The approximate donor metal-insulator threshold is often written $n_c^{1/3}{a}_B^{\ast }\simeq 0.25$. It estimates when overlap and screening destroy isolated donor localization. The numerical coefficient is empirical and not a universal exact result; disorder, compensation and anisotropy can shift the threshold.^[ref]

A perpendicular magnetic dipole $m\hat{z}$ a height $h$ above an ideal flux-excluding plane has image $-m\hat{z}$ at $-h$. The exterior field has zero normal component at the plane. Images are a mathematical construction outside the physical solution domain.

A tethered nucleophile attacking the ipso carbon of an activated aromatic ring can form a cyclic sigma-adduct sharing one atom with that ring. Rearomatization then cleaves the original aryl–heteroatom bond.

Deviation in inherited allele proportions from the Mendelian expectation. Embryonic viability selection can also distort offspring counts and must be separated from gametic transmission bias.

A modifier reducing a driver’s transmission advantage or harmful consequences. Selection depends on linkage, background and the costs of suppression.

Biased transmission associated with centromere differences during asymmetric meiosis, where only some meiotic products contribute to the next generation.

A median of a weighted probability distribution. With positive weights summing to one, at most half the weight lies strictly below a weighted median and at most half lies strictly above it. The lower weighted median chooses the first observed value whose cumulative weight reaches one half; the convention matters when the cumulative mass is exactly one half.

The length associated with relaxation of carrier momentum. In a parabolic band, electron mobility measures the momentum relaxation time, giving the displayed expression. Frequent weak small-angle scattering can make the single-particle scattering time shorter than the transport time, and electron phase coherence has its own lifetime.

Dinic's algorithm solves a maximum flow problem by repeatedly constructing the level graph of the residual network using breadth-first search, then sending a blocking flow through edges advancing one level. A blocking flow saturates at least one edge on every source-to-sink path in this level graph.

After a phase, no shortest residual path from the previous phase remains; the residual distance to the sink strictly increases if it remains reachable. Hence there are fewer than $V$ successful phases. With current-edge pointers and a standard blocking-flow implementation, the general bound is $O(V^{2}E)$. When the sink becomes unreachable, the reachable vertices define a minimum cut, certifying the maximum flow. Residual reverse edges must be maintained so later phases can undo earlier choices.

When a square matrix satisfies $M^2=\kappa I$, its powers reduce to $M^{2n}={\kappa }^{n}I$ and $M^{2n+1}={\kappa }^{n}M$. Absolute convergence of the matrix exponential allows even and odd subseries to be separated. Thus $C(\kappa )={\sum }_{n\ge 0}{\kappa }^n/(2n)!$ and $S(\kappa )={\sum }_{n\ge 0}{\kappa }^n/(2n+1)!$. For real $\kappa >0$, these are $\cosh \sqrt{\kappa }$ and $\sinh \sqrt{\kappa }/\sqrt{\kappa }$. For $\kappa <0$, they are $\cos \sqrt{-\kappa }$ and $\sin \sqrt{-\kappa }/\sqrt{-\kappa }$. At $\kappa =0$, $e^M=I+M$, including nonzero nilpotent matrices of square zero. This also explains the limiting sine-over-argument coefficients without dividing by zero.

For finite matrices, or formally on a common invariant domain where the expansion is meaningful, exponential conjugation is generated by the commutator map. Differentiating $F(s)=e^{sS}H{e}^{-sS}$ gives $F^{\prime }=[S,F]$, and its Taylor series yields the nested-commutator expansion. An anti-Hermitian $S$ makes this a unitary transformation. For perturbation theory this may be used order by order without asserting convergence for arbitrary unbounded operators.

Canonical free scalar and gauge-vector kinetic terms contain two derivatives and two fields, whereas the Dirac kinetic term contains one derivative and two spinor fields. Matching their mass dimension to $d$ in $d$ spacetime dimensions gives the displayed engineering dimensions. These are the dimensions used in power counting; interacting scaling dimensions can additionally contain anomalous contributions.

If $x_{\alpha }$ counts amphiphile molecules belonging to aggregates of size $\alpha$, the corresponding dimensionless aggregate count is $n_{\alpha }=x_{\alpha }/\alpha$. Its ideal translational chemical potential contains $\ln n_{\alpha }$. Confusing these two distributions drops a factor of $\alpha$ and changes both normalization and the chemical-potential interpretation.

Here $\mu$ is the common equilibrium chemical potential per amphiphile and ${\mu }_{\alpha }^0$ the standard free energy per amphiphile in an aggregate of size $\alpha$. A minimum of the aggregate free-energy balance at a finite size produces a micellar population around that size, in addition to monomers. Translational entropy leaves a small aggregate population below the nominal critical micelle concentration; a preferred finite size is not supplied by the mass-action equation alone.

The initial segment of a Markov chain Monte Carlo run discarded to reduce dependence on its starting state. A fixed discarded count is a choice to assess using Markov chain Monte Carlo convergence diagnostics, not a proof that the retained sample has reached its stationary distribution.

An autocorrelation time measures how long an observable retains dependence on its past. It depends on both the process and the chosen observable. For a stationary sequence with finite positive variance and autocorrelation function $\rho (k)$, the integrated autocorrelation time is ${\tau }_{\mathrm{int}}=1+2{\sum }_{k\ge 1}\rho (k)$ when the series is absolutely summable. It controls the leading variance of a long sample mean.

An exponential autocorrelation time instead describes tail decay, for example $\rho (k)\approx C{e}^{-k/{\tau }_{\exp }}$. These are different summaries. A stationary autoregressive process of order one with coefficient $0<a<1$ has $\rho (k)=a^k$, so ${\tau }_{\mathrm{int}}=(1+a)/(1-a)$ and ${\tau }_{\exp }=-1/\log a$. For a Markov chain with several decay modes, different observables can emphasize different timescales.

A Markov chain with at least one absorbing state, such that from each state an absorbing state can be reached. In a finite such chain, absorption occurs with probability 1. Expected times to absorption satisfy a linear system obtained by conditioning on the first transition.

On a finite state space, an acyclic Markov chain has no directed cycle among its nonabsorbing states. Order those states topologically by their possible transitions. Hitting probabilities and expected absorption times can then be calculated in reverse order, because a state's equations involve only later states.

For example, let $A$ go to $B$ or to an Absorbing state $C$ with equal probabilities, and let $B$ go directly to $C$. The expected absorption times are $t_C=0$, $t_B=1$ and $t_A=1+t_B/2=3/2$. This is dynamic programming on the transition graph, rather than a general simultaneous linear solve.

If a convention permits self-transitions while forbidding cycles between distinct nonabsorbing states, the holding probabilities must be retained. Provided $P_{ii}<1$, the same backward calculation usesA self-loop at $A$ of Probability $1/2$, with probabilities $1/4$ each to $B,C$, gives $t_A=5/2$ instead of $3/2$. The finite-state assumption is substantive: an infinite deterministic chain $i\mapsto i+1$ is acyclic but has no terminal Absorbing state and no finite reverse recursion of this kind.

For a linear isotropic unsaturated effective susceptibility contrast $\Delta \chi$, the induced-moment energy is $-V\Delta \chi B^2/(2{\mu }_0)$, giving $\mathbf{F}=V\Delta \chi \nabla B^2/(2{\mu }_0)$. The one-half differs from the fixed permanent-dipole energy because the applied field creates the moment. Demagnetization and saturation are absorbed into or replace the assumed response model.

For a tether with extension $\ell$ and tensile load $F$, a small transverse coordinate costs $F{x}^2/(2\ell )$. Equipartition gives $F=k_{B}T\ell /⟨x^2⟩$ for one Cartesian transverse coordinate. Using the sum of two transverse variances changes the factor of two. Geometry and measurement filtering must match the calibration model.