Invariant Lagrangians, mechanical connections and the Lagrange-Poincare equations

We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated to Lagrangians which have a kinetic energy function that is defined by a Riemannian metric. In this paper we extend this notion to arbitrary Lagrangians. We then derive the reduced Lagrange-Poincare equations in a new fashion and we show how solutions of the Euler-Lagrange equations can be reconstructed with the help of the mechanical connection. Illustrative examples confirm the theory.Comment: 22 pages, to appear in J. Phys. A: Math. Theor., D2HFest special issu

Reaction-diffusion models for biological pattern formation

We consider the use of reaction-diffusion equations to model biological pattern formation and describe the derivation of the reaction-terms for several illustrative examples. After a brief discussion of the Turing instability in such systems we extend the model formulation to incorporate domain growth. Comparisons are drawn between solution behaviour on growing domains and recent results on self-replicating patterns on domains of fixed size

The Berwald-type linearisation of generalised connections

We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalised connections on such bundles.Comment: 25 page

Mode doubling and tripling in reaction-diffusion patterns on growing domains: A piece-wise linear model

Reaction-diffusion equations are ubiquitous as models of biological pattern formation. In a recent paper [4] we have shown that incorporation of domain growth in a reaction-diffusion model generates a sequence of quasi-steady patterns and can provide a mechanism for increased reliability of pattern selection. In this paper we analyse the model to examine the transitions between patterns in the sequence. Introducing a piecewise linear approximation we find closed form approximate solutions for steady-state patterns by exploiting a small parameter, the ratio of diffusivities, in a singular perturbation expansion. We consider the existence of these steady-state solutions as a parameter related to the domain length is varied and predict the point at which the solution ceases to exist, which we identify with the onset of transition between patterns for the sequence generated on the growing domain. Applying these results to the model in one spatial dimension we are able to predict the mechanism and timing of transitions between quasi-steady patterns in the sequence. We also highlight a novel sequence behaviour, mode-tripling, which is a consequence of a symmetry in the reaction term of the reaction-diffusion system

Routh's procedure for non-Abelian symmetry groups

We extend Routh's reduction procedure to an arbitrary Lagrangian system (that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies) with a symmetry group which is not necessarily Abelian. To do so we analyse the restriction of the Euler-Lagrange field to a level set of momentum in velocity phase space. We present a new method of analysis based on the use of quasi-velocities. We discuss the reconstruction of solutions of the full Euler-Lagrange equations from those of the reduced equations.Comment: 30 pages, to appear in J Math Phy

A metabolite-sensitive, thermodynamically-constrained model of\ud cardiac cross-bridge cycling: Implications for force development during ischemia

We present a metabolically regulated model of cardiac active force generation with which we investigate the effects of ischemia on maximum forceproduction. Our model, based on the Rice et al. (2008) model of cross-bridge kinetics, reproduces many of the observed effects of MgATP, MgADP, Pi and H+ on force development while still retaining the force/length/Ca2+ properties of the original model. We introduce three new parameters to account for the competitive binding of H+ to the Ca2+ binding site on troponin C and the binding of MgADP within the cross-bridge cycle. These parameters along with the Pi and H+ regulatory steps within the cross-bridge cycle were constrained using data from the literature and validated using a range of metabolic and sinusoidal length perturbation protocols. The placement of the MgADP binding step between two strongly-bound and force-generating states leads to the emergence of an unexpected effect on the force-MgADP curve, where the trend of the relationship (positive or negative) depends on the concentrations of the other metabolites and [H+]. The model is used to investigate the sensitivity of maximum force production to changes in metabolite concentrations during the development of ischemia

Reaction and diffusion on growing domains: Scenarios for robust pattern formation

We investigate the sequence of patterns generated by a reaction—diffusion system on a growing domain. We derive a general evolution equation to incorporate domain growth in reaction—diffusion models and consider the case of slow and isotropic domain growth in one spatial dimension. We use a self-similarity argument to predict a frequency-doubling sequence of patterns for exponential domain growth and we find numerically that frequency-doubling is realized for a finite range of exponential growth rate. We consider pattern formation under different forms for the growth and show that in one dimension domain growth may be a mechanism for increased robustness of pattern formation

Spin-orbit splitting of image states

We quantify the effect of the spin-orbit interaction on the Rydberg-like series of image state electrons at the (111) and (001) surface of Ir, Pt and Au. Using relativistic multiple-scattering methods we find Rashba-like dispersions with Delta E(K)=gamma K with values of gamma for n=1 states in the range 38-88 meV Angstrom. Extending the phase-accumulation model to include spin-orbit scattering we find that the splittings vary like 1/(n+a)^3 where a is the quantum defect and that they are related to the probability of spin-flip scattering at the surface. The splittings should be observable experimentally being larger in magnitude than some exchange-splittings that have been resolved by inverse photoemission, and are comparable to linewidths from inelastic lifetimes.Comment: 10 pages, 4 figure

An embedding potential definition of channel functions

We show that the imaginary part of the embedding potential, a generalised logarithmic derivative, defined over the interface between an electrical lead and some conductor, has orthogonal eigenfunctions which define conduction channels into and out of the lead. In the case of an infinitely extended interface we establish the relationship between these eigenfunctions and the Bloch states evaluated over the interface. Using the new channel functions, a well-known result for the total transmission through the conductor system is simply derived.Comment: 14 pages, 2 figure