Proceedings. Mathematical, Physical, and Engineering Sciences

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 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...

In noise control applications, a perfect metasurface absorber would have the desirable traits of not only mitigating unwanted sound, but also being much thinner than the wavelengths of interest. Such deep-subwavelength performance is difficult to achieve technologically, yet moth wings, as natural metamaterials, offer functionality as efficient sound absorbers through the action of the numerous resonant scales that decorate their wing membrane. Here, we quantify the potential for moth wings to act as a sound-absorbing metasurface coating for acoustically reflective substrates...

A partially supervised approach to Structural Health Monitoring is proposed, to manage the cost associated with expert inspections and maximize the value of monitoring regimes. Unlike conventional data-driven procedures, the monitoring classifier is learnt online while making predictions-negating the requirement for complete data before a system is in operation (which are rarely available). Most critically, periodic inspections are replaced (or enhanced) by an automatic inspection regime, which only queries measurements that appear informative to the evolving model of the damage-sensitive features...

Understanding whether a population will survive or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction-diffusion equations, where migration is usually represented as a linear diffusion term, and birth-death is represented with a nonlinear source term. While linear diffusion is most commonly employed to study migration, there are several limitations of this approach, such as the inability of linear diffusion-based models to predict a well-defined population front...

The use of the complex velocity potential and the complex velocity is widely disseminated in the study of two-dimensional incompressible potential flows. The advantages of working with complex analytical functions made this representation of the flow ubiquitous in the field of theoretical aerodynamics. However, this representation is not usually employed in linear stability studies, where the representation of the velocity as real vectors is preferred by most authors, in order to allow the representation of the perturbation as the complex exponential function...

We introduce a correspondence between phylogenetic trees and Brauer diagrams, inspired by links between binary trees and matchings described by Diaconis and Holmes (1998 Proc. Natl Acad. Sci. USA 95 , 14 600-14 602. (doi:10.1073/pnas.95.25.14600)). This correspondence gives rise to a range of semigroup structures on the set of phylogenetic trees, and opens the prospect of many applications. We furthermore extend the Diaconis-Holmes correspondence from binary trees to non-binary trees and to forests, showing for instance that the set of all forests is in bijection with the set of partitions of finite sets...

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...

In this paper, we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot's embedding for growing radii. Specifically, we extract features from the persistent homology (PH) of the Vietoris-Rips complexes built from point clouds associated with knots. Statistical analysis of our data shows the existence of increasing correlations between geometric quantities associated with the embedding and PH-based features, as a function of the knots' lengths...

No abstract text is available yet for this article.

Sea-level variations spread over a very broad spectrum of spatial and temporal scales as a result of complex processes occurring in the Earth System in response to natural variability of the climate system, as well as to external forcing due to natural phenomena and anthropogenic factors. Here, we address contemporary sea-level changes, focusing on the satellite altimetry era (since the early 1990s), for which various observing systems from space and in situ allow precise monitoring of sea-level variations from global to local scales, as well as improved understanding of the components responsible for the observed variations...

We study the effects of elastic anisotropy on Landau-de Gennes critical points, for nematic liquid crystals, on a square domain. The elastic anisotropy is captured by a parameter, <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:math> , and the critical points are described by 3 d.f. We analytically construct a symmetric critical point for all admissible values of <mml:math xmlns:mml="https://www...

Recent observations indicate that the Universe is not transparent but partially opaque due to absorption of light by ambient cosmic dust. This implies that the Friedmann equations valid for the transparent universe must be modified for the opaque universe. This paper studies a scenario in which the opacity rises with redshift. In this case, the light-matter interactions become important, because cosmic opacity produces radiation pressure that counterbalances gravitational forces. The presented theoretical model assumes the Universe is expanding according to the standard FLRW metric but with the scale factor <mml:math xmlns:mml="https://www...

Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a spherical surface, curvature and compactness lead to non-trivial behaviour that finds uses in physics, computer science and geometry. A well-known idealized isotropic example is the Tammes problem of finding optimal non-intersecting packings of equal hard disks. The anisotropic case of elliptic particles remains, on the other hand, comparatively unexplored...

Application of pulsed electric current to solidifying conductive materials leads to a rapid and significant reduction of surface roughness. This is validated experimentally in the present work for multi-component oxides. The surface geometry of the casts without and with pulsating treatment was measured using a Leica DCM-3D Profiler. The pulsating treatment reduces the roughness of solidified materials by more than 50%. The unmeasurable points, which account for nearly 2% of total area, were interpolated using an artificial multi-layer neural network...

Hand hygiene is among the most fundamental and widely used behavioural measures to reduce the person-to-person spread of human pathogens and its effectiveness as a community intervention is supported by evidence from randomized trials. However, a theoretical understanding of the relationship between hand hygiene frequency and change in risk of infection is lacking. Using a simple model-based framework for understanding the determinants of hand hygiene effectiveness in preventing viral respiratory tract infections, we show that a crucial, but overlooked, determinant of the relationship between hand hygiene frequency and risk of infection via indirect transmission is persistence of viable virus on hands...

We introduce the concept of a handlebody decomposition of a three-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable three-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss a topological study of microphase separation of a block copolymer melt.

We introduce a mathematical model, based on networks, for the elasticity and plasticity of materials. We define the tension tensor for a periodic graph in a Euclidean space, and we show that the tension tensor expresses elasticity under deformation. Plasticity is induced by local moves on a graph. The graph is described in terms of the weights of edges, and we discuss how these weights affect the plasticity.

Research in modern data-driven dynamical systems is typically focused on the three key challenges of high dimensionality, unknown dynamics and nonlinearity. The dynamic mode decomposition (DMD) has emerged as a cornerstone for modelling high-dimensional systems from data. However, the quality of the linear DMD model is known to be fragile with respect to strong nonlinearity, which contaminates the model estimate. By contrast, sparse identification of nonlinear dynamics learns fully nonlinear models, disambiguating the linear and nonlinear effects, but is restricted to low-dimensional systems...

Sparse model identification enables the discovery of nonlinear dynamical systems purely from data; however, this approach is sensitive to noise, especially in the low-data limit. In this work, we leverage the statistical approach of bootstrap aggregating (bagging) to robustify the sparse identification of the nonlinear dynamics (SINDy) algorithm. First, an ensemble of SINDy models is identified from subsets of limited and noisy data. The aggregate model statistics are then used to produce inclusion probabilities of the candidate functions, which enables uncertainty quantification and probabilistic forecasts...