A researcher who coauthored experiments testing the contribution of a transferred reproductive peptide to female mating costs in Drosophila melanogaster. Comparing males differing in peptide production connected a particular component to lifetime performance.^[ref]

A researcher who coauthored comparative and experimental work on mating traits in water striders. The study related interacting male and female structures to sexual conflict.^[ref]

A researcher who coauthored comparative and experimental work on mating traits in water striders. The study related interacting male and female structures to sexual conflict.^[ref]

A researcher who coauthored experiments on parental care in great tits. Manipulating a partner’s provisioning independently of chick begging helped isolate the parent’s response to its partner.^[ref]

A researcher who coauthored experiments relating social opportunity to rapid changes in cichlid behavior and neural gene expression. Removal of a dominant individual created a controlled change in the social environment.^[ref]

A researcher who coauthored experiments on cooperation in cleaner wrasse. Responses to simulated clients helped test how removal of feeding opportunities and chasing alter the cleaner’s choices.^[ref]

A researcher who coauthored experiments on food donations among vampire bats. Controlled fasting and records of directed donations helped compare reciprocal assistance with relatedness and harassment.^[ref]

A researcher who coauthored experiments on food donations among vampire bats. Controlled fasting and records of directed donations helped compare reciprocal assistance with relatedness and harassment.^[ref]

For a thin sheet of initial face area $A$, compressed normally by force $F$ between frictionless plates, each lateral strain is $\nu F/(EA)$. To first order the area of either broad face increases by $2\nu F/E$; total two-face area changes twice as much.

The uniaxial strain modulus relates longitudinal normal stress to longitudinal strain when both transverse normal strains vanish. For isotropic linear elasticity, solving the transverse compliance equations gives the transverse stresses $\nu {\tau }_1/(1-\nu )$ and hence the displayed modulus in terms of Young's modulus and Poisson's ratio. It differs from Young's modulus, which applies when the transverse stresses vanish. Their difference is crucial for thin elastic layers bonded to rigid faces.

The elasticity tensor relates stress to infinitesimal strain by ${\sigma }_{ij}=C_{ijkl}{ϵ}_{kl}$. In conservative linear elasticity, the minor symmetries reflect symmetric stress and strain, while the major symmetry $C_{ijkl}=C_{klij}$ expresses the existence of an elastic energy density.

The lexicographic rank of an object is its position among the admissible objects in lexicographic order. The rank convention must specify whether the first object has rank zero or one.

For a fixed nonzero $a\in {\mathbb{R}}^n$, a nonzero level set $f(x)=h$ is the sphereThis follows by completing the square after multiplying by the positive denominator. Nonzero levels exist only for $|h|\le \Vert a\Vert /2$. The zero level is the hyperplane $a\cdot x=0$; the extreme levels shrink to points $x=\pm a/\Vert a\Vert$. The Cauchy-Schwarz inequality gives $|f(x)|\le \Vert a\Vert \Vert x\Vert /(1+\Vert x{\Vert }^2)\le \Vert a\Vert /2$, so these extrema are global. At them the Hessian matrix is respectively $-\Vert a\Vert I/2$ and $+\Vert a\Vert I/2$. In two dimensions the spheres are circles whose centres all lie on the axis spanned by $a$.

At speed $v\ll c$, an endpoint separation $L$ measured simultaneously in its own rest frame becomes $L\sqrt{1-v^2/c^2}$ in the moving frame, shorter by approximately $L{v}^2/(2c^2)$. The stable evaluation $L{\beta }^2/[1+\sqrt{1-{\beta }^2}]$ avoids subtracting nearly equal floating-point numbers.