Abstract:
Numerical simulations show that microscopic rod-like bodies suspended in a turbulent flow tend to align with the vorticity vector, rather than with the dominant eignevector of the strain-rate tensor. This paper investigates an analytically solvable limit of a model for alignment in a random velocity field with isotropic statistics. The vorticity varies very slowly and the isotropic random flow is equivalent to a pure strain with statistics which are axisymmetric about the direction of the vorticity. We analyse the alignment in a weakly fluctuating uniaxial strain field, as a function of the product of the strain relaxation time $\tau_{\rm s}$ and the angular velocity $\omega$ about the vorticity axis. We find that when $\omega\tau_{\rm s}\gg 1$, the rods are predominantly either perpendicular or parallel to the vorticity.

Abstract:
We consider the optimal covering of fractal sets in a two-dimensional space using ellipses which become increasingly anisotropic as their size is reduced. If the semi-minor axis is \epsilon and the semi-major axis is \delta, we set \delta=\epsilon^\alpha, where 0<\alpha<1 is an exponent characterising the anisotropy of the covers. For point set fractals, in most cases we find that the number of points N which can be covered by an ellipse centred on any given point has expectation value < N > ~ \epsilon^\beta, where \beta is a generalised dimension. We investigate the function \beta(\alpha) numerically for various sets, showing that it may be different for sets which have the same fractal dimension.

Abstract:
The ability to detect and integrate associations between unrelated items that are close in space and time is a key feature of human learning and memory. Learning sequential associations between non-adjacent visual stimuli (higher-order visuospatial dependencies) can occur either with or without awareness (explicit vs. implicit learning) of the products of learning. Existing behavioural and neurocognitive studies of explicit and implicit sequence learning, however, are based on conscious access to the sequence of target locations and, typically, on conditions where the locations for orienting, or motor, responses coincide with the locations of the target sequence.

Abstract:
New visuomotor skills can guide behaviour in novel situations. Prior studies indicate that learning a visuospatial sequence via responses based on manual key presses leads to effector- and response-independent knowledge. Little is known, however, about the extent to which new sequence knowledge can generalise, and, thereby guide behaviour, outside of the manual response modality. Here, we examined whether learning a visuospatial sequence either via manual (key presses, without eye movements), oculomotor (obligatory eye movements), or perceptual (covert reorienting of visuospatial attention) responses supported generalisation to direct and indirect tests administered either in the same (baseline conditions) or a novel response modality (transfer conditions) with respect to initial study. Direct tests measured the use of conscious knowledge about the studied sequence, whereas the indirect tests did not ostensibly draw on the study phase and measured response priming. Oculomotor learning supported the use of conscious knowledge on the manual direct tests, whereas manual learning supported generalisation to the oculomotor direct tests but did not support the conscious use of knowledge. Sequence knowledge acquired via perceptual responses did not generalise onto any of the manual tests. Manual, oculomotor, and perceptual sequence learning all supported generalisation in the baseline conditions. Notably, the manual baseline condition and the manual to oculomotor transfer condition differed in the magnitude of general skill acquired during the study phase; however, general skill did not predict performance on the post-study tests. The results demonstrated that generalisation was only affected by the responses used to initially code the visuospatial sequence when new knowledge was applied to a novel response modality. We interpret these results in terms of response-effect distinctiveness, the availability of integrated effector- and motor-plan based information, and discuss their implications for neurocognitive accounts of sequence learning.

Abstract:
We prove an obstruction at the level of rational cohomology in small degrees to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As an application, we provide evidence for a generalized conjecture of Hopf that says that no symmetric space of rank at least two admits a metric with positive curvature.

Abstract:
The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new tool is the action of the Steenrod algebra on cohomology.

Abstract:
In 1965, S.-S. Chern posed a question concerning the extent to which fundamental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar in 1998, but there are a number of positive results in the presence of symmetry. These classifications fall into categories according to the strength of their conclusions. We give an overview of these results in the case of torus symmetry and prove new results in each of these categories.

Abstract:
Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry.

Abstract:
This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds in dimensions up to 16. Many of the calculations are sharp and they require less symmetry than previous classifications. When restricted to certain classes of manifolds that admit non-negative curvature, these results imply diffeomorphism classifications. Also studied is a closely related family of manifolds called positively elliptic manifolds, and we prove the Halperin conjecture in this context for dimensions up to 16 or Euler characteristics up to 16.

Abstract:
Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include a small upper bound on the Euler characteristic and confirmations of famous conjectures by Hopf and Halperin under additional torus symmetry. We prove several cases (including all known even-dimensional examples of positively curved manifolds) of a conjecture by Wilhelm.