A superconductor with an s-wave orbital pairing component: isotropic zero orbital angular momentum in the rotationally invariant model, or the corresponding fully symmetric orbital representation in a crystal. Crystal anisotropy can still make the gap magnitude angle dependent. In the usual centrosymmetric spin-singlet model its orbital parity is even.

Pairing can establish a local binding or gap scale before a system develops phase-coherent transport. An actual superconducting transition temperature depends on both the pair scale and the phase stiffness. Low carrier density or weak interlayer coupling can make phase ordering the lower scale even when the pairing amplitude is appreciable. A minimum or smooth interpolation of two estimated scales is a qualitative guide, not a substitute for a self-consistent theory of the crossover.

A superconducting alloy used in magnet conductors. Its useful critical current depends on flux pinning and processing as well as the equilibrium upper field.

The superconducting intermetallic $\mathrm{N}{\mathrm{b}}_3\mathrm{Sn}$ used in high-field conductors. Maintaining current-carrying connectivity and controlling strain are distinct from achieving a large upper critical field.

A transition of part of a current-carrying superconducting system into a resistive state. Joule heating can spread the transition, and the stored magnetic energy must be handled safely by the system's protection design. Zero steady winding resistance does not remove the finite magnetic energy or the refrigeration requirement.

A limiting transport current per conductor cross-sectional area for specified temperature, magnetic field and dissipation criterion. Local current distribution, flux motion and weak links can make it distinct from an intrinsic depairing limit. Its variation with the operating conditions is summarized by a superconducting critical surface.

A supercapacitor whose principal storage mechanism is non-Faradaic charge separation at electrode–electrolyte interfaces. The two electrode interfaces contribute series capacitances.

Capacitor-like differential charge storage produced by sufficiently fast and reversible Faradaic processes over a useful potential interval. A redox-active nanomaterial is not necessarily pseudocapacitive.

Charge-storage response of separated ionic and electronic charge at an electrode-electrolyte interface. The differential capacitance is $C=dQ/dV$; it can vary with electrode potential, ion packing and surface structure.

A high-temperature alloy based on a nickel-rich disordered $\gamma$ matrix, commonly strengthened by ordered ${\gamma }^{\prime }$ precipitates and substitutional solutes.

A sum whose resulting value is a prime number. This is a condition on the value, rather than a requirement that the summation index run over prime numbers. Counting subsets with a prime-valued sum can use subset sum counts followed by a primality filter on the possible totals.

The Euler-Maclaurin formula relates a finite sum to an integral and endpoint derivatives. If $a<b$ are integers, $p\ge 1$ and $f$ has continuous derivatives through order $2p$ on $[a,b]$, thenHere $B_{2r}$ are Bernoulli numbers. With ${\tilde{B}}_{2p}(x)=B_{2p}(\{x\})$ the periodic Bernoulli function, the exact remainder isThis form rearranges the last endpoint correction in the NIST statement.^[ref]

For $f(x)=x^2$ on $[0,n]$ and $p=1$, the remainder vanishes because the second Bernoulli polynomial has integral zero over each unit interval. The result is ${\sum }_{k=1}^n{k}^2=n^3/3+n^2/2+n/6$, the square pyramidal number formula. For other functions, retaining or bounding $R_p$ makes the finite truncation a controlled approximation.

For an integer $p\ge 0$, Faulhaber's formula expresses a sum of powers as a polynomial in the upper limit:where $B_j(x)$ are the Bernoulli polynomials. The identity $B_{p+1}(x+1)-B_{p+1}(x)=(p+1)x^p$ makes the sum telescope. This is a discrete analogue of finding an antiderivative.^[ref]

For $p=1$ the result is $n(n+1)/2$; for $p=3$ it is $n^2(n+1{)}^2/4$. Although the displayed polynomial has rational coefficients, it takes integer values at nonnegative integers.

A cyclic sulfonate ester. Alkoxide attack at a suitable primary carbon bound to the ring oxygen can open the ring by carbon–oxygen cleavage, leaving a sulfonate group.

A BLOSUM matrix assigns amino-acid substitution scores estimated from conserved aligned blocks of proteins. Related sequences are clustered to reduce overrepresentation, and observed replacement frequencies are compared with background frequencies. Positive scores favour a pairing relative to that background; negative scores disfavor it. The matrix number identifies the clustering identity threshold, not the percentage identity of every sequence pair to which the matrix can be applied. ^[ref]

A convention for the small fractional reduction of azimuthal gas speed due to radial pressure support. With $c_s^2=P_g/{\rho }_g$, radial balance gives $v_{g\varphi }^2=v_K^2(1-2\eta )$ and hence $v_{g\varphi }\simeq v_K(1-\eta )$. Other authors absorb the factor of two into the definition; the convention must accompany a drift formula.

For test particles with linear drag, negligible radial gas flow and negligible dust feedback, Sub-Keplerian rotation removes solid-particle angular momentum. The displayed inward velocity uses Stokes number $\mathrm{St}={\Omega }_K{t}_s$ and the pressure-gradient parameter of a protoplanetary disk. Drift is fastest at $\mathrm{St}=1$; the physical particle size at that point depends on the local gas and drag law.

For $\omega >0$ on $x>0$, use the regular finite branch at zero, decay at infinity, and a finite quadratic-form energy. For trial functions with $[x{y}^2{]}_0^{\infty }=0$, integration by parts givesThe Rayleigh quotient is at least $\omega$, with equality exactly for a nonzero multiple of $e^{-\omega x}$. Direct substitution verifies that this is an eigenfunction with eigenvalue $\omega$, so the lower bound is attained. The zero coefficient $p(0)=0$ makes this a singular-endpoint Sturm-Liouville problem; regularity replaces a prescribed zero endpoint value. Continuous piecewise differentiable trials of finite form energy are allowed in the Rayleigh-Ritz method, even if their derivative has a jump away from the endpoint.