A model relating a stimulus to the inputs and discrimination capacity of a visual system. Receiver spectral sensitivity, illumination and background matter; human colour distances are not automatically perceptual distances for another animal.

For noninteracting particles with positive buoyant mass $m_b$ in a gravitational field, a dilute thermodynamic equilibrium profile is proportional to $e^{-z/{\ell }_g}$ with ${\ell }_g=k_{B}T/(m_{b}g)$. Uniformity over a vessel height $H$ requires $H\ll {\ell }_g$. Requiring only ${\ell }_g$ to exceed a particle radius is a weaker microscopic gravity-versus-diffusion condition.

For a sphere of radius $a$ and buoyant density difference $\Delta \rho$, define ${\mathrm{Pe}}_g=v_{s}a/D=m_{b}ga/(k_{B}T)$ using radius as the transport length. With Stokes law and Stokes–Einstein diffusion, ${\mathrm{Pe}}_g=4\pi \Delta \rho g{a}^4/(3k_{B}T)$. Replacing radius by diameter changes the convention by a factor of two. This local criterion does not guarantee a uniform suspension over a macroscopic vessel.

Contact inhibition of locomotion is a contact-triggered change of protrusions and movement direction. It is distinct from contact inhibition of proliferation. In neural crest experiments, cell collisions caused protrusion collapse and repolarization; blocking noncanonical Wnt signalling disrupted the response.^[ref]

A quantitative measure of coordinated cell velocities. A common external force can create correlation without proving direct cell-to-cell transmission of guidance.

For vacuum incidence on a sharp planar, collisionless plasma interface, the transmitted normal wavenumber obeys $k_z^2=({\omega }^2/c^2)({\cos }^2\theta -{\omega }_p^2/{\omega }^2)$. Negative $k_z^2$ produces an evanescent field and total reflection in this lossless model.

For a downward transverse wave in an isotropic plasma, $E_z/E_x=\tan {\theta }_t$ fixes the transmitted direction. Tangential phase matching gives $\sin {\theta }_i=n_p\sin {\theta }_t$. Continuity of tangential electric and magnetic fields then separates incident and reflected amplitudes. A measured transmitted electric amplitude alone cannot be identified with the incident one; mean power for peak phasors includes a factor of one half, and interface-normal flux includes a cosine.

Expand both states in the Fock basis and sum the exponential series to obtain the displayed overlap. In particular $⟨\alpha |-\alpha ⟩=e^{-2|\alpha {|}^2}$. Distinct coherent states are nonorthogonal at finite separation; dropping this overlap can give incorrect normalization and small-amplitude limits.

A lossless mode transformation maps a product of coherent states to another product of coherent states with transformed complex amplitudes. unitarity preserves the total mean particle number. It does not generally turn a fixed-number input into coherent states.

The normalized coherent state has number amplitude $e^{-|\alpha {|}^2/2}{\alpha }^n/\sqrt{n!}$, hence the displayed Poisson law with mean and variance $|\alpha {|}^2$. A product of coherent modes has factorized number probabilities.

A normalized bosonic field coherent state with $\bar{N}=\int |\psi {|}^2$ has $⟨\hat{\Psi }(\mathbf{r})⟩=\psi (\mathbf{r})$. For a normal-ordered contact-interacting Hamiltonian, its energy expectation is $\int [{\hslash }^2|\nabla \psi {|}^2/(2m)+V|\psi {|}^2+g|\psi {|}^4/2]d^{3}r$. Minimizing at fixed mean particle number gives the stationary Gross–Pitaevskii equation. This mean-field state has number fluctuations and does not include the pair correlations responsible for quantum depletion.

A normalized nonnegative probability measure over coherent states produces a classical mixture. The measure may contain Dirac delta distributions, so a pure coherent state is included. Conditioning on $\alpha$ gives Poisson distribution photon statistics; hence $\mathrm{Var}n=⟨n⟩+\mathrm{Var}(|\alpha {|}^2)\ge ⟨n⟩$. Each optical quadrature has variance at least its vacuum value because mixing adds nonnegative variance of conditional means. These inequalities provide simple nonclassicality witnesses.

The equilibrium water-vapor saturation ratio above a solution droplet combines a curvature effect, which raises vapor pressure, with dissolved-solute reduction of water activity. Its maximum defines a critical supersaturation for activation. In a dilute hygroscopicity approximation using droplet diameter $D$ and dry diameter $D_d$, $s(D)\simeq A_K/D-\kappa D_d^3/D^3$, where $A_K=4\sigma M_w/(RT{\rho }_w)$. Then $s_c\simeq \sqrt{4A_K^3/(27\kappa D_d^3)}$. These approximate forms assume the stated diameter convention and dilute solution; composition and surface tension must be consistent.^[ref]

Competition among labelled clones with equivalent average reproductive behaviour. Clone extinction can coexist with tissue maintenance, and surviving clones form a selected subset of all initial clones.

Production of more than one differentiated cell type from one confirmed progenitor. Several separately labelled unipotent cells can imitate this pattern if the label is not unique.

Labelling few enough founder cells to resolve individual clones. The acceptable density depends on spatial mixing, label diversity and the chance of independent activation.

The distribution of descendant counts per clone. Differences can arise from proliferation, survival, migration or sampling rather than altered fate choice alone.

An individual-nucleotide-resolution crosslinking-and-immunoprecipitation method. Recovering information from reverse-transcription termination helps locate RNA–protein contacts, linking a biochemical event with a sequence position.^[ref]

An enhanced crosslinking-and-immunoprecipitation method for mapping RNA-binding protein contacts. Its controls and library design improve interpretation of enriched sites, without making every recovered sequence a functional regulatory target.^[ref]