For independent geometric observations with mass $p^{r-1}(1-p)$ on $r=1,2,\dots$, where $0\le p<1$, the total $T$ is sufficient. Conditional on $T=t$, positive-integer compositions are uniformly distributed. The indicator $1_{\{X_1>1\}}$ is unbiased for $p$, and conditioning it on $T$ gives $\left(\genfrac{}{}{0ex}{}{t-2}{n-1}\right)/\left(\genfrac{}{}{0ex}{}{t-1}{n-1}\right)=(t-n)/(t-1)$. For one observation, use $1_{\{T>1\}}$ instead, including its zero value at $T=1$.

For independent uniform random variables on $(\theta ,3\theta )$ with $\theta >0$, the sample minimum and maximum form a sufficient statistic. Conditional on these endpoints, exchangeability and the uniform law of the remaining observations give $E[X_1\mid X_{(1)},X_{(n)}]=(X_{(1)}+X_{(n)})/2$. Applying the Rao-Blackwell theorem to $X_1/2$ gives the displayed unbiased estimator. Its variance is ${\theta }^2/[2(n+1)(n+2)]$, compared with ${\theta }^2/12$ for $X_1/2$.

In a one-brane warped background, the proper extra dimension can be infinite while the four-dimensional graviton zero mode has finite norm. The Einstein-Hilbert action weighs its four-dimensional curvature term by the warp factor integral shown above, finite for $k>0$. Thus the proper length is not the effective gravitational volume. This differs from an unwarped noncompact direction, whose volume integral diverges. It also differs from the two-brane hierarchy mechanism, where finite brane separation redshifts masses on the distant brane. The noncompact construction is established in the original single-brane warped-gravity paper.

For a spatially flat, tuned one-brane background without energy transfer to the bulk, the Israel junction conditions produce the displayed modified Friedmann equation. Positive brane tension $\lambda$ gives ordinary expansion when $\rho \ll \lambda$, while the quadratic density correction enhances expansion at high energy. Big Bang nucleosynthesis, early acoustic evolution and primordial spectra can constrain this change. Spatial curvature, a residual cosmological constant and bulk energy exchange require their additional terms rather than this special formula.

Let $R_k(3)$ be the least order of a complete graph whose every $k$-edge-colouring has a monochromatic triangle in a graph. At a vertex of a graph of order $k(R_{k-1}(3)-1)+2$, some colour joins at least $R_{k-1}(3)$ neighbours. An edge of that colour among these neighbours completes a triangle; otherwise the neighbours use at most $k-1$ colours and contain a triangle by induction. This proves the recurrence and the weaker explicit bound $R_k(3)\le 3k!$.

Every finite colouring of the positive integers contains a pair with nonzero square difference. One proof takes limits of the colour autocorrelations and represents the resulting positive-definite sequence by a positive spectral measure. Its atom at zero has mass at least the sum of squared colour densities. Averaging along squares kills irrational frequencies; a common multiple makes finitely many rational atomic frequencies equal to one. The remaining rational atomic mass can be made arbitrarily small, leaving a positive average correlation.

Radius stabilization fixes the size of a compactification dynamically rather than choosing it as an undetermined parameter. If the four-dimensional low-energy effective action contains a radion $R$ with positive kinetic coefficient $Z(R)$ and scalar potential $V(R)$, a stable radius $R_0$ requires $V^{\prime }(R_0)=0$ and $m_R^2=V^{\prime \prime }(R_0)/Z(R_0)>0$. This matters for the hierarchy problem: the volume or warp factor explains a hierarchy only if a consistent dynamics selects and maintains the required geometry.

An electrical substitution radiometer measures radiant flux by replacing absorbed optical power with calibrated Joule heating. At fixed temperature, feedback control holds the receiver equally warm with the beam blocked or admitted: after corrections, absorbed optical power equals the reduction in supplied electrical heating. Incident radiant flux additionally requires radiometric absorptance and entrance-path corrections. This connects radiometry to electrical standards.^[ref]

Picture a small absorbing cavity connected to a colder reference block by a radiometer thermal link. A heater supplies Joule heating, a thermometer senses the receiver, and feedback control adjusts the heater. When light supplies some of the heating, the controller supplies less. The readout is the change in heater power, rather than the beam's brightness or its number of photons. NIST's institutional description gives a concrete receiver with a conical interior, absorbing coating and weak Kapton support, operated with a receiver heater and thermometer.^[ref]

Irradiance measures incident radiant flux per actual receiving area. Its units $\mathrm{W}{\mathrm{m}}^{-2}$ match radiant exitance, but it counts the incoming side of the radiation exchange. Measure incoming angles from the unit normal pointing toward the incident hemisphere. ThenFor $2\mathrm{W}$ distributed uniformly over $0.010{\mathrm{m}}^2$, $E=200\mathrm{W}{\mathrm{m}}^{-2}$. If the receiving plane is tilted $60^{\circ }$ in the same broad uniform parallel beam, the incident power per actual area is halved to $100\mathrm{W}{\mathrm{m}}^{-2}$. This receiving-surface cosine factor is geometric and does not assume Lambertian reflectance. Together, irradiance and radiant exitance describe how radiation supplies and removes energy at a surface, while radiance preserves the directional information.

Radiant exitance is total outward emitted radiant flux per actual emitting area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$. Integrate radiance over the outward hemisphere, keeping the projected area factor:If the radiance is direction-independent, the spherical coordinates element is $d\Omega =\sin \theta d\theta d\phi$, soThe weighted solid angle is $\pi \mathrm{sr}$, although the whole hemisphere covers $2\pi \mathrm{sr}$. Directions near grazing contribute less because of the projected area. This geometric factor is the bridge between Lambert's cosine law and the total output of a diffuse patch. ^[ref].

For the uniform patch with $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, radiant exitance is $M=10\pi \mathrm{W}{\mathrm{m}}^{-2}$. Its area $0.020{\mathrm{m}}^2$ gives total radiant flux $\Phi =MA=0.20\pi \mathrm{W}\approx 0.628319\mathrm{W}$ over the outward hemisphere. This total differs from its radiant intensity $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$, which describes only one direction.

For an ideal blackbody, integrate Planck's law over frequency as well: $M=M_e=j^{\ast }=\sigma T^4$ by the Stefan–Boltzmann law. An ideal $6000\mathrm{K}$ emitting patch of area $1\mathrm{m}{\mathrm{m}}^2$ emits about $73.4881\mathrm{W}$ outward. Irradiance counts incoming radiation separately; for purely emitted output the net outward exchange per area is $M-E$. Reflected output must also be included if the surface reflects incident radiation.

Radiance resolves both position and direction: it is radiant flux per projected area and per solid angle. For a surface element of area $dA$, and a small outgoing cone $d\Omega$ at angle $0\le \theta <\pi /2$ to its outward unit normal,Here $\mathbf{x}$ locates the patch and $\mathbf{s}$ is a unit vector along the outgoing direction. The factor $dA\cos \theta$ is the patch's apparent area viewed along that direction. The same local definition applies to transmitted or received radiation, choosing the positive projected area for the crossing direction. ^[ref].

For a planar patch of area $A$ with spatially uniform, direction-independent radiance $L$, integrating over the patch givesTake $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$ and $A=0.020{\mathrm{m}}^2$. The radiant intensities are $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ and $I(60^{\circ })=0.10\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$. A sufficiently narrow $0.010\mathrm{sr}$ cone therefore carries approximately $2.0\mathrm{mW}$ or $1.0\mathrm{mW}$, respectively. Both views have the same radiance $L$: the oblique view presents half the projected area. This is Lambert's cosine law, not a loss of radiance with viewing angle.

Radiance is useful because it separates a source's local directional output from the geometry of an observation. Integrating over outgoing directions gives radiant exitance; integrating incoming radiance at a receiver gives irradiance. In the stationary vacuum setting of vacuum conservation of specific intensity, a resolved uniform source retains its radiance along each light ray, while the radiant flux received from the whole source can decrease as its apparent solid angle shrinks.

Radiant intensity describes how a source distributes its radiant flux over directions. A small cone of solid angle $\Delta \Omega$ carries approximately $I\Delta \Omega$ when $I$ varies little across it; integrating $I$ over all emitting directions recovers the total radiant flux. For a source with isotropy emitting $4\pi \mathrm{W}$ over a full sphere, $I=1\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ and a $0.01\mathrm{sr}$ cone carries $0.01\mathrm{W}$. A planar source satisfying Lambert's cosine law instead has direction-dependent radiant intensity, even though its radiance is direction-independent.

Radiant flux is the power carried by electromagnetic radiation: radiation energy transferred per unit time, measured in watts, $\mathrm{W}=\mathrm{J}{\mathrm{s}}^{-1}$. Specify what is counted: all radiation emitted by a source, radiation within one beam, or radiation intercepted by a photodetector. Writing $Q(t)$ for the cumulative energy crossing the specified boundary givesThe first expression is an instantaneous derivative; the second is an average over the stated time interval. If $0.020\mathrm{J}$ arrives during $0.010\mathrm{s}$, the mean radiant flux is $2\mathrm{W}$. The photodetector's absorbed power equals this incident radiant flux only if it absorbs all of the incident radiation.

Keeping radiant flux fixed while widening a beam reduces its irradiance, the incident power per receiving area. Radiance additionally resolves the propagation direction and has units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$; an optical camera can collect different radiant fluxes from sources with the same radiance but different apparent solid angles. Luminous flux, measured in lumens, weights the optical spectrum by the human eye's standardized response instead of counting energy equally at every wavelength. Equal radiant fluxes need not give equal luminous fluxes.^[ref]

Radiant power also differs from the optical power of an optical lens, which is reciprocal focal length in dioptres. Likewise, astronomical radiative flux is power per area. Stating the units prevents these shared words from hiding different physical quantities.

The effective recombination coefficient for a spectral line counts recombinations and subsequent radiative cascades producing that line. At sufficiently low density, its photon production rate per volume is $n_e{n}_{\mathrm{parent}}{\alpha }_{\ell }^{\mathrm{eff}}$. It depends on gas temperature, optical-depth assumptions such as Case B recombination, and sometimes density. It differs from the total rate coefficient removing the parent ion.

A recombination line is a spectral line produced after a free electron is captured, often through a subsequent radiative cascade among bound levels. Its emissivity is computed with an effective recombination coefficient, distinct from the free-bound recombination continuum.

A radiation-dominated universe has expansion controlled mainly by radiation in cosmology. When spatial curvature and the cosmological constant are negligible, the Friedmann equation and density scaling $\rho \propto a^{-4}$ give the displayed scale factor, with Hubble parameter $H=1/(2t)$.

A correlated pair of quarks. Its color, flavor, spin and orbital states are separate degrees of freedom and must collectively satisfy Fermi statistics. In flavor SU(3), the antisymmetric pair space is ${\Lambda }^{2}3\cong \overline{3}$; contraction with the invariant Levi-Civita symbol realizes the isomorphism. The antisymmetric $ud$ flavor pair is an isospin singlet with flavor hypercharge $2/3$. Combining two such flavor-antitriplet pairs with a strange antiquark permits the highest weight of the pentaquark antidecuplet; this flavor construction does not itself establish a bound state.

Partial-wave scattering decomposes a scattering state into partial waves with definite orbital angular momentum. For a central potential, each channel has its own scattering phase shift and Partial-wave S-matrix.

A scattering state describes an incoming preparation and its outgoing components after an interaction. In a stationary calculation, it is typically a generalized energy eigenstate with specified incoming boundary data, rather than a square-integrable bound state.

For a short-range potential, its distant form separates freely propagating incoming and scattered waves. The outgoing boundary condition selects how a localized disturbance radiates away, and the resulting relative coefficient determines a scattering amplitude. A physical normalizable packet is formed by superposing stationary scattering states over an energy or momentum range. This distinction allows finite particle probabilities and localized preparations without treating an infinite plane wave as a normalized state inside the physical Hilbert space.

For one-dimensional scattering from a real potential, write the left exterior wavefunction as $e^{ikx}+r{e}^{-ikx}$ with $k>0$. The reflected-to-incident probability-current ratio is $R=|r{|}^2$, since both components have the same speed and their currents have opposite signs. This probability differs from the complex reflection amplitude $r$. With transmission into a right exterior wavenumber $q$, the transmission probability is $T=(q/k)|t{|}^2$ and conservation of current gives $R+T=1$ when there is no absorption.