Panel collapse and its applications

We describe a procedure called panel collapse for replacing a CAT(0) cube complex $\Psi$ by a "lower complexity" CAT(0) cube complex $\Psi_\bullet$ whenever $\Psi$ contains a codimension-$2$ hyperplane that is extremal in one of the codimension-$1$ hyperplanes containing it. Although $\Psi_\bullet$ is not in general a subcomplex of $\Psi$, it is a subspace consisting of a subcomplex together with some cubes that sit inside $\Psi$ "diagonally". The hyperplanes of $\Psi_\bullet$ extend to hyperplanes of $\Psi$. Applying this procedure, we prove: if a group $G$ acts cocompactly on a CAT(0) cube complex $\Psi$, then there is a CAT(0) cube complex $\Omega$ so that $G$ acts cocompactly on $\Omega$ and for each hyperplane $H$ of $\Omega$, the stabiliser in $G$ of $H$ acts on $H$ essentially. Using panel collapse, we obtain a new proof of Stallings's theorem on groups with more than one end. As another illustrative example, we show that panel collapse applies to the exotic cubulations of free groups constructed by Wise. Next, we show that the CAT(0) cube complexes constructed by Cashen-Macura can be collapsed to trees while preserving all of the necessary group actions. (It also illustrates that our result applies to actions of some non-discrete groups.) We also discuss possible applications to quasi-isometric rigidity for certain classes of graphs of free groups with cyclic edge groups. Panel collapse is also used in forthcoming work of the first-named author and Wilton to study fixed-point sets of finite subgroups of $\mathrm{Out}(F_n)$ on the free splitting complex. Finally, we apply panel collapse to a conjecture of Kropholler, obtaining a short proof under a natural extra hypothesis.Comment: Revised according to referee comments. This version accepted in "Groups, Geometry, and Dynamics

Cultivar diversity as a means of ecologically intensifying dry matter production in a perennial forage stand

The relationship between genotypic diversity and productivity has not been adequately explored in perennial forage production systems despite strong theoretical and empirical evidence supporting diversity\u27s role in ecosystem functioning in other managed and unmanaged systems. We conducted a two-year field experiment with six cultivars of an agriculturally important forage grass, Lolium perenne L. (perennial ryegrass). Dry matter production of L. perenne and the weed community that emerged from the soil seed bank were measured each year in treatments that ranged from cultivar monocultures to three- and six-way cultivar mixtures, all sown at a constant seeding rate. Mean L. perenne dry matter production increased with increasing cultivar diversity and was highest in mixtures that contained cultivars representing the greatest additive trait range (calculated on rankings of three traits: winter hardiness, heading date, and tolerance to grazing). Mixtures had greater yields than those predicted by the mean of their component monoculture yields, but there was evidence that highly productive cultivars may have dampened over-yielding in mixtures. Weed abundance was correlated with L. perenne dry matter, but not L. perenne cultivar diversity. These results suggest that multi-cultivar mixtures may have utility as an approach to ecologically intensifying perennial forage production. Additional research will be necessary to determine the mechanisms responsible for the over-yielding observed in this study and the generality of these findings

Costs of colour change in fish: food intake and behavioural decisions

Many animals, particularly reptiles, amphibians, fish and cephalopods, have the ability to change their body colour, for functions including thermoregulation, signalling and predator avoidance. Many fish plastically darken their body colouration in response to dark visual backgrounds, and this functions to reduce predation risk. Here, we tested the hypotheses that colour change in fish (1) carries with it an energetic cost and (2) affects subsequent shoal and habitat choice decisions. We demonstrate that guppies (Poecilia reticulata) change colour in response to dark and light visual backgrounds, and that doing so carries an energetic cost in terms of food consumption. By increasing food intake, however, guppies are able to maintain growth rates and meet the energetic costs of changing colour. Following colour change, fish preferentially choose habitats and shoals that match their own body colouration, and maximise crypsis, thus avoiding the need for further colour change but also potentially paying an opportunity cost associated with restriction to particular habitats and social associates. Thus, colour change to match the background is complemented by behavioural strategies, which should act to maximise fitness in variable environments. © 2013. Published by The Company of Biologists Ltd

Modeling hydrodynamic self-propulsion with Stokesian Dynamics. Or teaching Stokesian Dynamics to swim

We develop a general framework for modeling the hydrodynamic self-propulsion (i.e., swimming) of bodies (e.g., microorganisms) at low Reynolds number via Stokesian Dynamics simulations. The swimming body is composed of many spherical particles constrained to form an assembly that deforms via relative motion of its constituent particles. The resistance tensor describing the hydrodynamic interactions among the individual particles maps directly onto that for the assembly. Specifying a particular swimming gait and imposing the condition that the swimming body is force- and torque-free determine the propulsive speed. The body’s translational and rotational velocities computed via this methodology are identical in form to that from the classical theory for the swimming of arbitrary bodies at low Reynolds number. We illustrate the generality of the method through simulations of a wide array of swimming bodies: pushers and pullers, spinners, the Taylor=Purcell swimming toroid, Taylor’s helical swimmer, Purcell’s three-link swimmer, and an amoeba-like body undergoing large-scale deformation. An open source code is a part of the supplementary material and can be used to simulate the swimming of a body with arbitrary geometry and swimming gait

Avalanche Size Scaling in Sheared Three-Dimensional Amorphous Solid

We have studied the statistics of plastic rearrangement events in a simulated amorphous solid at T=0. Events are characterized by the energy release and the ``slip volume'', the product of plastic strain and system volume. Their distributions for a given system size $L$ appear to be exponential, but a characteristic event size cannot be inferred, because the mean values of these quantities increase as $L^{\alpha}$ with $\alpha \sim 3/2$. In contrast to results obtained in 2D models, we do not see simply connected avalanches. The exponent suggests a fractal shape of the avalanches, which is also evidenced by the mean fractal dimension and participation ratio.Comment: Accepted for publication in Physical Review Letter