This article reassesses the social significance of Early Neolithic chambered tombs. It critically evaluates inferences about social organization drawn from tomb architecture and interpretations of kinship based on aDNA analyses of human remains from tombs. Adopting the perspective that kinship is a multifaceted and ongoing field of practice, it argues that the arrangement of tomb chambers was related to the negotiation of Early Neolithic kinship. Drawing together inferences about biological relatedness from aDNA analyses with interpretations of chamber arrangements, it suggests that variation in the architectural arrangements and sequential modification of chambered tombs relates to different ways of negotiating aspects of kinship, particularly descent and affinity. It presents interpretations of how kinship was negotiated at Early Neolithic tombs in different regions of Britain and Ireland and concludes that it is increasingly possible to gauge pattern and diversity in Neolithic negotiations of kinship, descent and affinity by combining different strands of evidence, including architectural arrangement.

For $(X,\,L)$ a polarized toric variety and $G\subset \mathrm {Aut}(X,\,L)$ a torus, denote by $Y$ the GIT quotient $X/\!\!/G$. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on $Y$ to the category of torus equivariant reflexive sheaves on $X$. We show, under a genericity assumption on $G$, that slope stability is preserved by these functors if and only if the pair $((X,\,L),\,G)$ satisfies a combinatorial criterion. As an application, when $(X,\,L)$ is a polarized toric orbifold of dimension $n$, we relate stable equivariant reflexive sheaves on certain $(n-1)$-dimensional weighted projective spaces to stable equivariant reflexive sheaves on $(X,\,L)$.

There are significant variations in the fuel consumption of aircraft during the descent phase of a flight. This paper uses aircraft flight data measurements to develop an improved understanding of these variations. A model of the aircraft engines is developed that is matched to flight data and shown to reproduce the time history of engine parameters. This model is used to determine the overall engine efficiency at each point during a descent. This enables an energy breakdown to be completed, in terms of mechanical energy from fuel, gravitational potential energy and kinetic energy. During descent, the aircraft engines operate at low overall pressure ratios corresponding to low fuel flow rates and low overall efficiencies. On average, the engine overall efficiency during descent is one-third of cruise efficiency. The airframe aerodynamic performance is deteriorated during descent with an average lift-to-drag ratio that is 87% of the average value at cruise. There are also large variations in air-track efficiency, and for the flights analysed the great circle descent distance was found to be 85% of the average descent air distance. To minimise fuel burn, flights should cruise as far as possible before starting descent and follow a trajectory with the shortest possible air distance. The descent air speed should be set to maximise the aircraft lift-to-drag ratio. Such descents could save up to 0.5% of the total aircraft mass in fuel.

Answering a question of H. Petersson, we provide a class of examples of a pair of octonion algebras over a ring having isometric norms.

We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author, sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg-categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.

The Neolithic court tombs of Ireland display variation in the ways that their component parts – concave courts, linear galleries and individual cells – were combined and arranged. This variation has been interpreted in the past in terms of both the diffusion of ideas and the design requirements of their builders. In this article it is suggested that the analysis of these formalized components points to central themes of the social and ritual discourse that accompanied the tombs' construction and use, the symbolism of the tombs expressing alleged lineage relationships between the living community, its ancestors and the land. It is suggested that these communities were based on one or more corporate descent groups, in some cases combining with other social units to appropriate and exploit territory, the relationships between them being symbolized and idealized in the spatial layout of the tombs' orthostatic structures.

Let X be the Fermat curve of degree q+1 over the field k of q$^2$ elements, where q is some prime power. Considering the Jacobian J of X as a constant abelian variety over the function field k(X), we calculate the multiplicities, in subfactors of the Shafarevich–Tate group, of representations associated with the action on X of a finite unitary group. J is isogenous to a power of a supersingular elliptic curve E, the structure of whose Shafarevich–Tate group is also described.

Given $E$, an elliptic curve defined over $K$, a field of positive characteristic, provided that $j$, the Weierstrass $j$-invariant, is not an element of $K^p$, we construct explicitly, that is, we give by a closed form formula, a non-trivial homomorphism, $\mu:E(K) \rightarrow K^+$, from the group of $K$-rational points of $E$ to $K^+$, the additive group of $K$. In the course of our analysis we discover a canonical differential, $\omega_q \in \Omega; K|{\bb F}_p$, associated to $E$ and we relate it to the differential $dq/q$ associated to the Tate curve. If the transcendence degree of $K$ over ${\bb F}_p$ is equal to one, as for example is the case for function fields in one variable, then $\mu$ is a $p$-descent map, that is, its kernel is equal to $pE(K)$ and the explicit formula for $\mu$ can be used to provide effective proofs of analogues of classical theorems on elliptic curves. For example, in the author's thesis at The University of Texas at Austin the analogue of Siegel's Theorem on the finiteness of integral points of $E(K)$ is proved effectively.