A linear polynomial in one variable over a field is $a_{1}x+a_0$ with $a_1\ne 0$, so its polynomial degree is one. It has the unique root $x=-a_0/a_1$. For example, $3x-6$ has root two.

As a function, it is affine; it is a linear map only when $a_0=0$. In several variables, an expression $a_0+{\sum }_i{a}_i{x}_i$ is a degree-one polynomial if some $a_i\ne 0$. Over a general ring, a nonzero leading coefficient need not be invertible, so the field root formula may be unavailable.

For polynomials $p$ and $q$ over a common commutative ring, their composition is $(p\circ q)(x)=p(q(x))$. It substitutes the whole polynomial $q$ for each occurrence of the indeterminate in $p$, rather than multiplying their coefficient arrays.

Error-prone PCR deliberately introduces sequence diversity during amplification by conditions or enzymes with reduced fidelity. The mutation burden and library composition affect how much useful sequence space can be explored.

The extent to which a solvent favors polymer–solvent contacts relative to polymer–polymer contacts.

Association of multiple polymer chains into a larger solution object.

A contiguous sequence of segments making contacts with an adsorbing surface. The adsorption energy gained per contact competes with the loss of chain configurations. A chain can have several trains separated by loops rather than lying wholly flat.

A nonadsorbed chain section whose two ends connect to surface-bound portions of the same adsorbed polymer. Loops extend into the solvent and contribute to the thickness, steric repulsion and conformational entropy of an adsorbed layer.

A nonadsorbed terminal section extending from a surface-bound part of an adsorbed chain to a free chain end. It has one surface connection, whereas an adsorbed loop has two.

The number of electrons allocated to the connected cluster framework after separating external bonds and appropriate lone pairs. For ${\mathrm{B}}_{\mathrm{n}}{\mathrm{H}}_{\mathrm{m}}^{\mathrm{q}-}$ with one terminal B–H bond per vertex it is $n+m+q$.

A closed deltahedral cluster described by $n+1$ skeletal electron pairs in the usual Wade model. borane examples include $[{\mathrm{B}}_{12}{\mathrm{H}}_{12}{]}^{2-}$.

An open cluster framework formally obtained by removing one vertex from a closo parent. In the usual electron-counting model, an $n$-vertex nido framework has $n+2$ skeletal pairs.

An open cluster framework formally obtained by removing two vertices from a closo parent. Its usual electron count is $n+3$ skeletal pairs; hydrogen positions and cage distortions require additional structural information.

A polyanionic main-group cluster whose electron count can support delocalized skeletal bonding. Localized lone pairs must be separated before applying a polyhedral count.

A recognition system preventing some self or genetically incompatible pollinations from completing fertilization. Its presence and mechanism must be tested rather than inferred from bisexual floral structure.

For a normal transition dipole of magnitude $D$, the Fermi's golden rule, one-polarization mode density $\omega /(2\pi c^{2}d)$ and single-photon squared field $\hslash \omega /(2{ϵ}_{0}V)$ give $A_{\perp }=D^2{\omega }_0^2/(2{ϵ}_0\hslash c^{2}d)$. Dividing by the free-space electric-dipole spontaneous emission rate gives the title. The factor three relative to the scalar density-of-states ratio comes from the isotropic free-space polarization average. This ideal weak coupling formula assumes wide plates and unperturbed atomic structure, with atomic size much smaller than $d$.

A radiative lifetime depends on the available electromagnetic modes as well as on the atomic transition. In the ideal parallel-perfect conductor geometry of polarization-selective spontaneous emission between conducting plates, an in-plane transition dipole has no coupled propagating electric dipole radiation mode below the relevant waveguide cutoff frequency.

If that idealized electric dipole transition is the only decay channel, its rate is zero and the corresponding lifetime diverges. This is a statement about a restricted channel and geometry, not a general prediction of an immortal excited atom. Finite plates, absorbing surfaces, dipole misalignment and other allowed transitions can restore decay. A normal transition dipole can couple to the surviving transverse electromagnetic mode.

An outgoing three-dimensional Green function behaves as $e^{ikr}/r=1/r+ik+O(r)$. Evaluating a point-coupled discrete-level equation at cutoff distance $\delta$ produces a real shift proportional to $1/\delta$ and an imaginary outgoing-wave term proportional to $ik$. The bare level must depend on the cutoff to keep the measured detuning finite.

A pluripotent state associated with the preimplantation mouse epiblast and a characteristic regulatory network. Naive cells differ from the postimplantation epiblast in signalling requirements, chromatin state and responsiveness to cell differentiation cues.

A transitional pluripotent state in which naive circuitry has been dismantled while competence for later lineage induction is being acquired. This is a state transition, not an additional germ layer.

A pluripotent state associated with later epiblast and readiness to respond to lineage-specifying signals. It is experimentally and molecularly distinct from naive pluripotency.