When two trains travel along the same track in the same direction, it is a common safety requirement that the trains must be separated by at least two signals. This means that there will always be at least one clear section of track between the two trains. If the safe-separation condition is violated, then the driver of the following train must adopt a revised strategy that will enable the train to stop at the next signal if necessary. One simple way to ensure safe separation is to define a prescribed set of latest allowed section exit times for the leading train and a corresponding prescribed set of earliest allowed section entry times for the following train. We will find strategies that minimize the total tractive energy required for both trains to complete their respective journeys within the overall allowed journey times and subject to the additional prescribed section clearance times. We assume that the drivers use a discrete control mechanism and show that the optimal driving strategy for each train is defined by a sequence of approximate speedholding phases at a uniquely defined optimal driving speed on each section and that the sequence of optimal driving speeds is a decreasing sequence for the leading train and an increasing sequence for the following train. We illustrate our results by finding optimal strategies and associated speed profiles for both trains in some elementary but realistic examples.

At the ANZIAM conference in Hobart in February 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components.

We consider the pricing of European options under a modified Black–Scholes equation having fractional derivatives in the “spatial” (price) variable. To be specific, the underlying price is assumed to follow a geometric Koponen–Boyarchenko–Levendorski process. This pure jump Lévy process could better capture the real behaviour of market data. Despite many difficulties caused by the “globalness” of the fractional derivatives, we derive an explicit closed-form analytical solution by solving the fractional partial differential equation analytically, using the Fourier transform technique. Based on the newly derived formula, we also examine, in theory, many basic properties of the option price under the current model. On the other hand, for practical purposes, we impose a reliable implementation method for the current formula so that it can be easily used in the trading market. With the numerical results, the impact of different parameters on the option price are also investigated.

Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviours in their homecoming, which are interesting to observe and to describe analytically. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group, and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account “natural parameters”, such as the eyesight of the penguins and their cruising speed. The model that we propose favours the formation of conglomerates of penguins that gather together, but, on the other hand, it also allows the possibility of isolated and exposed individuals.

The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behaviour of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a “stop-and-go” procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins remain isolated). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguin parade.

Recent years have seen a large increase in the popularity of Texas hold ’em poker. It is now the most commonly played variant of the game, both in casinos and through online platforms. In this paper, we present a simulation study for games of Texas hold ’em with between two and 23 players. From these simulations, we estimate the probabilities of each player having been dealt the winning hand. These probabilities are calculated conditional on both partial information (that is, the player only having knowledge of his/her cards) and also on fuller information (that is, the true probabilities of each player winning given knowledge of the cards dealt to each player). Where possible, our estimates are compared to exact analytic results and are shown to have converged to three significant figures.

With these results, we assess the poker strategies described in two recent pieces of popular culture. In comparing the ideas expressed in Taylor Swift’s song, New Romantics, and the betting patterns employed by James Bond in the 2006 film, Casino Royale, we conclude that Ms Swift demonstrates a greater understanding of the true probabilities of winning a game of Texas hold ’em poker.

Before wastewaters can be released into the environment, they must be treated to reduce the concentration of organic pollutants in the effluent stream. There is a growing concern as to whether wastewater treatment plants are able to effectively reduce the concentration of micropollutants that are also contained in their influent streams. We investigate the removal of micropollutants in treatment plants by analysing a model that includes biodegradation and sorption as the main mechanisms of micropollutant removal. For the latter a linear adsorption model is used in which adsorption only occurs onto particulates.

The steady-state solutions of the model are found and their stability is determined as a function of the residence time. In the limit of infinite residence time, we show that the removal of biodegradable micropollutants is independent of the processes of adsorption and desorption. The limiting concentration can be decreased by increasing the concentration of growth-related macropollutants. Although, in principle, it is possible that the concentration of micropollutants is minimized at a finite value of the residence time, this was found not to be the case for the particular biodegradable micropollutants considered.

For nonbiodegradable pollutants, we show that their removal is always optimized at a finite value of the residence time. For finite values of the residence time, we obtain a simple condition which identifies whether biodegradation is more or less efficient than adsorption as a removal mechanism. Surprisingly, we find that, for the micropollutants considered, adsorption is always more important than biodegradation, even when the micropollutant is classified as being highly biodegradable with low adsorption.

We study the problem of pseudostate and static output feedback stabilization for singular fractional-order linear systems with fractional order $\unicode[STIX]{x1D6FC}$ when $0<\unicode[STIX]{x1D6FC}<1$. All the results are given by linear matrix inequalities. First, a new sufficient and necessary condition for the admissibility of singular fractional-order systems is presented. Then based on the admissible result, not only are sufficient conditions for designing pseudostate and static output feedback controllers obtained, but also sufficient and necessary conditions are presented by using different methods that guarantee the admissibility of the closed-loop systems. Finally, the effectiveness of the proposed approach is demonstrated by numerical simulations and a real-world example.

We use an integral equation formulation approach to value shout options, which are exotic options giving an investor the ability to “shout” and lock in profits while retaining the right to benefit from potentially favourable movements in the underlying asset price. Mathematically, the valuation is a free boundary problem involving an optimal exercise boundary which marks the region between shouting and not shouting. We also find the behaviour of the optimal exercise boundary for one- and two-shout options close to expiry.

We derive an evolution equation for the free-surface dynamics of a thin film of a second-grade fluid over an unsteady stretching sheet using long-wave theory. For the numerical investigation of the viscoelastic effect on the thin-film dynamics, a finite-volume approach on a uniform grid with implicit flux discretization is applied. The present results are in excellent agreement with results available in the literature for a Newtonian fluid. We observe that the fluid thins faster with the rapid stretching rate of the sheet, but the second-grade parameter delays the thinning behaviour of the liquid film.

We analyse a scattering problem of electromagnetic waves by a bounded chiral conductive obstacle, which is surrounded by a dielectric, via the quasi-stationary approximation for the Maxwell equations. We prove the reciprocity relations for incident plane and spherical electric waves upon the scatterer. Mixed reciprocity relations have also been proved for a plane wave and a spherical wave. In the case of spherical waves, the point sources are located either inside or outside the scatterer. These relations are used to study the inverse scattering problems.

We focus on the convergence rate of the alternating direction method of multipliers (ADMM) in a complex domain. First, the complex form of variational inequality (VI) is established by using the Wirtinger calculus technique. Second, the $O(1/K)$ convergence rate of the ADMM in a complex domain is provided. Third, the ADMM in a complex domain is applied to the least absolute shrinkage and selectionator operator (LASSO). Finally, numerical simulations are provided to show that ADMM in a complex domain has the $O(1/K)$ convergence rate and that it has certain advantages compared with the ADMM in a real domain.

The effect of uniform wind flow on modulational instability of two crossing waves is studied here. This is an extension of an earlier work to the case of a finite-depth water body. Evolution equations are obtained as a set of three coupled nonlinear equations correct up to third order in wave steepness. Figures presented in this paper display the variation in the growth rate of instability of a pair of obliquely interacting uniform wave trains with respect to the changes in the air-flow velocity, depth of water medium and the angle between the directions of propagation of the two wave packets. We observe that the growth rate of instability increases with the increase in the wind velocity and the depth of water medium. It also increases with the decrease in the angle of interaction of the two wave systems.