Sensory information about bodily position, movement and force, contributing to coordination and relations between seen and felt movement.

Detection and processing of acoustic stimulation. Signal detectability depends on the receiver and background noise.

For a nondegenerate semiconductor with donor density $N_D$, no acceptors, and donor energy $E_D=E_c-E_b$, an occupied donor has two electron spin states while an empty donor has one. Its ionized fraction is therefore $N_D^{+}/N_D=[1+2e^{(E_F-E_D)/(k_{B}T)}{]}^{-1}$. The conduction-band relation $n=N_c{e}^{(E_F-E_c)/(k_{B}T)}$ gives $N_D^{+}=N_D/[1+(2n/N_c)e^{E_b/(k_{B}T)}]$. Combining charge neutrality, $n-p=N_D^{+}$, with the semiconductor mass-action law, $np=n_i^2$, proves the displayed equation.

Its left side increases strictly for $n>0$ and its right side decreases, so there is one physical solution, with $n\ge n_i$. In the freeze-out regime, holes are negligible and solving the quadratic gives $n=2N_D/[1+\sqrt{1+(8N_D/N_c)e^{E_b/(k_{B}T)}}]$. Strong dopant freeze-out reduces this to $n\simeq \sqrt{N_D{N}_c/2}{e}^{-E_b/(2k_{B}T)}$. An extrinsic plateau requires $(2N_D/N_c)e^{E_b/(k_{B}T)}\ll 1$ and $n_i\ll N_D$, not simply $k_{B}T\gg E_b$.

An amphoteric dopant can act as a donor impurity or an acceptor impurity according to its incorporated lattice site or defect configuration. Silicon in gallium arsenide is the standard example: substitution on the gallium sublattice gives a shallow donor, while substitution on the arsenic sublattice gives a shallow acceptor. Its electrical effect therefore depends on growth conditions as well as total silicon concentration.^[ref]

A compensated semiconductor contains both donor impurities and acceptor impurities. In a nondegenerate equilibrium model with both species fully ionized, charge neutrality gives $n-p=\Delta =N_D-N_A$, while the semiconductor semiconductor mass-action law gives $np=n_i^2$. Solving yieldsFor $\Delta >0$ and $n_i\ll \Delta$, $n\simeq \Delta$ and $p\simeq n_i^2/\Delta$. Numerically the minority concentration is better obtained as $n_i^2/n$ than by subtracting nearly equal numbers. If dopants are only partly ionized, replace $N_D,N_A$ in charge neutrality by their ionized concentrations and determine the chemical potential self-consistently.

Modulation doping places ionized dopants away from the region in which their supplied charge carriers move. In a semiconductor heterostructure, band alignment can transfer electrons from a doped layer into a nearby lower-energy channel.

This separation reduces scattering from nearby charged impurities while maintaining a substantial mobile-carrier population. It supports a two-dimensional electron gas and the channel of a high-electron-mobility transistor. Separation is not complete decoupling: the donors still set electrostatics, and remote impurities, interfaces and other scattering mechanisms can still limit charge carrier mobility.

Introduction of dopants by accelerating ions into a solid. Their depth distribution, lattice damage and electrical activation depend on implantation and subsequent annealing.

At sufficiently low temperature, thermal ionization of shallow dopants becomes inefficient, reducing the mobile carrier density. An acceptor-doped sample may then have many neutral acceptors and few mobile electron holes. Extracting an activation energy requires a model of dopant ionization and compensation; the mobile density need not have the same simple activation exponent in every compensation regime.

A heterostructure transistor with a high-mobility channel spatially separated from ionized dopants; the gate modulates channel population.

Preferential assembly of components with matching handedness into an individual assembly. A racemic sample can contain equal populations of homochiral enantiomeric cages.

Parental use or selection of substances that reduce disease in offspring. A benefit to offspring does not imply treatment or cure of the parent.

A molecular layer formed through adsorption and lateral organization at a substrate. Anchoring groups, chain packing and exposed terminal chemistry jointly determine its properties. Hydroxyl-terminated alkanethiol films on gold provide a nonbiological example with controllable water wetting.

Preferential reaction of one functional group or bond type in a molecule containing alternatives. It depends on reagents, conditions and reaction time rather than being an absolute property of a functional group.

In the nonrelativistic electric-dipole approximation, the dipole operator does not act on spin, so transitions between different total-spin states vanish. Spin–orbit coupling can give such transitions weak intensity.

Lazy propagation delays updating a segment tree node's descendants while keeping the node's own aggregate current. Before inspecting only part of its interval, apply its pending tag to both children, compose that tag with each child's pending work, and clear the parent's tag. Whole-interval queries can use the current aggregate without this descent.

Composition order matters. For affine maps $x\mapsto ax+b$, represent a tag by $(a,b)$. Applying $(a_1,b_1)$ first and $(a_2,b_2)$ later givesOn an interval aggregate $(s,\ell )$ it acts as $(as+b\ell ,\ell )$. These formulas prove that the same combined tag works for a parent and its children. For example, doubling then adding three gives $2x+3$, whereas adding three then doubling gives $2x+6$.

The lazy segment tree contract therefore needs both a correct aggregate action and correct ordered function composition. A pending tag is deferred descendant work, not permission for the parent's aggregate to be stale.^[ref]

A lazy segment tree stores interval aggregates in a segment tree while representing compatible deferred updates by tags. The aggregate combination must be a monoid operation, and each tag must act on aggregates consistently with combining intervals. Tags include an identity operation and are closed under function composition.^[ref]

For range addition and range sums, store $(s,\ell )$, the sum and length of the interval. Combining children adds both components. Adding $d$ transforms it into $(s+d\ell ,\ell )$, so a covered node can update its stored sum immediately without visiting every descendant. Its pending tag accumulates the addition for later lazy propagation.

For example, adding three to all four entries of $[2,1,4,0]$ changes the root sum from seven to nineteen. A query for the whole interval returns nineteen immediately. A query for just the first entry must first propagate the pending update along that entry's path. Constant-time aggregate and tag operations give $O(\log n)$ interval updates and queries; an arbitrary update rule need not admit such a representation.

Computational reproducibility means obtaining consistent results from the same input data, computational procedure and stated conditions. A reproducible analysis identifies the actual inputs and their data provenance, the software versions, parameters and relevant environment. Exact bitwise agreement is stronger than agreement within a justified numerical tolerance.^[ref]

For example, changing the order of a floating point sum can change its last few digits through rounding error. A useful check states the acceptable absolute error or relative error before comparing results. Successfully rerunning the same code also reruns any undetected mistake in it, so consistency is evidence about the computational process rather than a proof that the physical model is correct.

Testing a scientific finding with newly collected data or an independently conducted study. Terminology varies, so the operational meaning should be stated.

For a spherically symmetric potential energy $V(r)$, separation of variables with $\psi =R(r)Y_{lm}(\theta ,\varphi )$ reduces the Schrödinger equation to the displayed radial equation. Writing $u=rR$ instead removes the first derivative:The angular contribution acts as a centrifugal barrier. Boundary behavior and wavefunction normalization must still be imposed; an arbitrary radial function is not an energy eigenfunction.