On the spectral properties of L_{+-} in three dimensions

This paper is part of the radial asymptotic stability analysis of the ground state soliton for either the cubic nonlinear Schrodinger or Klein-Gordon equations in three dimensions. We demonstrate by a rigorous method that the linearized scalar operators which arise in this setting, traditionally denoted by L_{+-}, satisfy the gap property, at least over the radial functions. This means that the interval (0,1] does not contain any eigenvalues of L_{+-} and that the threshold 1 is neither an eigenvalue nor a resonance. The gap property is required in order to prove scattering to the ground states for solutions starting on the center-stable manifold associated with these states. This paper therefore provides the final installment in the proof of this scattering property for the cubic Klein-Gordon and Schrodinger equations in the radial case, see the recent theory of Nakanishi and the third author, as well as the earlier work of the third author and Beceanu on NLS. The method developed here is quite general, and applicable to other spectral problems which arise in the theory of nonlinear equations

Domains in Melts of Comb-Coil Diblock Copolymers: Superstrong Segregation Regime

Conditions for the crossover from the strong to the superstrong segregation regime are analyzed for the case of comb-coil diblock copolymers. It is shown that the critical interaction energy between the components required to induce the crossover to the superstrong segregation regime is inversely proportional to mb = 1 + n/m, where n is the degree of polymerization of the side chain and m is the distance between successive grafting points. As a result, the superstrong segregation regime, being rather rare in the case of ordinary block copolymers, has a much better chance to be realized in the case of diblock copolymers with combs grafted to one of the blocks.

Strong-Segregation Theory of Bicontinuous Phases in Block Copolymers

We compute phase diagrams for $A_nB_m$ starblock copolymers in the strong-segregation regime as a function of volume fraction $\phi$, including bicontinuous phases related to minimal surfaces (G, D, and P surfaces) as candidate structures. We present the details of a general method to compute free energies in the strong segregation limit, and demonstrate that the gyroid G phase is the most nearly stable among the bicontinuous phases considered. We explore some effects of conformational asymmetry on the topology of the phase diagram.Comment: 14 pages, latex, 21 figures, to appear in Macromolecule

Inversion of the Diffraction Pattern from an Inhomogeneously Strained Crystal using an Iterative Algorithm

The displacement field in highly non uniformly strained crystals is obtained by addition of constraints to an iterative phase retrieval algorithm. These constraints include direct space density uniformity and also constraints to the sign and derivatives of the different components of the displacement field. This algorithm is applied to an experimental reciprocal space map measured using high resolution X-ray diffraction from an array of silicon lines and the obtained component of the displacement field is in very good agreement with the one calculated using a finite element model.Comment: 5 pages, 4 figure

Forced Symmetry Breaking from SO(3) to SO(2) for Rotating Waves on the Sphere

We consider a small SO(2)-equivariant perturbation of a reaction-diffusion system on the sphere, which is equivariant with respect to the group SO(3) of all rigid rotations. We consider a normally hyperbolic SO(3)-group orbit of a rotating wave on the sphere that persists to a normally hyperbolic SO(2)-invariant manifold $M(\epsilon)$. We investigate the effects of this forced symmetry breaking by studying the perturbed dynamics induced on $M(\epsilon)$ by the above reaction-diffusion system. We prove that depending on the frequency vectors of the rotating waves that form the relative equilibrium SO(3)u_{0}, these rotating waves will give SO(2)-orbits of rotating waves or SO(2)-orbits of modulated rotating waves (if some transversality conditions hold). The orbital stability of these solutions is established as well. Our main tools are the orbit space reduction, Poincare map and implicit function theorem

Finite to infinite steady state solutions, bifurcations of an integro-differential equation

We consider a bistable integral equation which governs the stationary solutions of a convolution model of solid--solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is varied to examine the transition from an infinite number of steady states to three for the continuum limit of the semi--discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem

Primordial helium recombination III: Thomson scattering, isotope shifts, and cumulative results

Upcoming precision measurements of the temperature anisotropy of the cosmic microwave background (CMB) at high multipoles will need to be complemented by a more complete understanding of recombination, which determines the damping of anisotropies on these scales. This is the third in a series of papers describing an accurate theory of HeI and HeII recombination. Here we describe the effect of Thomson scattering, the $^3$He isotope shift, the contribution of rare decays, collisional processes, and peculiar motion. These effects are found to be negligible: Thomson and $^3$He scattering modify the free electron fraction $x_e$ at the level of several $\times 10^{-4}$. The uncertainty in the $2^3P^o-1^1S$ rate is significant, and for conservative estimates gives uncertainties in $x_e$ of order $10^{-3}$. We describe several convergence tests for the atomic level code and its inputs, derive an overall $C_\ell$ error budget, and relate shifts in $x_e(z)$ to the changes in $C_\ell$, which are at the level of 0.5% at $\ell =3000$. Finally, we summarize the main corrections developed thus far. The remaining uncertainty from known effects is $\sim 0.3$ in $x_e$.Comment: 19 pages, 15 figures, to be submitted to PR

Single Stranded DNA Translocation Through A Nanopore: A Master Equation Approach

We study voltage driven translocation of a single stranded (ss) DNA through a membrane channel. Our model, based on a master equation (ME) approach, investigates the probability density function (pdf) of the translocation times, and shows that it can be either double or mono-peaked, depending on the system parameters. We show that the most probable translocation time is proportional to the polymer length, and inversely proportional to the first or second power of the voltage, depending on the initial conditions. The model recovers experimental observations on hetro-polymers when using their properties inside the pore, such as stiffness and polymer-pore interaction.Comment: 7 pages submitted to PR

Dimensional analysis using toric ideals: Primitive invariants

© 2014 Atherton et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.Classical dimensional analysis in its original form starts by expressing the units for derived quantities, such as force, in terms of power products of basic units M, L, T etc. This suggests the use of toric ideal theory from algebraic geometry. Within this the Graver basis provides a unique primitive basis in a well-defined sense, which typically has more terms than the standard Buckingham approach. Some textbook examples are revisited and the full set of primitive invariants found. First, a worked example based on convection is introduced to recall the Buckingham method, but using computer algebra to obtain an integer K matrix from the initial integer A matrix holding the exponents for the derived quantities. The K matrix defines the dimensionless variables. But, rather than this integer linear algebra approach it is shown how, by staying with the power product representation, the full set of invariants (dimensionless groups) is obtained directly from the toric ideal defined by A. One candidate for the set of invariants is a simple basis of the toric ideal. This, although larger than the rank of K, is typically not unique. However, the alternative Graver basis is unique and defines a maximal set of invariants, which are primitive in a simple sense. In addition to the running example four examples are taken from: a windmill, convection, electrodynamics and the hydrogen atom. The method reveals some named invariants. A selection of computer algebra packages is used to show the considerable ease with which both a simple basis and a Graver basis can be found.The third author received funding from Leverhulme Trust Emeritus Fellowship (1-SST-U445) and United Kingdom EPSRC grant: MUCM EP/D049993/1

Directed motion emerging from two coupled random processes: Translocation of a chain through a membrane nanopore driven by binding proteins

We investigate the translocation of a stiff polymer consisting of M monomers through a nanopore in a membrane, in the presence of binding particles (chaperones) that bind onto the polymer, and partially prevent backsliding of the polymer through the pore. The process is characterized by the rates: k for the polymer to make a diffusive jump through the pore, q for unbinding of a chaperone, and the rate q kappa for binding (with a binding strength kappa); except for the case of no binding kappa=0 the presence of the chaperones give rise to an effective force that drives the translocation process. Based on a (2+1) variate master equation, we study in detail the coupled dynamics of diffusive translocation and (partial) rectification by the binding proteins. In particular, we calculate the mean translocation time as a function of the various physical parameters.Comment: 22 pages, 5 figures, IOP styl