Combining equal-intensity beams with complex degree of first-order coherence $g$ gives a fringe modulation proportional to $\mathrm{Re}(e^{i\phi }g)$. Maximizing and minimizing over observation phase gives visibility $V=|g|$. A symmetric source can make $g$ real, but it may still be negative. Its sign changes fringe phase by $\pi$; it does not represent negative physical contrast.

A fringe shift measures a change of relative phase between interfering waves in units of a full fringe spacing. A changing optical path difference $\Delta \ell$ gives a shift $\Delta N=\Delta \ell /\lambda$ for monochromatic light.

Absorption in which a photon excites an electron between different energy bands, requiring an occupied initial state, an available final state and a nonzero transition matrix element. Indirect band-edge transitions also need momentum transfer, commonly through a phonon. Such absorption can prevent optical transmission even above a metal's free-carrier plasma edge.

A photon-induced interband transition that approximately preserves the electron's crystal wavevector because the photon momentum is small compared with reciprocal-lattice momenta. Its strength depends on the dipole matrix element and the joint density of states. A direct band gap permits such a transition between the band-edge states, but optical selection rules can still suppress it.

An optical interband transition that requires another momentum source, commonly a phonon, to connect initial and final electronic wavevectors. Near an indirect band gap, phonon emission gives a threshold photon energy $E_g+\hslash \Omega$, while phonon absorption gives $E_g-\hslash \Omega$ and requires an occupied phonon mode. The temperature dependence of the absorption therefore contains phonon-population information in addition to electronic occupations.

Overlap delayed replicas in a sufficiently thin phase-matched second-order crystal and collect their cross-generated sum-frequency signal. Its integrated energy gives the displayed autocorrelation under undepleted, broadband-response assumptions. Temporal resolution depends on gate overlap, group-velocity mismatch and spectral acceptance. Duration inference requires a pulse-shape assumption, and arbitrary temporal phase is not recovered.

A two-replica second-harmonic measurement retaining carrier interference fringes. The formula assumes an instantaneous broad-band second-harmonic response and includes the carrier in $E$. The extra interference information does not in general determine an arbitrary complex pulse uniquely.

An integer root bound brackets an integer nth root using exact inequalities. If $N>0$ has bit length $\ell$ and $k\ge 1$, thenIndeed, $2^{\ell -1}\le N<2^{\ell }$; taking positive real $k$th roots gives the bounds. They supply an initial interval for binary search.

For $N=1000$ and $k=3$, the bit length is ten, giving $8\le \lfloor \sqrt[3]{1000}\rfloor <16$. Testing nearby powers then finds the exact value $10$. Using inequalities rather than rounded roots avoids excluding a perfect power at a search boundary.

If birth counts obey $dN/dm\propto m^{-s}$ and individual luminosity scales as $m^q$, multiply by mass or luminosity and by $dm=md\ln m$ to obtain the displayed logarithmic weights. With $s=2.35$ and $q=4$, birth mass has weight $m^{-0.35}$ while light has weight $m^{2.65}$. Finite mass cutoffs are essential; stellar lifetimes also alter the luminosity weighting of an evolved population.

The up game is the short partizan game $\uparrow =\{0\mid \ast \}$, where $\ast =\{0\mid 0\}$ is the star game. If Left starts, moving to zero wins. If Right starts, the only move leaves star to Left, who again moves to zero and wins. Thus up is strictly positive in the game ordering.

Its value is nevertheless smaller than every positive real number: it is an infinitesimal combinatorial game. This makes it useful for separating numerical advantage from a tiny advantage that can decide an otherwise balanced disjunctive sum. Reversing the players gives down, $\downarrow =\{\ast \mid 0\}=-\uparrow$.

The impartial game $\ast =\{0\mid 0\}$, so either player can move to zero. It is equivalent to a Nim heap of size one. It is incomparable with zero, and $\ast +\ast =0$ under the disjunctive sum of games.

For a uniform ball of radius $R>0$ and positive mass $M$, let $G$ be its centre of mass. Spherical symmetry gives the second-moment matrix $\int u{u}^{T}dm=(M{R}^2/5)I$, so the central inertia tensor is $(2M{R}^2/5)I$. The tensor parallel-axis theorem gives the displayed shifted tensor. For $X\ne 0$ its eigenvector along $X$ has eigenvalue $2M{R}^2/5$, while the two-dimensional orthogonal complement has repeated eigenvalue $M(2R^2/5+|X{|}^2)$. Thus it is an isotropic tensor exactly at $P=G$. Displacement along a chosen axis does not change that axis's moment; the transverse moments increase by $M|X{|}^2$.

Two uniform discs of mass m and radius a stick at first edge contact. One initially translates at speed v and the other rests, with neither initially spinning. If the incoming velocity makes angle $\theta$ to the line of centres, the composite axial moment of inertia about its centre of mass is $3m{a}^2$. Conserved angular-momentum magnitude is $mav|\sin \theta |$, so the angular velocity magnitude is $v|\sin \theta |/(3a)$. The centre-of-mass speed is v/2. Splitting kinetic energy into translation and rotation gives $K_f=m{v}^2(1/4+{\sin }^2\theta /6)$. The difference from $m{v}^2/2$ is dissipated internally; it cannot be removed from angular-momentum conservation.

For a reversible device with flux linkage $\Psi$, electrical power is $I\dot{\Psi }$, so stored energy changes by $dU=Id\Psi$. This reduces to $U=L{I}^2/2$ only for a fixed linear inductance. Hysteresis makes energy path dependent and introduces loss.

For one three-level unit with energies $-\mu B,0,\mu B$, the high-temperature heat capacity is $C\simeq 2{\mu }^2{B}^2/(3k_B{T}^2)$. A measured coefficient $C=a/T^2$ therefore gives the displayed magnetic moment. If the heat capacity instead represents $N$ independent units, replace $a$ by $a/N$; the normalization must be established before extracting a single-unit moment.

The inradius is the radius of an incircle. For a triangle of area $A$ and semiperimeter $s$, it is $r=A/s$: joining the incenter to the three vertices partitions the triangle into three triangles with common altitude $r$.

For $a_{\mathrm{out}}=\alpha a_{\mathrm{in}}+\beta b_{\mathrm{in}}^{†}$ with real canonical coefficients, the inverse is $a_{\mathrm{in}}=\alpha a_{\mathrm{out}}-\beta b_{\mathrm{out}}^{†}$. The in-vacuum annihilation condition therefore has coefficient $v=-\beta$ when written as $(u{a}_{\mathrm{out}}+v{b}_{\mathrm{out}}^{†})|0_{\mathrm{in}}⟩=0$. Forward and inverse maps must not be confused.

A Hermitian impulse $U\delta (t)$ with zero diagonal has kick operator $e^{-iU/\hslash }$. If $u=U_{21}$, its transition probability is ${\sin }^2(|u|/\hslash )$, reducing to $|u{|}^2/{\hslash }^2$ for a weak impulse. The integrated operator has action units, and a pulse integrated across its full support has no extra half-delta factor.

For the ideal uniform solid sphere with negligible table impulse during the cue strike, immediate rolling requires $u_0=a{\Omega }_0$, hence the horizontal cue line lies $2a/5$ above the centre. A different inertia distribution changes this height.

A load matched to a source receives the maximum available power in the stated circuit model. For a real source resistance $R$ and open-circuit root mean square voltage $V_{\mathrm{oc}}$, a load $R$ receives voltage $V_{\mathrm{oc}}/2$ and power $V_{\mathrm{oc}}^2/(4R)$. For complex impedances, maximum power transfer instead uses conjugate matching under the usual linear-source assumptions.