Objective Priors for Invariant e-Values in the Presence of Nuisance Parameters.

This paper aims to contribute to refining the e-values for testing precise hypotheses, especially when dealing with nuisance parameters, leveraging the effectiveness of asymptotic expansions of the posterior. The proposed approach offers the advantage of bypassing the need for elicitation of priors and reference functions for the nuisance parameters and the multidimensional integration step. For this purpose, starting from a Laplace approximation, a posterior distribution for the parameter of interest is only considered and then a suitable objective matching prior is introduced, ensuring that the posterior mode aligns with an equivariant frequentist estimator. Consequently, both Highest Probability Density credible sets and the e-value remain invariant. Some targeted and challenging examples are discussed.

Asymptotics of the meta-atom: plane wave scattering by a single Helmholtz resonator.

Using a combination of multipole methods and the method of matched asymptotic expansions, we present a solution procedure for acoustic plane wave scattering by a single Helmholtz resonator in two dimensions. Closed-form representations for the multipole scattering coefficients of the resonator are derived, valid at low frequencies, with three fundamental configurations examined in detail: the thin-walled, moderately thick-walled and extremely thick-walled limits. Additionally, we examine the impact of dissipation for extremely thick-walled resonators, and also numerically evaluate the scattering, absorption and extinction cross-sections (efficiencies) for representative resonators in all three wall thickness regimes. In general, we observe strong enhancement in both the scattered fields and cross-sections at the Helmholtz resonance frequencies. As expected, dissipation is shown to shift the resonance frequency, reduce the amplitude of the field, and reduce the extinction efficiency at the fundamental Helmholtz resonance. Finally, we confirm results in the literature on Willis-like coupling effects for this resonator design, and connect these findings to earlier works by several of the authors on two-dimensional arrays of resonators, deducing that depolarizability effects (off-diagonal terms) for a single resonator do not ensure the existence of Willis coupling effects (bianisotropy) in bulk. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.

Equations of motion for weakly compressible point vortices.

Equations of motion for compressible point vortices in the plane are obtained in the limit of small Mach number, M, using a Rayleigh-Jansen expansion and the method of Matched Asymptotic Expansions. The solution in the region between vortices is matched to solutions around each vortex core. The motion of the vortices is modified over long time scales [Formula: see text] and [Formula: see text]. Examples are given for co-rotating and co-propagating vortex pairs. The former show a correction to the rotation rate and, in general, to the centre and radius of rotation, while the latter recover the known result that the steady propagation velocity is unchanged. For unsteady configurations, the vortex solution matches to a far field in which acoustic waves are radiated. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Information Geometry of the Exponential Family of Distributions with Progressive Type-II Censoring.

In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the α-connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the α-connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.

Analytical approximation and numerical simulations for periodic travelling water waves.

We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity.This article is part of the theme issue 'Nonlinear water waves'.

High Cooperativity in Negative Feedback can Amplify Noisy Gene Expression.

Burst-like synthesis of protein is a significant source of cell-to-cell variability in protein levels. Negative feedback is a common example of a regulatory mechanism by which such stochasticity can be controlled. Here we consider a specific kind of negative feedback, which makes bursts smaller in the excess of protein. Increasing the strength of the feedback may lead to dramatically different outcomes depending on a key parameter, the noise load, which is defined as the squared coefficient of variation the protein exhibits in the absence of feedback. Combining stochastic simulation with asymptotic analysis, we identify a critical value of noise load: for noise loads smaller than critical, the coefficient of variation remains bounded with increasing feedback strength; contrastingly, if the noise load is larger than critical, the coefficient of variation diverges to infinity in the limit of ever greater feedback strengths. Interestingly, feedbacks with lower cooperativities have higher critical noise loads, suggesting that they can be preferable for controlling noisy proteins.

The computation of biomarkers in pharmacokinetics with the aid of singular perturbation methods.

In pharmacokinetics, exact solutions to one-compartment models with nonlinear elimination kinetics cannot be found analytically, if dosages are assumed to be administered repetitively through extravascular routes (Tang and Xiao in J Pharmacokinet Pharmacodyn 34(6):807-827, 2007). Hence, for the corresponding impulsed dynamical system, alternative methods need to be developed to find approximate solutions. The primary purpose of this paper is to use the method of matched asymptotic expansions (Holmes Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012), a singular perturbation method (Holmes, Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012; Keener Principles of Applied Mathematics, Addison-Wesley, Boston, 1988), to obtain approximate solutions. With this method, we are able to rigorously determine conditions under which there is a stable periodic solution of the model equations. Furthermore, typical important biomarkers that enable the design of practical, efficient and safe drug delivery protocols, such as the time the drug concentration reaches the peak and the peak concentrations, are theoretically estimated by the perturbation method we employ.

Optimal Flat Functions in Carleman-Roumieu Ultraholomorphic Classes in Sectors.

We construct optimal flat functions in Carleman-Roumieu ultraholomorphic classes associated to general strongly nonquasianalytic weight sequences, and defined on sectors of suitably restricted opening. A general procedure is presented in order to obtain linear continuous extension operators, right inverses of the Borel map, for the case of regular weight sequences in the sense of Dyn'kin. Finally, we discuss some examples (including the well-known q-Gevrey case) where such optimal flat functions can be obtained in a more explicit way.

Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays.

We present a novel multipole formulation for computing the band structures of two-dimensional arrays of cylindrical Helmholtz resonators. This formulation is derived by combining existing multipole methods for arrays of ideal cylinders with the method of matched asymptotic expansions. We construct asymptotically close representations for the dispersion equations of the first band surface, correcting and extending an established lowest-order (isotropic) result in the literature for thin-walled resonator arrays. The descriptions we obtain for the first band are accurate over a relatively broad frequency and Bloch vector range and not simply in the long-wavelength and low-frequency regime, as is the case in many classical treatments. Crucially, we are able to capture features of the first band, such as low-frequency anisotropy, over a broad range of filling fractions, wall thicknesses and aperture angles. In addition to describing the first band we use our formulation to compute the first band gap for both thin- and thick-walled resonators, and find that thicker resonator walls correspond to both a narrowing of the first band gap and an increase in the central band gap frequency.

Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays.

We present a solution method which combines the technique of matched asymptotic expansions with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonator arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness becomes very large compared with the aperture width (the extremely thick-walled limit). In this regime, the existing treatment in Part I (Smith & Abrahams, 2022 Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays), with updated parameters, is found to return spurious spectral behaviour. We derive a regularized system which overcomes this issue and also derive compact asymptotic descriptions for the low-frequency dispersion equation in this setting. We find that the matched-asymptotic system is able to recover the first few bands over the entire Brillouin zone with ease, when suitably truncated. A homogenization treatment is outlined for describing the effective bulk modulus and effective density tensor of the resonator array for all wall thicknesses. We demonstrate that extremely thick-walled resonators are able to achieve exceptionally low Helmholtz resonant frequencies, and present closed-form expressions for determining these explicitly. We anticipate that the analytical expressions and the formulation outlined here may prove useful in designing metamaterials for industrial and other applications.

Robustness of subwavelength devices: a case study of cochlea-inspired rainbow sensors.

We derive asymptotic formulae describing how the properties of subwavelength devices are changed by the introduction of errors and imperfections. As a demonstrative example, we study a class of cochlea-inspired rainbow sensors. These are graded metamaterials which have been designed to mimic the frequency separation performed by the cochlea. The device considered here has similar dimensions to the cochlea and has a resonant spectrum that falls within the range of audible frequencies. We show that the device's properties (including its role as a signal filtering device) are stable with respect to small imperfections in the positions and sizes of the resonators. Additionally, under suitable assumptions, if the number of resonators is sufficiently large, then the device's properties are stable under the removal of a resonator.

Boundary-layer effects on electromagnetic and acoustic extraordinary transmission through narrow slits.

We study the problem of resonant extraordinary transmission of electromagnetic and acoustic waves through subwavelength slits in an infinite plate, whose thickness is close to a half-multiple of the wavelength. We build on the matched-asymptotics analysis of Holley & Schnitzer (2019 Wave Motion 91, 102381 (doi:10.1016/j.wavemoti.2019.102381)), who considered a single-slit system assuming an idealized formulation where dissipation is neglected and the electromagnetic and acoustic problems are analogous. We here extend that theory to include thin dissipative boundary layers associated with finite conductivity of the plate in the electromagnetic problem and viscous and thermal effects in the acoustic problem, considering both single-slit and slit-array configurations. By considering a distinguished boundary-layer scaling where dissipative and diffractive effects are comparable, we develop accurate analytical approximations that are generally valid near resonance; the electromagnetic-acoustic analogy is preserved up to a single parameter that is provided explicitly for both scenarios. The theory is shown to be in excellent agreement with GHz-microwave and kHz-acoustic experiments in the literature.

Mathematical modelling of the vitamin C clock reaction.

Chemical clock reactions are characterized by a relatively long induction period followed by a rapid 'switchover' during which the concentration of a clock chemical rises rapidly. In addition to their interest in chemistry education, these reactions are relevant to industrial and biochemical applications. A substrate-depletive, non-autocatalytic clock reaction involving household chemicals (vitamin C, iodine, hydrogen peroxide and starch) is modelled mathematically via a system of nonlinear ordinary differential equations. Following dimensional analysis, the model is analysed in the phase plane and via matched asymptotic expansions. Asymptotic approximations are found to agree closely with numerical solutions in the appropriate time regions. Asymptotic analysis also yields an approximate formula for the dependence of switchover time on initial concentrations and the rate of the slow reaction. This formula is tested via 'kitchen sink chemistry' experiments, and is found to enable a good fit to experimental series varying in initial concentrations of both iodine and vitamin C. The vitamin C clock reaction provides an accessible model system for mathematical chemistry.

On the role of vector modeling in a minimalistic epidemic model.

The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type II functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.

On the homogenization of the acoustic wave propagation in perforated ducts of finite length for an inviscid and a viscous model.

The direct numerical simulation of the acoustic wave propagation in multiperforated absorbers with hundreds or thousands of tiny openings would result in a huge number of basis functions to resolve the microstructure. One is, however, primarily interested in effective and so homogenized transmission and absorption properties and how they are influenced by microstructure and its endpoints. For this, we introduce the surface homogenization that asymptotically decomposes the solution in a macroscopic part, a boundary layer corrector close to the interface and a near-field part close to its ends. The effective transmission and absorption properties are expressed by transmission conditions for the macroscopic solution on an infinitely thin interface and corner conditions at its endpoints to ensure the correct singular behaviour, which are intrinsic to the microstructure. We study and give details on the computation of the effective parameters for an inviscid and a viscous model and show their dependence on geometrical properties of the microstructure for the example of Helmholtz equation. Numerical experiments indicate that with the obtained macroscopic solution representation one can achieve an high accuracy for low and high porosities as well as for viscous boundary conditions while using only a small number of basis functions.

Error bounds for the asymptotic expansion of the Hurwitz zeta function.

In this paper, we reconsider the large-a asymptotic expansion of the Hurwitz zeta function ζ(s,a). New representations for the remainder term of the asymptotic expansion are found and used to obtain sharp and realistic error bounds. Applications to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes G-function and the s-derivative of the Hurwitz zeta function ζ(s,a) are provided. A detailed discussion on the sharpness of our error bounds is also given.

Coffee extraction kinetics in a well mixed system.

The extraction of coffee solubles from roasted and ground coffee is a complex operation, the understanding of which is key to the brewing of high quality coffee. This complexity stems from the fact that brewing of coffee is achieved through a wide variety of techniques each of which depends on a large number of process variables. In this paper, we consider a recent, experimentally validated model of coffee extraction, which describes extraction from a coffee bed using a double porosity model. The model incorporates dissolution and transport of coffee in the coffee bed. The model was shown to accurately describe extraction of coffee solubles from grains in two situations: extraction from a dilute suspension of coffee grains and extraction from a packed coffee bed. The full model equations can only be solved numerically. In this work we consider asymptotic solutions, based on the dominant mechanisms, in the case of coffee extraction from a dilute suspension of coffee grains. Extraction in this well mixed system, can be described by a set of ordinary differential equations. This allows analysis of the extraction kinetics from the coffee grains independent of transport processes associated with flow through packed coffee beds. Coffee extraction for an individual grain is controlled by two processes: a rapid dissolution of coffee from the grain surfaces in conjunction with a much slower diffusion of coffee through the tortuous intragranular pore network to the grain surfaces. Utilising a small parameter resulting from the ratio of these two timescales, we construct asymptotic solutions using the method of matched asymptotic expansions. The asymptotic solutions are compared with numerical solutions and data from coffee extraction experiments. The asymptotic solutions depend on a small number of dimensionless parameters, so the solutions facilitate quick investigation of the influence of various process parameters on the coffee extraction curves.

Heat kernel asymptotic expansions for the Heisenberg sub-Laplacian and the Grushin operator.

The sub-Laplacian on the Heisenberg group and the Grushin operator are typical examples of sub-elliptic operators. Their heat kernels are both given in the form of Laplace-type integrals. By using Laplace's method, the method of stationary phase and the method of steepest descent, we derive the small-time asymptotic expansions for these heat kernels, which are related to the geodesic structure of the induced geometries.