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A quasi-isometric embedding satisfies the two-sided coarse distance inequality in the definition of quasi-isometry, without requiring its image to be coarsely dense.

For constant buoyancy frequency $N$ on a beta plane, three-dimensional quasi-geostrophic potential vorticity can be written

The quasi-geostrophic streamfunction defines horizontal velocity by $u=-{\psi }_y$ and $v={\psi }_x$. For a one-layer free-surface model, geostrophic balance gives $\psi =g\eta /f_0$.

For an attractive potential $-\alpha /|x{|}^n$ with $n>2$, dilations make the negative potential energy grow faster than kinetic energy, sending the Rayleigh quotient to minus infinity.

For a square potential barrier of width $2a$ and height $U_0=2E$, putThe oscillatory wavenumber outside and decay constant inside are both $k$. Matching the wavefunction and its derivative at both faces gives transmission amplitudeand transmission probability ${\mathrm{sech}}^2(2ka)$.

If $f$ marks only $x_0$, phase kickback realizes $I_{x_0}=I-2|x_0⟩⟨x_0|$ on the search register.

For $f(x)=a\cdot x\oplus b$, phase kickback followed by a Walsh-Hadamard transform maps the uniform phase state toThe hidden linear string $a$ is therefore recovered with certainty.

For the exact sample $c=sN/r$, reducingrecovers denominator $r/\gcd (s,r)$. Thus one sample reveals the full period exactly when $s$ is coprime to $r$; otherwise it reveals only a proper divisor and the algorithm must obtain more information.

If a measured ratio approximates a reduced $k/r$ within $1/(2r^2)$, continued-fraction convergents recover the candidate denominator $r$.

The Fourier amplitude of equally spaced basis states is a finite geometric sum with peaks where the Fourier phase increment is nearly one.

In a covariant gauge, the photon propagator is the inverse of the gauge-fixed Maxwell kinetic operator.

For an on-shell amplitude $\mathcal{M}={ϵ}_{\mu }(k){\mathcal{M}}^{\mu }$ with an external photon, gauge invariance implies $k_{\mu }{\mathcal{M}}^{\mu }=0$.

Photon vacuum polarization is the one-particle-irreducible photon two-point function generated by charged-particle loops.

A Markovian continuous-time quantum channel obeys

A Stinespring dilation represents a quantum channel by an isometry $V={\sum }_i{K}_i\otimes |i⟩$ followed by tracing out the environment.

A finite-dimensional quantum channel is primitive when some power maps every nonzero positive operator to a positive-definite operator. Equivalently, eigenvalue one is simple and no other eigenvalue lies on the unit circle.

A quantum channel is strictly contractive in trace distance when it reduces the distance between every pair of distinct density operators by a uniform factor smaller than one.

A quantum channel is unital when $T(I)=I$. Every random unitary channel is unital, while the converse fails in dimension at least three.

A random unitary channel has $T(\rho )={\sum }_j{p}_j{U}_j\rho U_j^{†}$. Its Choi matrix is a convex combination of maximally entangled pure states.

For normalized maximally entangled $|\varphi ⟩=d^{-1/2}{\sum }_j|jj⟩$, the Choi matrix of $T$ is $C_T=(T\otimes \mathrm{id})(|\varphi ⟩⟨\varphi |)$. Choi's theorem says $T$ is completely positive exactly when $C_T\ge 0$.