Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation

We demonstrate the conjunction of new exponential-asymptotic effects in the context of a second order nonlinear ordinary differential equation with a small parameter. First, we show how to use a hyperasymptotic, beyond-all-orders approach to seed a numerical solver of a nonlinear ordinary differential equation with sufficiently accurate initial data so as to track a specific solution in the presence of an attractor. Second, we demonstrate the necessary role of a higher order Stokes phenomenon in analytically tracking the transition between asymptotic behaviours in a heteroclinic solution. Third, we carry out a double resummation involving both subdominant and sub-subdominant transseries to achieve the two-dimensional (in terms of the arbitrary constants) uniform approximation that allows the exploration of the behaviour of a two parameter set of solutions across wide regions of the independent variable. This is the first time all three effects have been studied jointly in the context of an asymptotic treatment of a nonlinear ordinary differential equation with a parameter. This paper provides an exponential asymptotic algorithm for attacking such problems when they occur. The availability of explicit results would depend on the individual equation under study

Exponentially-improved asymptotics and numerics for the (un)perturbed first Painlevé equation

The solutions of the perturbed first Painlev\'e equation $y"=6y^2-x^\mu$, $\mu>-4$, are uniquely determined by the free constant $C$ multiplying the exponentially small terms in the complete large $x$ asymptotic expansions. Full details are given, including the nonlinear Stokes phenomenon, and the computation of the relevant Stokes multipliers. We derive asymptotic approximations, depending on $C$, for the locations of the singularities that appear on the boundary of the sectors of validity of these exponentially-improved asymptotic expansions. Several numerical examples illustrate the power of the approximations. For the tri-tronqu\'ee solution of the unperturbed first Painlev\'e equation we give highly accurate numerics for the values at the origin and the locations of the zeros and poles.Comment: 13 pages, 3 figure

The effect of ceria co-doping on chemical stability and fracture toughness of Y-TZP

The fracture toughness and ageing resistance of yttria, ceria-stabilized tetragonal zirconia polycrystals (Y, Ce-TZP) were evaluated as a function of grain size and ceria content. Very fine grained, fully dense materials could be produced by sinter forging at relatively low temperatures (1150–1200 °C). The ageing resistance in hot water (185 °C) of 2 mol% Y2O3-stabilized TZP is strongly enhanced by alloying with ceria. The ceria content necessary to avoid degradation completely, decreases with grain size. The toughness of fully dense Y, Ce-TZP is 7–9 MPa m1/2 for grain sizes down to 0.2 mgrm. No or very little transformation took place during fracturing and no clear variation with grain size was observed for the toughness at grain sizes up to 0.8 mgrm. Reversible transformation and crack deflection may explain the observed toughness values

The North Wyke Farm Platform: Eddy Covariance Greenhouse Gas Data

The North Wyke Farm Platform (NWFP) was established in 2010 to study and improve grassland livestock production at the farm-scale. The NWFP uses a combination of environmental sensors, routine field and lab-based measurements, and detailed management records to monitor livestock and crop production, emissions to water, emissions to air, soil health, and biodiversity. The rich NWFP datasets help researchers to evaluate the effectiveness of different grassland (and arable) farming systems, which in turn, contributes to the development of sustainable, resilient and net zero land management strategies. This document serves as a user guide to the greenhouse gas data captured using Eddy Covariance analysers and is associated with other dedicated user guides that detail the design, establishment and development of the NWFP, field events, and the quality control process of datasets

Theory of tunneling spectroscopy of normal metal/ferromagnet/spin-triplet superconductor junctions

We study the tunneling conductance of a ballistic normal metal / ferromagnet / spin-triplet superconductor junction using the extended Blonder-Tinkham-Klapwijk formalism as a model for a $c$-axis oriented Au / SrRuO$_{3}$ / Sr$_{2}$RuO$_{4}$ junction. We compare chiral $p$-wave (CPW) and helical $p$-wave (HPW) pair potentials, combined with ferromagnet magnetization directions parallel and perpendicular to the interface. For fixed $\theta_{M}$, where $\theta_{M}$ is a direction of magnetization in the ferromagnet measured from the $c$-axis, the tunneling conductance of CPW and HPW clearly show different voltage dependencies. It is found that the cases where the $d$-vector is perpendicular to the magnetization direction (CPW with $\theta_{M} = \pi/2$ and HPW with $\theta_{M} = 0$) are identical. The obtained results serve as a guide to determine the pairing symmetry of the spin-triplet superconductor Sr$_{2}$RuO$_{4}$.Comment: 12 pages, 7 figures. There is also a supplementary (not uploaded

Stokes phenomenon and matched asymptotic expansions

This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

Tunable critical field in Rashba superconductor thin-films

The upper critical field in type II superconductors is limited by the Pauli paramagnetic limit. In superconductors with strong Rashba spin-orbit coupling this limit can be overcome by forming a helical state. Here we quantitatively study the magnetic field-temperature phase diagram of finite-size superconductors with Rashba spin-orbit coupling. We discuss the effect of finite size and shape anisotropy. We demonstrate that the critical field is controllable by intrinsic parameters such as spin-orbit coupling strength and tunable parameters such as sample geometry and applied field direction. Our study opens new avenues for the design of superconducting spin-valves.Comment: 5 pages, 4 figures, supplemental material