Equality, Diversity and Inclusion (EDI) questions have taken on increasing importance within the mathematics community recently and generated substantial debate. This article focuses on data collection, data analysis and evidence-based policy for EDI in the context of the UK mathematics higher education sector. It proposes that targeted latitudinal surveys and longitudinal studies would be useful to quantify experiences of disadvantage or discrimination but cautions against hasty conclusions from the data obtained. It notes that topics related to equality and diversity are politically controversial and emphasizes the importance of academic freedom in data analysis. It argues that EDI action should be based on evidence of disadvantage or discrimination rather than on the existence of quantitative disparities in outcomes. Finally, it reflects upon the importance of gathering balanced data on the effects of EDI policies in order to measure their costs and benefits.

Droplet impact on oscillating substrates is important for both natural and industrial processes. Recognizing the importance of the dynamics that arises from the interplay between droplet transport and substrate motion, in this work, we present an experimental investigation of the spreading of a droplet impacting a sinusoidally oscillating hydrophobic substrate. We focus particularly on the maximum spread of droplets as a function of various parameters of substrate oscillation. We first quantify the maximum spreading diameter attained by the droplets as a function of frequency, amplitude of vibration, and phase at the impact for various impact velocities. We highlight that there can be two stages of spreading. Stage I, which is observed at all impact conditions, is controlled by the droplet inertia and affected by the substrate oscillation. For certain conditions, a Stage II spreading is also observed, which occurs during the retraction process of Stage I due to additional energies imparted by the substrate oscillation. Subsequently, we derive scaling analyses to predict the maximum spreading diameters and the time for this maximum spread for both Stage I and Stage II. Furthermore, we identify the necessary condition for Stage II spreading to be greater than Stage I spreading. The results will enable optimization of the parameters in applications where substrate oscillation is used to control the droplet spread, and thus heat and mass transfer between the droplet and the substrate.

The dynamics of an evaporating droplet in an unsteady flow is of practical interest in many industrial applications and natural processes. To investigate the transport and evaporation dynamics of such droplets, we present a numerical study of an isolated droplet in an oscillating gas-phase flow. The study uses a one-way coupled two-phase flow model to assess the effect of the amplitude and the frequency of a sinusoidal external flow field on the lifetime of a multicomponent droplet containing a non-volatile solute dissolved in a volatile solvent. The results show that the evaporation process becomes faster with an increase in the amplitude or the frequency of the gas-phase oscillation. The liquid-phase transport inside the droplet also is influenced by the unsteadiness of the external gas-phase flow. A scaling analysis based on the response of the droplet under the oscillating drag force is subsequently carried out to unify the observed evaporation dynamics in the simulations under various conditions. The analysis quantifies the enhancement in the droplet velocity and Reynolds number as a function of the gas-phase oscillation parameters and predicts the effects on the evaporation rate.

This work reports experimental observation and theoretical explanation of the dynamics and morphology of a droplet passing through a soap film. During the process, the film undergoes four sequential responses: (1) film deformation upon droplet impact; (2) drop–film detachment; (3) coalescence of the film shell with the drop; (4) peel-off of the film shell. Physical models and the corresponding analytical expressions are developed to reveal the underlying physics for the observed four responses. It is identified that the film is an elongated catenoid under continuous stretch by the droplet, and that they separate at the fixed height of 5.8 times of the droplet radius while the detach point is located at the centre of the height. After separation, the droplet is wrapped with a film shell, which is then punctured by the ring tip of the converging surface wave at the impacting Weber number range of [45, 225]. The film shell then coalesces with the droplet, falls off with a fixed velocity and is eventually ejected as a bubble leaving the droplet with a transplanted surface of the soap solution.

Let F be a Siegel cusp form of degree $2$ , even weight $k \ge 2$ , and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of F at fundamental matrices S (i.e., with $-4\det (S)$ equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with $\det (S) \asymp X$ , the sequence $a(F,S)$ has at least $X^{1-\varepsilon }$ sign changes and takes at least $X^{1-\varepsilon }$ ‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound $\lvert a(F,S)\rvert \ll _{F, \varepsilon } \frac {\det (S)^{\frac {k}2 - \frac {1}{2}}}{ \left (\log \lvert \det (S)\rvert \right )^{\frac 18 - \varepsilon }}$ for fundamental matrices S.

It has been suggested that a cellularly unstable laminar flame, which is freely propagating in unbounded space, can accelerate and evolve into a turbulent flame with the neighbouring flow exhibiting the basic characteristics of turbulence. Famously known as self-turbulization, this conceptual transition in the flow regime, which arises from local interactions between the propagating wrinkled flamefront and the flow, is critical in extreme events such as the deflagration-to-detonation transition (DDT) leading to supernova explosions. Recognizing that such a transition in the flow regime has not been conclusively demonstrated through experiments, in this work, we present experimental measurements of flow characteristics of flame-generated ‘turbulence’ for expanding cellular laminar flames. The energy spectra of such ‘turbulence’ at different stages of cellular instability are analysed. A subsequent scaling analysis points out that the observed energy spectra are driven by the fractal topology of the cellularly unstable flamefront.

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

We herein report an experimental study to explore the effects of impact inertia, film thickness and viscosity on the dynamics of shape deformation of a drop impacting a liquid film. We have identified that the spreading dynamics shows a weak dependence on impact inertia, but strongly depends on the film thickness. For a thick film, the liquid surface deforms and absorbs part of the impact energy, and hence inhibits spreading of the drop. For a thin film, the drop motion is restricted by the bottom solid substrate, promoting spreading. The periodicity of the capillary controlled shape oscillation, on the other hand, is found to be independent of impact inertia and film thickness. The damping of the shape oscillation shows strong dependence on the film thickness, in that the oscillation decays faster for smaller film thicknesses, due to the enhanced viscous loss.

We herein report an experimental study on the morphological evolution of a vortex ring formed inside a liquid pool after it is impacted and penetrated by a coalescing drop of the same liquid. The dynamics of the penetrating vortex ring along with the deformation of the pool surface has been captured using simultaneous high-speed laser induced fluorescence and shadowgraph techniques. It is identified that the motion of such a vortex ring can be divided into three stages, during which inertial, capillary and viscous effects alternatingly play dominant roles to modulate the penetration process, resulting in linear, non-monotonic and decelerating motion in these three stages respectively. Furthermore, we also evaluate the relevant time and length scales of these three stages and subsequently propose a unified description of the downward motion of the penetrating vortex ring. Finally, we use the experimental data for a range of drop diameters and impact speeds to validate the proposed scaling.

In this paper we study the essential role of Darrieus–Landau (DL), hydrodynamic, cellular flame-front instability in the propagation of expanding turbulent flames. First, we analyse and compare the characteristic time scales of flame wrinkling under the simultaneous actions of DL instability and turbulent eddies, based on which three turbulent flame propagation regimes are identified, namely, instability dominated, instability–turbulence interaction and turbulence dominated regimes. We then perform experiments over an extensive range of conditions, including high pressures, to promote and manipulate the DL instability. The results clearly demonstrate the increase in the acceleration exponent of the turbulent flame propagation as these three regimes are traversed from the weakest to the strongest, which are respectively similar to those of the laminar cellularly unstable flame and the turbulent flame without flame-front instability, and thus validating the scaling analysis. Finally, based on the scaling analysis and the experimental results, we propose a modification of the conventional turbulent flame regime diagram to account for the effects of DL instability.

We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

We study the effect of acoustic streaming on nanoparticle motion and morphological evolution inside an acoustically levitated droplet using an analytical approach coupled with experiments. Nanoparticle migration due to internal recirculation forms a density stratification, the location of which depends on initial particle concentration. The time scale of density stratification is similar to that of perikinetic-driven agglomeration of particle flocculation. The density stratification ultimately leads to force imbalance leading to a unique bowl-shaped structure. Our analysis shows the mechanism of bowl formation and how it is affected by particle size, concentration, internal recirculation and fluid viscosity.

We give a short and “soft” proof of the asymptotic orthogonality of Fourier coefficients of Poincaré series for classical modular forms as well as for Siegel cusp forms, in a qualitative form.