Heisenberg symmetry and hypermultiplet manifolds

We study the emergence of Heisenberg (Bianchi II) algebra in hyper-Kähler and quaternionic spaces. This is motivated by the rôle these spaces with this symmetry play in N = 2 hypermultiplet scalar manifolds. We show how to construct related pairs of hyper-Kähler and quaternionic spaces under general symmetry assumptions, the former being a zooming-in limit of the latter at vanishing scalar curvature. We further apply this method for the two hyper-Kähler spaces with Heisenberg algebra, which is reduced to U (1) × U (1) at the quaternionic level. We also show that no quaternionic spaces exist with a strict Heisenberg symmetry – as opposed to Heisenberg U (1). We finally discuss the realization of the latter by gauging appropriate Sp(2, 4) generators in N = 2 conformal supergravity.

The Effects of Density, Spatial Pattern, and Competitive Symmetry on Size Variation in Simulated Plant Populations

Patterns of size inequality in crowded plant populations are often taken to be indicative of the degree of size asymmetry of competition, but recent research suggests that some of the patterns attributed to size‐asymmetric competition could be due to spatial structure. To investigate the theoretical relationships between plant density, spatial pattern, and competitive size asymmetry in determining size variation in crowded plant populations, we developed a spatially explicit, individual‐based plant competition model based on overlapping zones of influence. The zone of influence of each plant is modeled as a circle, growing in two dimensions, and is allometrically related to plant biomass. The area of the circle represents resources potentially available to the plant, and plants compete for resources in areas in which they overlap. The size asymmetry of competition is reflected in the rules for dividing up the overlapping areas. Theoretical plant populations were grown in random and in perfectly uniform spatial patterns at four densities under size‐asymmetric and size‐symmetric competition. Both spatial pattern and size asymmetry contributed to size variation, but their relative importance varied greatly over density and over time. Early in stand development, spatial pattern was more important than the symmetry of competition in determining the degree of size variation within the population, but after plants grew and competition intensified, the size asymmetry of competition became a much more important source of size variation. Size variability was slightly higher at higher densities when competition was symmetric and plants were distributed nonuniformly in space. In a uniform spatial pattern, size variation increased with density only when competition was size asymmetric. Our results suggest that when competition is size asymmetric and intense, it will be more important in generating size variation than is local variation in density. Our results and the available data are consistent with the hypothesis that high levels of size inequality commonly observed within crowded plant populations are largely due to size‐asymmetric competition, not to variation in local density.

Sandbox experiments on basin inversion: testing the influence of basin orientation and basin fill

Analogue modelling experiments using brittle materials are performed to study the inversion of extensional structures. Asymmetric grabens of two different orientations are first created during a phase of extension and progressively filled. They are subsequently shortened in the same direction. The aim of our experiments is to determine factors affecting the style of deformation during inversion. We specifically investigate variations in thickness and distribution of strong and weak layers constituting the graben fill and in initial basin orientation. The main advantage of our experimental set-up is that we have a complete control on graben location, width, infill and orientation before inversion. The experiments show that shortening results only in limited reactivation of pre-existing normal faults. In general, forward thrusts and backthrusts cut across normal faults into the footwall of the graben. The forward thrusts either propagate parallel to the enveloping surface of faulted blocks or they cut across basin-limiting normal faults at various angles. The graben fill is mechanically extruded by displacement along forward thrusts that accommodate most of the shortening. Both pre-existing faults and weak graben fill act as zones of weakness during inversion and determine the orientation and location of both backthrusts and forward thrusts. The results of our experiments conform well to natural examples of inverted graben structures.