The special lead article in this issue of EJAM, by Profs. John Ockendon and Brian Sleeman, is dedicated to Prof. Joseph B. Keller (referred to as Joe by his friends), Emeritus professor of Stanford University, who was one of the greatest applied mathematicians of our time. Joe died in September 2016. A workshop in honour of Joe was held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, in March 2017, focussing on some of the astonishingly wide range of topics in the mathematical sciences and continuum mechanics that Joe made substantial contributions to over his long career. This article discusses some of these pioneering contributions, describes some modern developments as discussed by the workshop speakers, and provides a more personal tribute to Joe, his history, and to his large influence on the international applied mathematics community. May the spirit of Joe, and his flavour of applied mathematics, continue to be reflected in EJAM.

Over the two days 2–3 March 2017, about 80 mathematicians and friends gathered in Cambridge to celebrate the life and work of Joseph Bishop Keller (1923–2016), one of the pre-eminent applied mathematicians of the 20th century. Joe, as he was known throughout the world, made pioneering contributions to a wide range of natural phenomena and developed fundamental mathematical techniques with which to understand them. Twenty-four talks were presented at the meeting, given by mathematicians who have either worked with Joe or have been influenced by his work. Rather than summarise each presentation, we have collated all the contributions under the headings of waves, fluids, solids, chemistry and biology, and finally some history.

When a towed sonar array is straight, it has the difficulty that it cannot distinguish a contact on the left from one at the same angle on the right. When the array is not straight and its shape known, we calculate the probability that the left–right ambiguity is resolved correctly, using the Neyman–Pearson hypothesis testing framework, assuming a delay-sum beamformer, a single-frequency contact, and Gaussian noise. We also initially consider the noise field to be uncorrelated and show that the evaluation of the probability of correct resolution reduces to evaluating a one-dimensional integral. We find, as expected, that the probability increases, as the signal-to-noise ratio and the lateral deviation of the array from straight increase. For demonstration purposes, we also evaluate the probability of correct resolution for two representative shapes the array might assume in practice. Finally, we consider a more realistic, correlated noise field and we show that the initial assumption of an uncorrelated noise field provides a good approximation when the lateral deviation of the array is sufficiently large.

We consider a non-local version of the Cahn–Hilliard equation characterized by the presence of a fractional diffusion operator, and which is subject to fractional dynamic boundary conditions. Our system generalizes the classical system in which the dynamic boundary condition was used to describe any relaxation dynamics of the order-parameter at the walls. The proposed fractional dynamic boundary condition appears to be more general in the sense that it incorporates non-local effects which were completely ignored in the classical approach. We aim to deduce well-posedness and regularity results as well as to establish the existence of finite-dimensional attractors for this system.

In this article, we use Mellin transforms to derive alternative results for option pricing and implied volatility estimation when the underlying asset price is governed by jump-diffusion dynamics. The current well known results are restrictive since the jump is assumed to follow a predetermined distribution (e.g., lognormal or double exponential). However, the results we present are general since we do not specify a particular jump-diffusion model within the derivations. In particular, we construct and derive an exact solution to the option pricing problem in a general jump-diffusion framework via Mellin transforms. This approach of Mellin transforms is further extended to derive a Dupire-like partial integro-differential equation, which ultimately yields an implied volatility estimator for assets subjected to instantaneous jumps in the price. Numerical simulations are provided to show the accuracy of the estimator.

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian type reaction–diffusion equation of non-Newtonian elastic filtration:

$$\begin{equation*} u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ p>2, \beta >0. \end{equation*}$$

We study functions whose time-frequency content are concentrated in a compact region in phase space using time-frequency localization operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators, as well as a local Gabor system covering the region of interest. These would allow the construction of modified time-frequency dictionaries concentrated in the region.