In 2017 the Scottish Government passed the Child Poverty (Scotland) Act with the commitment to significantly reduce the relative child poverty rate from the current prevailing level of around 25% to 10% by 2030/31. In response, the government introduced the Scottish Child Payment (SCP) that provides a direct transfer to households at a fixed rate per eligible child – currently £25 per week. In this paper we explore, using a micro to macro modelling approach, the effectiveness of using the SCP to achieve the Scottish child poverty targets. While we find that the ambitious child poverty targets can technically be met solely using the SCP, the necessary payment of £165 per week amounting to a total government cost of £3 billion per year, makes the political and economy-wide barriers significant. A key issue with only using the SCP is the non-linearity in the response to the payment; as the payment increases, the marginal gain in the reduction of child poverty decreases – this is particularly evident after payments of £80 per week. A ‘policy-mix’ option combining the SCP, targeted cash transfers and other policy levels (such as childcare provision) seems the most promising approach to reaching the child poverty targets.

Although piecewise isometries (PWIs) are higher-dimensional generalizations of one-dimensional interval exchange transformations (IETs), their generic dynamical properties seem to be quite different. In this paper, we consider embeddings of IET dynamics into PWI with a view to better understanding their similarities and differences. We derive some necessary conditions for existence of such embeddings using combinatorial, topological and measure-theoretic properties of IETs. In particular, we prove that continuous embeddings of minimal 2-IETs into orientation-preserving PWIs are necessarily trivial and that any 3-PWI has at most one non-trivially continuously embedded minimal 3-IET with the same underlying permutation. Finally, we introduce a family of 4-PWIs, with an apparent abundance of invariant non-smooth fractal curves supporting IETs, that limit to a trivial embedding of an IET.

The structure and origin of twin defects have been studied over the past half-century. Recently, there has been renewed interest in investigating the mechanisms by which twin defects facilitate the growth of bulk and nanoscale systems. This article reviews our understanding and experimental advances to unravel the complex role that twin defects play during crystal growth. The following topics are addressed: growth promotion at single and multiple, parallel and antiparallel twin boundaries; the role of {100} and {111} solid–liquid interfaces during crystallization; the application of realtime imaging to the study of crystal growth in the presence of twin defects; and suggested future research needed to shed light on the driving forces for twin-related phenomena. By providing a broad survey of the existing literature on twin-assisted crystal growth, we anticipate that our review will aid researchers in deciphering various growth forms that arise in materials processing applications.

There are various notions of attractor in the literature, including measure (Milnor) attractors and statistical (Ilyashenko) attractors. In this paper we relate the notion of statistical attractor to that of the essential ω-limit set and prove some elementary results about these. In addition, we consider the convergence of time averages along trajectories. Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables. For non-ergodic systems, time averages may not exist even for almost all trajectories. However, averages of some observables may converge; we characterize conditions on observables that ensure convergence of time averages even in non-ergodic systems.

We introduce a class of two-dimensional piecewise isometries on the plane that we refer to as cone exchange transformations (CETs). These are generalizations of interval exchange transformations (IETs) to 2D unbounded domains. We show for a typical CET that boundedness of orbits is determined by ergodic properties of an associated IET and a quantity we refer to as the ‘flux at infinity’. In particular we show, under an assumption of unique ergodicity of the associated IET, that a positive flux at infinity implies unboundedness of almost all orbits outside some bounded region, while a negative flux at infinity implies boundedness of all orbits. We also discuss some examples of CETs for which the flux is zero and/or we do not have unique ergodicity of the associated IET; in these cases (which are of great interest from the point of view of applications such as dual billiards) it remains an outstanding problem to find computable necessary and sufficient conditions for boundedness of orbits.

In 1998 a circle of timber posts within the intertidal zone on the north Norfolk coast was brought to the attention of the Norfolk County Council Archaeological Service. A subsequent programme of archaeological recording and dating revealed that the structure was constructed in the spring or early summer of 2049 BC, during the Early Bronze Age. Because of the perceived threat of damage and erosion from the sea a rescue excavation was undertaken during the summer months of 1999. The structure was entirely excavated, involving the removal of the timbers and a programme of stratigraphic recording and environmental analysis. A survey was also undertaken within the environs of the site which has identified further timber structures dating from the Bronze Age. Detailed examination of the timber from the circle has produced a wealth of unexpected information which has added greatly to our understanding of Early Bronze Age woodworking, organisation of labour and the layout and construction of timber ritual monuments.

We consider robust relative homoclinic trajectories (RHTs) for G-equivariant vector fields. We give some conditions on the group and representation that imply existence of equivariant vector fields with such trajectories. Using these results we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that imply existence of RHTs, we construct some new examples of robust relative homoclinic trajectories. We also classify RHTs of the dihedral and low order symmetric groups by means of their symmetries.

We consider coupled sets of identical cells and address the problem of which symmetries are permissible in such networks. For example, n linearly coupled cells with one independent variable in each cell cannot be constructed with the symmetry group An, the alternating group on n symbols. Using a graphical technique, we show that it is possible to construct cell networks with any desired finite group of symmetries. In particular, we show that any subgroup of Sn can be realized as the symmetries of a group of n cells. Special forms of coupling (especially low order polynomial coupling) are shown to restrict the possible symmetries. We give some upper and lower bounds for the degree of polynomial required to realize several classes of subgroups of Sn.