Different sex-specific optima for a trait influenced by shared genetic variation. Sex-limited expression can reduce the conflict without requiring all genetic effects to become independent.

Conflict mediated by interacting traits encoded at different loci in the sexes, including manipulative traits and resistance.

A seminal protein studied in Drosophila melanogaster whose transfer affects female physiology and behaviour. Its evolutionary costs and benefits depend on mating regime and measured time horizon.

In a planar good-solvent solution, a correlation blob of $g$ monomers has size $\xi \sim \ell g^{3/4}$. Its area fraction is $\varphi \sim g{\ell }^2/{\xi }^2\sim g^{-1/2}$, giving $g\sim {\varphi }^{-2}$ and $\xi \sim \ell {\varphi }^{-3/2}$. Each blob contributes order $k_{B}T$ to the osmotic free energy, so the surface osmotic pressure is $\Pi \sim k_{B}T/{\xi }^2\sim (k_{B}T/{\ell }^2){\varphi }^3$. This regime requires many blobs per chain and area fraction below close packing. In the dilute regime entire chains are the independently translating objects and $\Pi \sim k_{B}T\varphi /(N{\ell }^2)$. The two expressions match at the two-dimensional polymer overlap fraction ${\varphi }^{\ast }\sim N^{-1/2}$.

In a nondegenerate semiconductor, the semiconductor mass-action law is $np=N_c{N}_v{e}^{-E_g/(k_{B}T)}$. Its prefactors count states, so spin degeneracy and valley degeneracy must be included once each.

For equivalent parabolic conduction valleys with one-valley density-of-states effective mass $m_d$, $N_c=g_s{g}_v(m_d{k}_{B}T/(2\pi {\hslash }^2){)}^{3/2}$. The analogous valence expression must sum distinct relevant bands when one effective band is insufficient. If a tabulated effective mass or effective density of states already incorporates a degeneracy, multiplying by it again overcounts states. This changes the mass-action prefactor even though the exponential gap factor is unchanged.

For chemical potential at the conduction edge, divide the exact product by the classical prediction. The ratio is $F_{1/2}(0)e^x{F}_{1/2}(-x)$, where $x=E_g/(k_{B}T)$. It is less than one, approaches $F_{1/2}(0)$ as $T\to 0$, and approaches its square in the formal high-temperature parabolic model.

The distance for an order-one self-phase modulation at peak power $P$ in a lossless Kerr envelope model. With attenuation, the phase accumulates over an effective interaction length instead.

A test star inside an infinite planar slab of uniform mass density $\rho$ experiences acceleration $\ddot{z}=-4\pi G\rho z$. Reflection symmetry gives zero vertical gravitational field at the mid-plane, and integrating the gravitational Poisson equation gives the field linear in height. The interior orbit is therefore simple harmonic motion; release from rest at any height within the slab gives its first mid-plane crossing after $\pi /(2\sqrt{4\pi G\rho })$. Finite radial extent, a varying vertical mass density or leaving the slab changes this result.

A broad focusing beam can amplify transverse intensity perturbations. Linearizing $i{A}_z+{\nabla }_{\perp }^{2}A/(2k)+\gamma |A{|}^{2}A=0$ about its uniform solution gives the displayed growth rate for transverse wave number $q$. For $\gamma >0$, a finite band grows; noise can develop into filaments. A finite beam and saturating response change the ideal plane-background prediction.

Detection and usable measurement of an extended source depend on its surface brightness and angular extent as well as its total radiative flux. A diffuse galaxy can be missed even if its integrated flux exceeds a nominal point-source limit. Artificial-source injection over a range of sizes and profiles tests this form of survey incompleteness.

Measurement scatter acting on a steep source-count distribution changes the observed abundance near a threshold. If faint objects are much more numerous, more scatter into a bright selected sample than scatter out of it. This is distinct from the accessible-volume weighting of Malmquist bias, which can occur without measurement noise. Forward convolution of the population with the error model and astronomical selection function treats both effects consistently.

Surface-wave tomography combines path-dependent measurements of seismic surface-wave dispersion to infer lateral and depth variation in elastic properties. Joint Rayleigh wave and Love wave measurements can help constrain seismic anisotropy.

For surface phase speed $c$ on a spherical Earth, the conserved angular ray parameter is $p=R_E/c$. The local horizontal slowness at radius $r$ is $p/r$, giving squared radial shear wavenumber $q_r^2={\omega }^2[\beta (r{)}^{-2}-p^2/r^2]$ in ray theory. The displayed inequality makes it negative. A jump in shear speed can turn a propagating wave above the interface into an evanescent wave below it. This is a local condition, not proof of evanescence at every greater depth.

In horizontally layered isotropic media, horizontal slowness is conserved along a ray. The incidence angle $i$ is measured from vertical. A smooth turning depth satisfies $pv=1$. The range and travel time are $x=2\int pvdz/\sqrt{1-p^2{v}^2}$ and $T=2\int dz/[v\sqrt{1-p^2{v}^2}]$ for a symmetric surface-to-surface turning ray.

An ideal uniform shallow basin, closed at one end and held at zero displacement at its open end, has its lowest standing wave with a displacement node at the opening and an antinode at the closed end. Its wavelength is four times its length. Real mouths need not satisfy this perfectly reflecting idealization.

The capacity of a seed lot to germinate and establish under specified conditions, distinguished from viability alone.

In the number-density virial expansion, $p/(n{k}_{B}T)=1+B_{2}n+\cdots$, so $B_2$ has units of volume. With molar concentration instead, the coefficient is $B_{2,\mathrm{mol}}=N_A{B}_2$, where $N_A$ is the Avogadro constant.

For a steep soft repulsion, the effective excluded radius decreases with temperature. An inverse-twelfth-power core gives a positive second-virial contribution proportional to $T^{-1/4}$. A hard core instead has a fixed positive excluded-volume limit. These asymptotics explain why different inert-gas model sketches have different high-temperature tails.

For a nondegenerate lowest state, every excited-state denominator $E_0-E_k$ is negative. Since each numerator $|V_{k0}{|}^2$ is nonnegative, their sum is nonpositive, and strictly negative if a coupled higher state exists. The statement assumes the perturbative expansion and appropriate sum or spectral integral are defined.