A value of the form $a^5$, the product of five copies of $a$.

The step at the beginning of $E_0{e}^{-t/\tau }\Theta (t)$ retains high-frequency weight. For fixed nonzero transition frequency, $|\tilde{E}|\to E_0/|\omega |$ as $\tau \to \infty$. A slowly decaying pulse with an abrupt onset is therefore not an adiabatically switched perturbation. The diagonal accumulated phase can separately grow with duration.

For $A=\Omega T/2$, $B=\Phi /2$, the ordered product of a $z$ rotation after an $x$ rotation has $C=\cos A\cos B$ and $\mathbf{v}=(\cos A\sin B,\sin A\sin B,\sin A\cos B)$. Since $C^2+|\mathbf{v}{|}^2=1$, its angle is $2atan2(|\mathbf{v}|,C)$ and its axis is $\mathbf{v}/|\mathbf{v}|$. At $\mathbf{v}=0$ the group element is central and its axis is undetermined.

The reduced band mass for relative electron–hole motion, ${\mu }^{\ast }=m_e^{\ast }{m}_h^{\ast }/(m_e^{\ast }+m_h^{\ast })$. It differs from the total mass that controls centre-of-mass motion.

The attractive electrostatic interaction between an electron and an electron hole. In a screened effective-mass approximation its potential has the displayed form. It can bind an exciton and reduce its optical excitation energy below the corresponding unbound-pair threshold.

A characteristic transport length before exciton decay, commonly $L_D=\sqrt{D_X{\tau }_X}$ in a diffusion-decay convention. A root-mean-square displacement over a fixed time has dimension-dependent factors and is not automatically this same convention.

A nonradiative transition between electronic states of different spin multiplicity, enabled by interactions such as spin–orbit coupling. A constant intersystem-crossing channel contributes to the unquenched excited-state lifetime.

Double exchange couples localized magnetic moments through actual carrier hopping between sites of different valence. Strong local Hund's coupling makes hopping energetically easier when neighboring core spins align, lowering the carrier kinetic energy. In the classical strong-coupling limit, a common effective hopping is $t_{\mathrm{eff}}=t\cos (\theta /2)$ for relative spin angle $\theta$; aligned cores maximize its magnitude. Unlike insulating superexchange, this mechanism directly connects carrier motion and magnetic alignment.^[ref]

A localized magnetic moment can polarize a metal's conduction electrons, whose response changes the energy of a second moment. This indirect exchange interaction can be long ranged and oscillate in sign with separation because of the Fermi surface. In the three-dimensional isotropic free-electron asymptotic limit its range dependence is proportional to $\cos (2k_{F}r)/r^3$, with coupling-dependent prefactor and sign convention. The original calculation for nuclear moments used their contact hyperfine interaction with conduction electrons; it was motivated by nuclear magnetic resonance line widths in metallic silver.^[ref]

For localized orbitals with an exponential tail $e^{-\kappa r}$, overlap-mediated hopping over separation $d$ inherits an exponentially small factor $e^{-\kappa d}$, multiplied by geometry-dependent polynomial factors. A kinetic-exchange scale then falls approximately as $e^{-2\kappa d}$. Comparisons must use actual nearest-neighbour distances, not interchange lattice constants between different crystal structures. The estimate is not a replacement for a screened solid-state orbital calculation.

Conditional reduction of benefits to a partner that supplies a poor service, potentially changing the returns to exploitation. An immediate physiological response and its evolutionary explanation should be distinguished.

Suppression of worker reproduction by other workers, including discriminatory removal of worker-laid eggs. Genetic incentives and effects on colony productivity can both matter.

A conditional strategy prescribes actions according to specified observations, states, or histories. The conditioning information must be available to the player when the action is chosen; the rule does not require conscious calculation.

In an iterated prisoner's dilemma, Tit for tat conditions on the opponent's previous action, whereas a grim trigger strategy remembers whether any defection has occurred. Both can choose the same action in the first round and then respond differently to the same most recent move because their retained histories differ. A finite-state strategy records the relevant past information in an automaton state. In an extensive-form game, the prescription must be constant across an information set, since its nodes cannot be distinguished when choosing an action.

A signalling equilibrium is a stable pattern of signal production and response in a game where one participant has information another lacks. A sender's strategy selects a signal according to its type; the receiver's strategy chooses a response after observing that signal. Equilibrium requires both choices to be best responses under consistent beliefs or a specified evolutionary stability criterion.

A separating pattern uses different signals for different types, while a pooling pattern uses the same signal. A costly signal can support separation only if each type prefers its assigned signal and response to mimicking another type. In evolutionary game theory, stability concerns resistance to invading strategies; it is a stronger, separately specified condition than merely observing a correlation between signals and types.

The Eulerian polynomial packages the Eulerian numbers asHere $\mathrm{des}(\pi )$ counts permutation descents. For example, $A_3(t)=1+4t+t^2$. In particular $A_n(1)=n!$, because setting every weight to one counts all permutations. The Eulerian number recurrence gives $A_n(t)=(1+(n-1)t)A_{n-1}(t)+t(1-t)A_{n-1}^{\prime }(t)$ for $n\ge 1$. Conventions using $t^{\mathrm{des}(\pi )+1}$ multiply this polynomial by $t$.

Taking the trace of the position-space imaginary-time kernel identifies its endpoints and integrates over the common position. The resulting closed paths have imaginary-time length $\beta \hslash$. Bosonic coherent fields are periodic, while fermionic coherent fields are antiperiodic. This temporal condition follows from the trace, independently of the spatial boundary conditions.

A local $d$-dimensional quantum field at finite temperature can often be represented by an anisotropic classical-looking action in $d+1$ dimensions, with a finite extra imaginary-time extent. The zero-temperature limit makes that extent infinite. A real nonnegative statistical weight is an additional requirement: fermionic signs and imaginary terms can obstruct interpretation as an ordinary classical probability distribution.

Lattice reduction replaces a basis of a Euclidean lattice by another basis of the same lattice with shorter, more nearly orthogonal vectors. If the basis columns form $B$, every replacement $BU$ with an integer unimodular matrix $U$ preserves the lattice. Arbitrary real changes of basis generally do not.

For columns $(1,0)$ and $(100,1)$, subtracting one hundred times the first from the second gives the orthogonal basis $(1,0),(0,1)$. In two dimensions Lagrange lattice reduction repeatedly performs such integer corrections and swaps. In higher dimensions the LLL algorithm gives a useful efficiently computable reduction condition; it does not in general return a globally shortest basis.