We discuss the emergence of common mathematical patterns governing the timing and severity of insurgent and terrorist attacks, across geographic scales and including cyberspace. We present mathematical models that provide a generative explanation of these patterns. Despite wide variations in the underlying settings and circumstances, the ubiquity of these patterns suggests there is a common way in which groups of humans fight each other. Our empirical findings follow from the analysis of myriad state-of-the-art datasets with resolution at the level of individual attacks, while our mathematical modelling involves numerical and analytical solutions of fission–fusion dynamics together with progress curve analysis.

This paper is concerned with scattering resonances of a 1D photonic crystal of finite extent. We propose a general perturbation approach to study the resonances that are close to the bound-state frequency of the infinite structure when some defect is embedded in the interior. It is shown that near bound-state resonances exist on the complex plane and the distance between the resonance and the associated bound-state frequency decays exponentially as a function of the number of periodic cells. A numerical approach based upon the perturbation theory is also proposed to calculate the near bound-state resonances accurately.

Outflow from a young star might be regarded as approximately equivalent to flow from a point source. If the fluid consists of charged particles, then the magnetic fields produced are governed by Faraday's law. This simple first approximation yields a linear partial differential equation in spherical polar coordinates, and its solution may be represented as the product of a Legendre polynomial with some function of the radial coordinate. This radial function is shown to involve orthogonal polynomials. Their properties are investigated and recurrence formulae for them are derived. Some of the magnetic fields generated by this simple model are illustrated.

In this paper, a closed-form pricing formula in the form of an infinite series for European call options is derived for the Heston stochastic volatility model under a chosen martingale measure. Given that markets with the stochastic volatility are incomplete, there exists a number of equivalent martingale measures and consequently investors face a problem of making a choice of appropriate measure when they price options. The one we adopt here is the so-called minimal entropy martingale measure shown to be related to the expected utility maximization theory (Frittelli 2000 Math. Finance10(1), 39–52) and the financial rationality for choosing this measure will be further illustrated in this paper. A great advantage of our newly-derived pricing formula is that the convergence of the solution in series form can be proved theoretically; such a proof of the convergence is also complemented by some numerical examples to demonstrate the speed of convergence. To further show the validity of our formula, a comparison of prices calculated through the newly derived formula is made with those obtained directly from the Monte Carlo simulation as well as those from solving the PDE (partial differential equation) with the finite difference method.

In this paper, a unique dataset of improvised explosive device attacks during “The Troubles” in Northern Ireland (NI) is analysed via a Hawkes process model. It is found that this past dependent model is a good fit to improvised explosive device attacks yielding key insights about the nature of terrorism in NI. We also present a novel approach to quantitatively investigate some of the sociological theory surrounding the Provisional Irish Republican Army which challenges previously held assumptions concerning changes seen in the organisation. Finally, we extend our use of the Hawkes process model by considering a multidimensional version which permits both self and mutual-excitations. This allows us to test how the Provisional Irish Republican Army responded to past improvised explosive device attacks on different geographical scales from which we find evidence for the autonomy of the organisation over the six counties of NI and Belfast. By incorporating a second dataset concerning British Security Force (BSF) interventions, the multidimensional model allows us to test counter-terrorism (CT) operations in NI where we find subsequent increases in violence.

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov–Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

In this paper, we are interested in the nilpotent centre problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centres. More general objects are considered as the formal inverse integrating factors. However, the existence of a formal inverse integrating factor is not sufficient to describe all the nilpotent centres. For the family studied in this paper, it is enough.

An integro-differential model for evolutionary dynamics with mutations is investigated by improving the understanding of its behaviour using numerical simulations. The proposed numerical approach can handle also density dependent fitness, and gives new insights about the role of mutation in the preservation of cooperation.

We examine a problem in which a line sink causes a disturbance to an otherwise uniform flowing stream of infinite depth. We consider the fully non-linear problem with the inclusion of surface tension and find the maximum sink strength at which steady solutions exist for a given stream flow, before examining non-unique solutions. The addition of surface tension allows for a more thorough investigation into the characteristics of the solutions. The breakdown of steady solutions with surface tension appears to be caused by a curvature singularity as the flow rate approaches the maximum. The non-uniqueness in solutions is shown to occur for a range of parameter values in all cases with non-zero surface tension.

This work looks at the asymptotic behaviour of the solution to the Helmholtz equation in a penetrable domain of $\mathbb{R}$3 with a thin layer of thickness δ which tends to 0. We use the method of multi-scale expansion to derive and justify an asymptotic expansion of the solution with respect to the thickness δ up to any order. We then provide approximate transmission conditions of order two defined on an interface located inside the thin layer, with accuracy up to O(δ2), which allow one to take into account the influence of the thin layer.

Using mathematical methods to understand and model crime is a recent idea that has drawn considerable attention from researchers during the last five years. From the plethora of models that have been proposed, perhaps the most successful one has been a diffusion-type differential equations model that describes how the number of criminals evolves in a specific area. We propose a more detailed form of this model that allows for two distinct criminal types associated with major and minor crime. Additionally, we examine a stochastic variant of the model that represents more realistically the ‘generation’ of new criminals. Numerical solutions from both models are presented and compared with actual crime data for the Greater Manchester area. Agreement between simulations and actual data is satisfactory. A preliminary statistical analysis of the data also supports the model's potential to describe crime.

A new eigenvalue ℝ-linear problem arisen in the theory of metamaterials and neutral inclusions is reduced to integral equations. The problem is constructively investigated for circular non-overlapping inclusions. An asymptotic formula for eigenvalues is deduced when the radii of inclusions tend to zero. The nodal domains conjecture related to univalent eigenfunctions is posed. Demonstration of the conjecture allows to justify that a set of inclusions can be made neutral by surrounding it with an appropriate coating.

In this paper, we present a distributed Proportional-Integral (PI) strategy with self-tuning adaptive gains for reaching asymptotic consensus in networks of non-identical linear agents under constant disturbances. Alternative adaptive strategies are presented, based on global or local measures of the agents' disagreement. The proposed approaches are validated on a representative numerical example. Preliminary analytical results further confirm the viability of the self-tuning strategies.

In this article, we provide a numerical algorithm to reconstruct a convex sound-soft scatterer from phaseless backscattering data, assuming sufficiently high frequency. Certain uniqueness and existence results for the case of circular scatterers are given as well, based on the asymptotic expansion for the normal derivative of the total field.

The ability to directly record human face-to-face interactions increasingly enables the development of detailed data-driven models for the spread of directly transmitted infectious diseases at the scale of individuals. Complete coverage of the contacts occurring in a population is however generally unattainable, due for instance to limited participation rates or experimental constraints in spatial coverage. Here, we study the impact of spatially constrained sampling on our ability to estimate the epidemic risk in a population using such detailed data-driven models. The epidemic risk is quantified by the epidemic threshold of the SIRS model for the propagation of communicable diseases, i.e. the critical value of disease transmissibility above which the disease turns endemic. We verify for both synthetic and empirical data of human interactions that the use of incomplete data sets due to spatial sampling leads to the underestimation of the epidemic risk. The bias is however smaller than the one obtained by uniformly sampling the same fraction of contacts: it depends non-linearly on the fraction of contacts that are recorded, and becomes negligible if this fraction is large enough. Moreover, it depends on the interplay between the timescales of population and spreading dynamics.

Intra-specific competition in insect and amphibian species is often experienced in completely different ways in their distinct life stages. Competition among larvae is important because it can impact on adult traits that affect disease transmission, yet mathematical models often ignore larval competition. We present two models of larval competition in the form of delay differential equations for the adult population derived from age-structured models that include larval competition. We present a simple prototype equation that models larval competition in a simplistic way. Recognising that individual larvae experience competition from other larvae at various stages of development, we then derive a more complex equation containing an integral with a kernel that quantifies the competitive effect of larvae of age ā on larvae of age a. In some parameter regimes, this model and the famous spruce budworm model have similar dynamics, with the possibility of multiple co-existing equilibria. Results on boundedness and persistence are also proved.

Crime reduction and prevention strategies are vital for policymakers and law enforcement to face inevitable increases in urban crime rates as a side effect of the projected growth of urban population by the year 2030. Studies conclude that crime does not occur uniformly across urban landscapes but concentrates in certain areas. This phenomenon has drawn attention to spatial crime analysis, primarily focusing on crime hotspots, areas with disproportionally higher crime density. In this paper, we present CrimeTracer1, a personalized random walk-based approach to spatial crime analysis and crime location prediction outside of hotspots. We propose a probabilistic model of spatial behaviour of known offenders within their activity spaces. Crime Pattern Theory concludes that offenders, rather than venture into unknown territory, frequently select targets in or near places they are most familiar with as part of their activity space. Our experiments on a large real-world crime dataset show that CrimeTracer outperforms all other methods used for location recommendation we evaluate here.

Within the online media universe, there are many underlying communities. These may be defined, for example, through politics, location, health, occupation, extracurricular interests or retail habits. Government departments, charities and commercial organisations can benefit greatly from insights about the structure of these communities; the move to customer-centred practices requires knowledge of the customer base. Motivated by this issue, we address the fundamental question of whether a sub-network looks like a collection of individuals who have effectively been picked at random from the whole, or instead forms a distinctive community with a new, discernible structure. In the former case, to spread a message to the intended user base it may be best to use traditional broadcast media (TV, billboard), whereas in the latter case a more targeted approach could be more effective. In this work, we therefore formalise a concept of testing for sub-structure and apply it to social interaction data. First, we develop a statistical test to determine whether a given sub-network (induced sub-graph) is likely to have been generated by sampling nodes from the full network uniformly at random. This tackles an interesting inverse alternative to the more widely studied “forward” problem. We then apply the test to a Twitter reciprocated mentions network where a range of brand name based sub-networks are created via tweet content. We correlate the computed results against the independent views of 16 digital marketing professionals. We conclude that there is great potential for social media based analytics to quantify, compare and interpret online brand allegiances systematically, in real time and at large scale.

In this paper, we explore some of the various issues that may occur in attempting to fit a dynamical systems (either agent- or continuum-based) model of urban crime to data on just the attack times and locations. We show how one may carry out a regression analysis for the model described by Short et al. (2008, Math. Mod. Meth. Appl. Sci.) by using simulated attack data from the agent-based model. It is discussed how one can incorporate the attack data into the partial differential equations for the expected attractiveness to burgle and the criminal density to predict crime rates between attacks. Using this predicted crime rate, we derive a likelihood function that one can maximise in order to fit parameters and/or initial conditions for the model. We focus on carrying out data assimilation for two different parameter regions, namely in the case where stationary and non-stationary crime hotspots form. It is found that the likelihood function is ‘flat’ for large ranges of parameters, and that this has major implications for crime forecasting. Hence, we look at how one might carry out a goodness-of-fit and forecasting analysis for crime rates given the range of parameter fits. We show how one can use the Kolmogorov–Smirnov statistic to assess the goodness-of-fit. The dynamical systems analysis of the partial differential equations proves invaluable to understanding how the crime rate forecasts depend on the parameters and their sensitivity. Finally, we outline several interesting directions for future research in this area where we believe that the combination of dynamical systems modelling, analysis, and data assimilation can prove effective in developing policing strategies for urban crime.

In the social sciences, networks are used to represent relationships between social actors, be they individuals or aggregates. The structural importance of these actors is assessed in terms of centrality indices which are commonly defined as graph invariants. Many such indices have been proposed, but there is no unifying theory of centrality. Previous attempts at axiomatic characterization have been focused on particular indices, and the conceptual frameworks that have been proposed alternatively do not lend themselves to mathematical treatment.

We show that standard centrality indices, although seemingly distinct, can in fact be expressed in a common framework based on path algebras. Since, as a consequence, all of these indices preserve the neighbourhood-inclusion pre-order, the latter provides a conceptually clear criterion for the definition of centrality indices.