For positive real numbers $a,b$, the beta integral represents the beta function by $B(a,b)={\int }_0^1{x}^{a-1}(1-x{)}^{b-1}dx$. In particular, for nonnegative integers $m,n$, it gives ${\int }_0^1{x}^m(1-x{)}^{n}dx=m!n!/(m+n+1)!$, connecting integration to binomial coefficients.

The Bell test reported by Bas Hensen and collaborators used electron spins separated by $1.3\mathrm{km}$, efficient spin readout and rapid random measurement settings. Those features addressed detection and locality loopholes together. Its result tests Bell local causality with the stated experimental assumptions, rather than ruling out every possible hidden-variable theory.^[ref]

A bound on the correlations attainable by measurement in quantum measurements for a specified Bell expression. The CHSH Tsirelson bound is $2\sqrt{2}$; the number depends on which Bell expression is being optimized.

A linear time-invariant circuit sends each sinusoidal input to an output at the same frequency, multiplied by its complex transfer function. Two inputs therefore give the sum of two phase-shifted sinusoids. If their output amplitudes are equal, the sum factors into a carrier at the mean frequency and a signed cosine envelope at half the frequency difference. The envelope magnitude repeats at the full difference frequency. Unequal amplitudes give a nonzero minimum; a nonlinear detector is needed to generate a separate difference-frequency signal.

For two identical bosonic input populations of $N$ in a balanced two-mode mixer, only even occupancies occur at either output. Expanding $[(b_1^{†}{)}^2-(b_2^{†}{)}^2{]}^N$ and retaining normalized number-state factorials gives the displayed law. Its sum is one by the generating function $(1-4z{)}^{-1/2}(1-4z{)}^{-1/2}=(1-4z{)}^{-1}$.

An elementary excitation of the paired mean-field state whose electron operator has mixed creation and annihilation components. Its positive excitation energy is $E_k=\sqrt{{\xi }_k^2+|{\Delta }_k{|}^2}$. At zero temperature the BCS ground state has zero occupation of these positive-energy modes, even though physical electrons have a smooth nontrivial momentum occupation.

The empty and doubly occupied states of a paired momentum block form a two-level space. Define $S_k^{+}=a_{k\uparrow }^{†}{a}_{-k\downarrow }^{†}$, $S_k^{-}=a_{-k\downarrow }{a}_{k\uparrow }$ and $S_k^z=(n_{k\uparrow }+n_{-k\downarrow }-1)/2$. They obey the spin algebra on that space. The normalized factor $u_k|0⟩+v_k|\mathrm{pair}⟩$ is a spin-one-half coherent state; the product over blocks is the BCS ground state. Pauli exclusion prevents interpreting a block as an unrestricted harmonic-oscillator pair mode.

For a fixed-density balanced two-spin Fermi gas with masses $(m,m,\gamma m)$, the Fermi energy scales as ${\gamma }^{-1/3}$ while the single-spin Fermi-level density of states scales as ${\gamma }^{1/3}$. A constant attractive interaction $g$ within a narrow symmetric energy window ${\epsilon }_c\ll {\epsilon }_F$ therefore gives $1={\lambda }_0{\gamma }^{1/3}\mathrm{arsinh}({\epsilon }_c/\Delta )$. Solving gives $\Delta ={\epsilon }_c/\sinh [1/({\lambda }_0{\gamma }^{1/3})]$, whose weak-coupling limit is displayed. Extrapolating to arbitrarily large anisotropy violates the assumptions that the Fermi energy dominates the window and that the pairing coupling is small.

In a conventional isotropic spin-singlet superconductor, the nuclear spin-lattice relaxation rate can rise just below the transition before falling at lower temperature. The BCS coherence factors and enhanced near-gap density of states produce the coherence enhancement. Nuclear relaxation and ultrasonic attenuation have different coherence factors, so their contrasting temperature dependences test pairing structure rather than only the existence of a gap.^[ref]

For a uniform isotropic weak-gap BCS state, the zero-temperature electron occupation is $n(\xi )=(1-\xi /\sqrt{{\xi }^2+{\Delta }^2})/2$. The number above the normal-state Fermi surface is approximately $D(\mu ){\int }_0^{\infty }n(\xi )d\xi$. The integral equals $\Delta /2$, while a three-dimensional parabolic band including both spins has $D(\mu )=3N/(2\mu )$. This gives the displayed leading fraction.

The result measures redistribution relative to a normal Fermi sea, not bosonic single-mode condensate depletion or quasiparticle occupation. A different band dimension or anisotropic gap requires its actual density-of-states or momentum integral. Material estimates need specified gap and chemical-potential scales.

A posterior moment is a moment computed under a posterior distribution. Its first raw moment is the posterior mean; its second central moment is the posterior variance.

For independent and identically distributed observations from a normal distribution with known variance ${\sigma }^2$, an exponential distribution prior of rate $\lambda$ on their nonnegative mean gives this truncated normal distribution posterior. Complete the square in $-n(\mu -\bar{x}{)}^2/(2{\sigma }^2)-\lambda \mu$; the truncation remains essential even when the untruncated normal mean is negative.

For a normalized uniform prior on $[a,b]$, Bayesian model evidence is $Z=(b-a{)}^{-1}{\int }_a^b\mathcal{L}(\theta )d\theta$. It averages the likelihood over the prior range. Extending a well-placed interval into low-likelihood regions reduces that average, but moving or widening a poorly placed interval can improve it. A model prior is separate: posterior model odds equal the evidence ratio multiplied by prior model odds.

Suppose exactly one of $N$ equally likely candidates is the source of an observation. A test is always positive for that source and is positive with probability $p$ for any nonsource candidate. For one randomly selected candidate, Bayes' theorem gives $P(G\mid +)=(1/N)/[(1/N)+p(1-1/N)]$, proving the displayed result. Even a small false positive rate can leave a substantial chance of a different source when there are many alternatives. The model concerns one selected candidate with the stated prior; searching many candidates and selecting a match requires accounting for that selection procedure.

The difference between baryon number and lepton number. Standard electroweak sphalerons violate $B+L$ while conserving $B-L$. Consequently an early baryon asymmetry with zero $B-L$ need not survive equilibration; a surviving nonzero $B-L$ can be redistributed between baryons and leptons. Additional lepton-number-violating interactions can change this conclusion.

The epoch at which momentum exchange with photons ceases to drag baryons efficiently. The sound horizon at this epoch calibrates the late-time baryon acoustic oscillation ruler. It is related to, but not identical with, photon last scattering.

In the uniform-current beta-plane model, $U=\beta /(k_2^2+l^2)$ with $k_2=2/(a\cos \varphi )$. This is a model-dependent balance between eastward advection and westward intrinsic propagation, not a unique speed set by zonal wavenumber alone.

For a stationary $n=2$, $l=0$ wave in a uniform zonal current, a perturbation $n>2$ has phase speed $U(1-4/n^2)$ and zonal group speed $U(1+4/n^2)$. Phase speed relative to the air is westward; packet propagation relative to the ground is faster downstream.

A four-terminal bend resistance uses adjacent current contacts and the other adjacent pair as voltage probes. It can be negative when a ballistic injector preferentially populates the opposite voltage probe. Its sign is a nonlocal routing property, not the sign of the total electrical dissipation.