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

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

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

Alfv\'en Reflection and Reverberation in the Solar Atmosphere

Magneto-atmospheres with Alfv\'en speed [a] that increases monotonically with height are often used to model the solar atmosphere, at least out to several solar radii. A common example involves uniform vertical or inclined magnetic field in an isothermal atmosphere, for which the Alfv\'en speed is exponential. We address the issue of internal reflection in such atmospheres, both for time-harmonic and for transient waves. It is found that a mathematical boundary condition may be devised that corresponds to perfect absorption at infinity, and, using this, that many atmospheres where a(x) is analytic and unbounded present no internal reflection of harmonic Alfv\'en waves. However, except for certain special cases, such solutions are accompanied by a wake, which may be thought of as a kind of reflection. For the initial-value problem where a harmonic source is suddenly switched on (and optionally off), there is also an associated transient that normally decays with time as O(t-1) or O(t-1 ln t), depending on the phase of the driver. Unlike the steady-state harmonic solutions, the transient does reflect weakly. Alfv\'en waves in the solar corona driven by a finite-duration train of p-modes are expected to leave such transients.Comment: Accepted by Solar Physic

Amine functionalization of cholecyst-derived extracellular matrix with generation 1 PAMAM dendrimer

This document is the unedited author's version of a Submitted Work that was subsequently accepted for publication in Biomacromolecules, copyright © American Chemical Society after peer review. To access the final edited and published work, see http://pubs.acs.org/doi/pdf/10.1021/bm701055k.A method to functionalize cholecyst-derived extracellular matrix (CEM) with free amine groups was established in an attempt to improve its potential for tethering of bioactive molecules. CEM was incorporated with Generation-1 polyamidoamine (G1 PAMAM) dendrimer by using N-(3-dimethylaminopropyl)-N′-ethylcarbodiimide and N-hydroxysuccinimide cross-linking system. The nature of incorporation of PAMAM dendrimer was evaluated using shrink temperature measurements, Fourier transform infrared (FTIR) assessment, ninhydrin assay, and swellability. The effects of PAMAM incorporation on mechanical and degradation properties of CEM were evaluated using a uniaxial mechanical test and collagenase degradation assay, respectively. Ninhydrin assay and FTIR assessment confirmed the presence of increasing free amine groups with increasing quantity of PAMAM in dendrimer-incorporated CEM (DENCEM) scaffolds. The amount of dendrimer used was found to be critical in controlling scaffold degradation, shrink temperature, and free amine content. Cell culture studies showed that fibroblasts seeded on DENCEM maintained their metabolic activity and ability to proliferate in vitro. In addition, fluorescence cell staining and scanning electron microscopy analysis of cell-seeded DENCEM showed preservation of normal fibroblast morphology and phenotype

Black Hole Scattering from Monodromy

We study scattering coefficients in black hole spacetimes using analytic properties of complexified wave equations. For a concrete example, we analyze the singularities of the Teukolsky equation and relate the corresponding monodromies to scattering data. These techniques, valid in full generality, provide insights into complex-analytic properties of greybody factors and quasinormal modes. This leads to new perturbative and numerical methods which are in good agreement with previous results.Comment: 28 pages + appendices, 2 figures. For Mathematica calculation of Stokes multipliers, download "StokesNotebook" from https://sites.google.com/site/justblackholes/techy-zon

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

Lerch Φ asymptotics

We use a Mellin-Barnes integral representation for the Lerch transcendent Φ(z,s,a) to obtain large z asymptotic approximations. The simplest divergent asymptotic approximation terminates in the case that s is an integer.For non-integer s the asymptotic approximations consists of the sum of two series. The first one is in powers of (lnz)−1 and the second one is in powers of z−1. Although the second series converges, it is completely hidden in the divergent tail of the first series. We use resummation and optimal truncation to make the second series visible