A refined estimate for the topological degree

We sharpen an estimate of Bourgain, Brezis, and Nguyen for the topological degree of continuous maps from a sphere $\mathbb{S}^d$ into itself in the case $d \ge 2$. This provides the answer for $d \ge 2$ to a question raised by Brezis. The problem is still open for $d=1$

General README

This general README file provides an overall description of the other files present in this repository

Behavior

Data related to behavior experiments presented in the manuscript

HighSpeed

Data related to high-speed (limb kinematics) experiments presented in the manuscript

OverallMovement

Data related to overall movement experiments presented in the manuscript

Table S1, Figure S1 and Figure S2 from Walking like an ant: a quantitative and experimental approach to understanding locomotor mimicry in the jumping spider <i>Myrmarachne formicaria</i>

Protective mimicry, in which a palatable species avoids predation by being mistaken for an unpalatable model, is a remarkable example of adaptive evolution. These complex interactions between mimics, models and predators can explain similarities between organisms beyond the often-mechanistic constraints typically invoked in studies of convergent evolution. However, quantitative studies of protective mimicry typically focus on static traits (e.g. colour and shape) rather than on dynamic traits like locomotion. Here, we use high-speed cameras and behavioural experiments to investigate the role of locomotor behaviour in mimicry by the ant-mimicking jumping spider <i>Myrmarachne formicaria</i>, comparing its movement to that of ants and non-mimicking spiders. Contrary to previous suggestions, we find mimics walk using all eight legs, raising their forelegs like ant antennae only when stationary. Mimics exhibited winding trajectories (typical wavelength = 5–10 body lengths), which resemble the winding patterns of ants specifically engaged in pheromone-trail following, although mimics walked on chemically inert surfaces. Mimics also make characteristically short (approx. 100 ms) pauses. Our analysis suggests that this makes mimics appear ant-like to observers with slow visual systems. Finally, behavioural experiments with predatory spiders yield results consistent with the protective mimicry hypothesis. These findings highlight the importance of dynamic behaviours and observer perception in mimicry

Measuring and Manipulating the Adhesion of Graphene

We present a technique to precisely measure the surface energies between two-dimensional materials and substrates that is simple to implement and allows exploration of spatial and chemical control of adhesion at the nanoscale. As an example, we characterize the delamination of single-layer graphene from monolayers of pyrene tethered to glass in water and maximize the work of separation between these surfaces by varying the density of pyrene groups in the monolayer. Control of this energy scale enables high-fidelity graphene-transfer protocols that can resist failure under sonication. Additionally, we find that the work required for graphene peeling and readhesion exhibits a dramatic rate-independent hysteresis, differing by a factor of 100. This work establishes a rational means to control the adhesion of 2D materials and enables a systematic approach to engineer stimuli-responsive adhesives and mechanical technologies at the nanoscale

Presentation of friction coefficient as a function of the Sommerfeld number reveals viscous lubrication by HA.

<p>(Top) Friction coefficient as a function of sliding speed for three different lubricants with a wide range of viscosities provided a range of friction coefficients that span almost an order of magnitude (n = 5). (Middle) When the friction coefficients from the top panel are presented as a function of the Sommerfeld number (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143415#pone.0143415.e001" target="_blank">Eq 1</a>) instead of sliding speed, the mechanisms of lubrication become apparent and are fit to a model curve (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143415#pone.0143415.e002" target="_blank">Eq 2</a>) to determine alterations of the elastoviscous transition. (Bottom) Serial dilutions of the HA solution provided overlapping data sets confirming HA had no influence on the boundary friction coefficient, and increased load (4.2, 5.2, and 6.2 N; all at 0.1 mm/s) and decreased concentration (2 mg/ml) tests with HYADD4 revealed convergence of HA and HYADD4 in the transition region (HYADD4 2 mg/ml and high load, n = 4, all others n = 5) (data points represent mean±SEM).</p

(A) Altering the presence of lubricin at the articular surface and in the lubricant solution revealed distinct elastoviscous transition curves. (B-D) The curve fit parameters revealed the importance of lubricin in boundary lubrication and also its importance in facilitating the transition away from boundary mode lubrication when present with HA. (E-H) Replication of the HA curves by dextran revealed that both the viscous lubrication by HA and the synergy with lubricin are not specific to the chemistry of HA. (n = 5 for unaltered, n = 3 for lubricin removed and lubricin in solution; data points represent mean±SEM).

<p>(A) Altering the presence of lubricin at the articular surface and in the lubricant solution revealed distinct elastoviscous transition curves. (B-D) The curve fit parameters revealed the importance of lubricin in boundary lubrication and also its importance in facilitating the transition away from boundary mode lubrication when present with HA. (E-H) Replication of the HA curves by dextran revealed that both the viscous lubrication by HA and the synergy with lubricin are not specific to the chemistry of HA. (n = 5 for unaltered, n = 3 for lubricin removed and lubricin in solution; data points represent mean±SEM).</p

A classical Stribeck curve (dotted line) maps the transition of friction from high (boundary) friction to low (hydrodynamic) friction.

<p>For soft, permeable contacts like articular cartilage, a divergence from the classical curve is described by an elastoviscous transition [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143415#pone.0143415.ref026" target="_blank">26</a>] where contact compliance and permeability hinder the transition to hydrodynamic lubrication.</p