-electric-field electronic states of Er3+. These results are consistent with a local phase separation effects. Possible mechanisms of the ESR line broadening at lower temperatures are discussed. Keywords: YBCO; EPR; ESR; Electron spin-lattice relaxation time, T ; Crystalline-electric-fieldComment: 6 pages, 4 figure

Influence of shear flow on vesicles near a wall: a numerical study

We describe the dynamics of three-dimensional fluid vesicles in steady shear flow in the vicinity of a wall. This is analyzed numerically at low Reynolds numbers using a boundary element method. The area-incompressible vesicle exhibits bending elasticity. Forces due to adhesion or gravity oppose the hydrodynamic lift force driving the vesicle away from a wall. We investigate three cases. First, a neutrally buoyant vesicle is placed in the vicinity of a wall which acts only as a geometrical constraint. We find that the lift velocity is linearly proportional to shear rate and decreases with increasing distance between the vesicle and the wall. Second, with a vesicle filled with a denser fluid, we find a stationary hovering state. We present an estimate of the viscous lift force which seems to agree with recent experiments of Lorz et al. [Europhys. Lett., vol. 51, 468 (2000)]. Third, if the wall exerts an additional adhesive force, we investigate the dynamical unbinding transition which occurs at an adhesion strength linearly proportional to the shear rate.Comment: 17 pages (incl. 10 figures), RevTeX (figures in PostScript

The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.Comment: To appear in Communications in Mathematical Physic

The McKean-Vlasov Equation in Finite Volume

We study the McKean--Vlasov equation on the finite tori of length scale $L$ in $d$--dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in \cite{GP} and \cite{KM}. Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, $\theta^{\sharp}$ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for $\theta < \theta^{\sharp}$ and prove, abstractly, that a {\it critical} transition must occur at $\theta = \theta^{\sharp}$. However for this system we show that under generic conditions -- $L$ large, $d \geq 2$ and isotropic interactions -- the phase transition is in fact discontinuous and occurs at some \theta\t < \theta^{\sharp}. Finally, for H--stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the \theta\t(L) tend to a definitive non--trivial limit as $L\to\infty$

Matrix factorisations and D-branes on K3

D-branes on K3 are analysed from three different points of view. For deformations of hypersurfaces in weighted projected space we use geometrical methods as well as matrix factorisation techniques. Furthermore, we study the D-branes on the T^4/\Z_4 orbifold line in conformal field theory. The behaviour of the D-branes under deformations of the bulk theory are studied in detail, and good agreement between the different descriptions is found.Comment: 35 pages, no figure

Obstructions and lines of marginal stability from the world-sheet

The behaviour of supersymmetric D-branes under deformations of the closed string background is studied using world-sheet methods. We explain how lines of marginal stability and obstructions arise from this point of view. We also show why N=2 B-type branes may be obstructed against (cc) perturbations, but why such obstructions do not occur for N=4 superconformal branes at c=6, i.e. for half-supersymmetric D-branes on K3. Our analysis is based on a field theory approach in superspace, as well as on techniques from perturbed conformal field theory.Comment: 32 page

A Mathematical Model of Liver Cell Aggregation In Vitro

The behavior of mammalian cells within three-dimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium, and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cell-ECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work

Antiferromagnetic Domains and Superconductivity in UPt3

We explore the response of an unconventional superconductor to spatially inhomogeneous antiferromagnetism (SIAFM). Symmetry allows the superconducting order parameter in the E-representation models for UPt3 to couple directly to the AFM order parameter. The Ginzburg-Landau equations for coupled superconductivity and SIAFM are solved numerically for two possible SIAFM configurations: (I) abutting antiferromagnetic domains of uniform size, and (II) quenched random disorder of `nanodomains' in a uniform AFM background. We discuss the contributions to the free energy, specific heat, and order parameter for these models. Neither model provides a satisfactory account of experiment, but results from the two models differ significantly. Our results demonstrate that the response of an E_{2u} superconductor to SIAFM is strongly dependent on the spatial dependence of AFM order; no conclusion can be drawn regarding the compatibility of E_{2u} superconductivity with UPt3 that is independent of assumptions on the spatial dependence of AFMComment: 12 pages, 13 figures, to appear in Phys. Rev.