For preselected $|i⟩$ and nonorthogonal postselected $|f⟩$, the displayed transition-amplitude ratio controls the first-order conditional pointer response to a von Neumann measurement interaction. It can be complex or outside an observable's eigenvalue range. Expanding $⟨f|\exp (-igA\otimes p/\hslash )|i⟩$ to first order gives $⟨f|i⟩[1-ig{A}_{w}p/\hslash ]$, proving that interpretation within the expansion's validity regime.^[ref]

The ratio of transmitted to incident normal time-averaged power for a specified wave and interface. Converting a field-amplitude transmission coefficient into power requires the appropriate impedances or flux normalization. For a displacement wave on a string with impedance $Z$, $\mathcal{T}=(Z_2/Z_1)|t{|}^2$; using an electric-field amplitude in electromagnetic waves instead gives the corresponding electromagnetic flux ratio. Amplitude squared alone is sufficient only when the relevant incident and transmitted normalizations agree.

A string of length $L$ with fixed endpoints, initially displaced into a triangular tent of height $k$ at its midpoint and released from rest, has a Fourier sine series with the coefficients in the title. Integrating the two linear halves gives $a_n$, and symmetry cancels the even modes. Each coefficient evolves as $a_n\cos ({\omega }_{n}t)$ under the wave equation. The displacement vanishes everywhere at $t=L/(2c)$, reverses sign at $t=L/c$, and returns after $2L/c$. A straight configuration at an intermediate instant does not mean the string is at rest: its kinetic energy is then nonzero. The initial slope corner gives a finite-energy weak solution whose corner lines propagate and reflect at the ends.

A physical sheet under uniform isotropic tension per unit length T and with mass per unit area $\rho$ has small transverse displacement satisfying ${\partial }_t^2\psi =c^2{\nabla }^2\psi$, where $c^2=T/\rho$. Fixed edges impose zero displacement. This mechanical membrane model neglects bending stiffness, large slopes and dissipation.

Alkaline conditions that hydrolyse acetate side groups can also promote elimination from chlorinated backbone units. Dechlorination, unsaturation, changed solubility and possible subsequent cross-linking complicate isolation of an ideal vinyl chloride-vinyl alcohol copolymer. The competition is condition-dependent: analytical work used staged treatment and precipitation to restrain dehydrochlorination, so difficulty under generic conditions is not a proof that every hydroxyl-containing chloride copolymer is impossible.^[ref]

A vertebrate belonging to the lineage of four-limbed vertebrates, including descendants whose limbs were reduced or lost.

A member of the lineage uniting living hagfish and lampreys. Jawlessness alone cannot assign an extinct vertebrate to this lineage.

A member of the jawed-vertebrate lineage. Crown usage includes the ancestor of living cartilaginous and bony vertebrates and all its descendants; a total-group usage additionally includes jawless stem relatives. The usage must be stated when discussing early fossils.

A traditional assemblage of armoured jawless fossil vertebrates. It is not a complete clade, and many members are stem relatives of gnathostomes, rather than early cyclostomes.

A Chengjiang fossil animal important to interpretations of early vertebrate anatomy.

A Chengjiang fossil animal described with Haikouichthys in studies of early vertebrate organization.

A Cambrian fossil animal with evidence of segmented musculature, sensory anatomy and serial branchial structures.

FLOWERING LOCUS C, or FLC, encodes a floral repressor in Arabidopsis thaliana. Prolonged cold can establish a persistently silenced state, connecting chromatin regulation to the memory underlying vernalization.

Natural variation in FLC silencing contributes to differences in cold requirements. Measuring later flowering and the persistence of repression tests memory, whereas measuring expression only during cold does not establish that the regulatory change survives warming.^[ref]

The Arabis alpina FLC orthologue, whose seasonal reactivation contributes to perennial flowering behavior.

Let a receiver be initially at distance $D$ from a stationary responder and approach at speed $\beta c$. A light signal takes $D/c$ to reach the responder, which waits time $\delta$ and replies. Intersecting the reply's null worldline with the receiver's worldline givesThe receiver's proper time is ${\tau }_D$. These expressions describe reception before arrival when the reply is sent before the receiver reaches the responder; otherwise the trajectory after arrival must be specified.