Geometric constraints explain much of the species richness pattern in African birds

The world contains boundaries (e.g., continental edge for terrestrial taxa) that impose geometric constraints on the distribution of species ranges. Thus, contrary to traditional thinking, the expected species richness pattern in absence of ecological or physiographical factors is unlikely to be uniform. Species richness has been shown to peak in the middle of a bounded one-dimensional domain, even in the absence of ecological or physiographical factors. Because species ranges are not linear, an extension of the approach to two dimensions is necessary. Here we present a two-dimensional null model accounting for effects of geometric constraints. We use the model to examine the effects of continental edge on the distribution of terrestrial animals in Africa and compare the predictions with the observed pattern of species richness in birds endemic to the continent. Latitudinal, longitudinal, and two-dimensional patterns of species richness are predicted well from the modeled null effects alone. As expected, null effects are of high significance for wide ranging species only. Our results highlight the conceptual significance of an until recently neglected constraint from continental shape alone and support a more cautious analysis of species richness patterns at this scale.

Convex polytopes and quantization of symplectic manifolds

Quantum mechanics associate to some symplectic manifolds M a quantum model Q(M), which is a Hilbert space. The space Q(M) is the quantum mechanical analogue of the classical phase space M. We discuss here relations between the volume of M and the dimension of the vector space Q(M). Analogues for convex polyhedra are considered.

Geometric incompatibility in a fault system.

Interdependence between geometry of a fault system, its kinematics, and seismicity is investigated. Quantitative measure is introduced for inconsistency between a fixed configuration of faults and the slip rates on each fault. This measure, named geometric incompatibility (G), depicts summarily the instability near the fault junctions: their divergence or convergence ("unlocking" or "locking up") and accumulation of stress and deformations. Accordingly, the changes in G are connected with dynamics of seismicity. Apart from geometric incompatibility, we consider deviation K from well-known Saint Venant condition of kinematic compatibility. This deviation depicts summarily unaccounted stress and strain accumulation in the region and/or internal inconsistencies in a reconstruction of block- and fault system (its geometry and movements). The estimates of G and K provide a useful tool for bringing together the data on different types of movement in a fault system. An analog of Stokes formula is found that allows determination of the total values of G and K in a region from the data on its boundary. The phenomenon of geometric incompatibility implies that nucleation of strong earthquakes is to large extent controlled by processes near fault junctions. The junctions that have been locked up may act as transient asperities, and unlocked junctions may act as transient weakest links. Tentative estimates of K and G are made for each end of the Big Bend of the San Andreas fault system in Southern California. Recent strong earthquakes Landers (1992, M = 7.3) and Northridge (1994, M = 6.7) both reduced K but had opposite impact on G: Landers unlocked the area, whereas Northridge locked it up again.