In the ideal cooled-shell limit with negligible continuing interior pressure, swept-up mass in a uniform medium scales as $R^3$. Constant radial shell momentum then gives $R^3\dot{R}=\mathrm{const}$ and $R\propto t^{1/4}$. This differs from an energy-conserving blast wave or a pressure-driven snowplough.

For a small fixed-current loop with moment $m=I\pi R^2$ and $R\ll h$, the image field gives upward force $3{\mu }_0{m}^2/(32\pi h^4)$. Differentiating the external field holds the image position fixed; moving both source and image in that derivative would double the force incorrectly.

An experiment converting the relative phase of a superconducting pairing amplitude into a measurable interference or flux signal. Orthogonal-face Josephson junctions and tricrystal rings are examples. Junction asymmetry, trapped flux, loop inductance and comparison geometries are important controls.

A convergent margin where one plate descends beneath another. In a simple junction-kinematic model the trench normal motion is tied to the overriding plate; rollback or distributed deformation can modify that rule.

The upper plate above a subducting plate. Its motion determines the normal velocity of a non-rolling-back trench in the idealized triple-junction compatibility construction.

A traceless Hermitian $3\times 3$ adjoint scalar field transforms by $\Phi \mapsto U\Phi U^{†}$. A diagonal vacuum with eigenvalues $({\lambda }_1,{\lambda }_2,{\lambda }_3)$, summing to zero, leaves its centralizer unbroken. The adjoint scalar gauge-boson mass matrix assigns mass $g|{\lambda }_i-{\lambda }_j|$ to the vector fields linking the $i$th and $j$th eigenspaces. Coincident eigenvalues preserve non-Abelian rotations of their eigenspace; distinct eigenvalues preserve only a maximal torus.

Strong and electromagnetic interactions conserve net strangeness; a decay changing it requires the weak interaction. A strange baryon decay into a less-strange baryon and a pion is an example. The weak interaction does not impose a parity selection rule, so a final orbital partial wave cannot by itself determine the parent's parity. Qualitative lifetime classes do not provide an exact lifetime without an amplitude calculation or measurement.

For a uniform exchange field with cost $N{M}^2/(2U)$ and spin-band energies ${ϵ}_k-\sigma M$, the grand potential is even in M. Its curvature at zero field is $N[U^{-1}-{\chi }_0]$, with ${\chi }_0=\int \nu (-n_F^{\prime })dϵ$ and spin-summed density of states per site. Negative curvature proves local paramagnetic instability. It does not by itself exclude a first-order transition before that instability, nor determine the stable finite-field amplitude.

The exponential inertial relaxation time ${\tau }_p=m/\zeta$ for linear drag. A no-slip sphere has ${\tau }_p={\rho }_p{d}^2/(18\eta )$; particle density enters this time, whereas fluid density enters the fluid Reynolds number.

Drag on a small particle when the gas mean free path is not negligible relative to its size. Continuum no-slip Stokes friction then needs a slip or kinetic correction depending on pressure and gas-surface interactions.

For integer $N\ge 2$, write $\ln (N!)={\sum }_{k=2}^N\ln k$. Monotonicity gives ${\int }_{k-1}^k\ln xdx\le \ln k\le {\int }_k^{k+1}\ln xdx$. Summing and evaluating the endpoint integrals proves the displayed bounds. They also hold with equality when $N=1$. The lower integral starts at one and the upper at two; confusing those endpoints loses the constant in the upper bound. The bounds imply $\ln (N!)=N\ln N-N+O(\ln N)$, while Stirling's formula determines the logarithmic and constant corrections.

The reference concentration $c^{\circ }$ used to make concentration activities dimensionless. Changing this reference changes the numerical dimensionless equilibrium constant when the reaction changes the total number of solute or gas molecules.

The reference pressure used to define a standard gas activity; a common convention is one bar. Normalizing pressure by this reference keeps a thermodynamic equilibrium constant dimensionless.

A spring $E_0$ in series with a parallel spring-dashpot branch $E_1,\eta$ has creep compliance $J(t)=1/E_0+(1/E_1)(1-e^{-E_{1}t/\eta })$. It includes immediate and delayed recoverable strain, but no unbounded viscous flow. Large strain, damage or anisotropy require additional constitutive structure.

A SQUID with two Josephson junctions in a superconducting loop. For equal junctions, negligible self-inductance and no intrinsic phase offset, adding their sinusoidal currents gives the displayed flux-dependent critical current. An rf SQUID instead has one junction and uses loop inductance.

A superconducting loop containing one Josephson junction and coupled inductively to a radio-frequency resonator. Its flux-dependent nonlinear inductive response changes the resonator signal. A DC SQUID instead uses two junctions and can turn their critical-current interference into a voltage modulation under bias. Inductance, junction dynamics, noise and feedback determine the practical transfer function.

A single optical mode in a squeezed vacuum state. In principal optical quadratures, its variances are $e^{-2r}/4$ and $e^{2r}/4$, while its mean photon number is ${\sinh }^{2}r$. Its mean field vanishes. A degenerate optical parametric amplifier generates the state by pair creation with phase fixed by the pump. Loss admixes unsqueezed vacuum and limits the measured variance reduction.

photon addition to an optical squeezed vacuum gives $a^{†}S|0⟩=\cosh rS|1⟩$; subtraction gives $aS|0⟩=-\sinh rS|1⟩$. After successful-event normalization these are the same state up to phase, with mean occupation $1+3{\sinh }^{2}r$ and odd photon parity. At zero squeezing subtraction never succeeds; the normalized limiting state is not an output at that exact parameter value.

Two square roots of the SWAP gate, separated and followed by local $z$ rotations, implement a controlled phase gate. Local $y$ rotations of its target convert it into a CNOT gate. The construction assumes isotropic exchange and independent one-qubit control; pulse errors and extra Hamiltonian terms change the implemented operation.

For point masses joined by unstretched central springs obeying Hooke's law, let $n_{ij}$ be the equilibrium unit vector along an edge and $u_i$ the small displacement of its endpoint. The leading extension is $n_{ij}\cdot (u_j-u_i)$, so its quadratic elastic potential energy is $k_{ij}[n_{ij}\cdot (u_j-u_i){]}^2/2$. Consequently the block stiffness matrix has off-diagonal blocks $K_{ij}=-k_{ij}{n}_{ij}{n}_{ij}^T$ and diagonal blocks $K_{ii}={\sum }_j{k}_{ij}{n}_{ij}{n}_{ij}^T$, with absent edges contributing zero. It is symmetric and positive semidefinite. The normal modes solve $Kq={\omega }^{2}Mq$, where $M$ is the mass matrix. Rigid translations and infinitesimal rotations are rigid-motion zero modes of a spring framework. A prestressed spring can add transverse quadratic terms, so the unstretched assumption matters.