Mathematics: Essential Tensions

The vivacity of mathematics results (partly) from the fact that mathematics is stretched between several poles, not being committed to any one. The paper presents the following polarities: realism - idealism, the finite - the infinite, the discrete - the continuous, the approximate - the exact, certitute - probability, simplicity - complexity, unity - multiplicity.     

Anisotropy of Earth's D' layer and stacking faults in the MgSiO3 post-perovskite phase

The post-perovskite phase of (Mg,Fe)SiO3 is believed to be the main mineral phase of the Earth's lowermost mantle (the D' layer). Its properties explain numerous geophysical observations associated with this layer-for example, the D' discontinuity, its topography and seismic anisotropy within the layer. Here we use a novel simulation technique, first-principles metadynamics, to identify a family of low-energy polytypic stacking-fault structures intermediate between the perovskite and post-perovskite phases. Metadynamics trajectories identify plane sliding involving the formation of stacking faults as the most favourable pathway for the phase transition, and as a likely mechanism for plastic deformation of perovskite and post-perovskite. In particular, the predicted slip planes are {010} for perovskite (consistent with experiment) and {110} for post-perovskite (in contrast to the previously expected {010} slip planes). Dominant slip planes define the lattice preferred orientation and elastic anisotropy of the texture. The {110} slip planes in post-perovskite require a much smaller degree of lattice preferred orientation to explain geophysical observations of shear-wave anisotropy in the D' layer.     

The ergodic hierarchy, randomness and Hamiltonian chaos

Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called ‘ergodic hierarchy’ (EH), which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion of randomness requires clarification. The mathematical definition of EH does not make explicit appeal to randomness; nor does the usual way of presenting EH involve a specification of the notion of randomness that is supposed to underlie the hierarchy. In this paper we argue that EH is best understood as a hierarchy of random behaviour if randomness is explicated in terms of unpredictability. We then show that, contrary to common wisdom, EH is useful in characterising the behaviour of Hamiltonian dynamical systems.