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A subgroup is quasiconvex when it is a quasiconvex subset of a Cayley graph. In a hyperbolic group, a finitely generated subgroup is quasiconvex exactly when it is quasi-isometrically embedded.

A sequential two-binding model can produce the reduced reaction rateHere $u=s/s_{\ast }$ is the free substrate concentration relative to a chosen scale $s_{\ast }$, $p$ is product concentration, and $A$ has units of concentration per time. The dimensionless parameters must be obtained from a specified enzyme kinetics mechanism; the rational function alone does not identify a mechanism involving allostery.

For example, suppose a single substrate species binds twice in sequence, producing three enzyme states $E,ES,E{S}_2$, and only $E{S}_2$ makes product. If binding relaxes rapidly compared with catalysis, take the approximate binding balancesThe total enzyme concentration $e_T=[E]+[ES]+[E{S}_2]$ givesMultiplying by the turnover number $k_{\mathrm{cat}}$ yields the stated law with $A=k_{\mathrm{cat}}{e}_T\beta /\gamma$. These rapid-binding balances are an additional model assumption within a quasi-steady-state approximation.

At low substrate concentration, $dp/dt\sim (A/\alpha )u^2$ because the active state requires two binding events. At high substrate concentration, the rate tends to $A\gamma /\beta =k_{\mathrm{cat}}{e}_T$. Such limiting behavior helps compare candidate mechanisms with measured reaction rates; evidence for cooperative binding requires examining how the binding constants depend on occupancy.

A quasigeodesic is a quasi-isometric embedding of an interval into a metric space.

A bilipschitz equivalence distorts all distances by multiplicative constants bounded away from zero and infinity. Every bilipschitz equivalence is a quasi-isometry.

A finitely generated subgroup $H\le G$ is quasi-isometrically embedded when its inclusion, equipped with word metrics from finite generating sets, is a quasi-isometric embedding. This property is independent of those generating sets.

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.

The Lindblad equation describes the generator of a time-homogeneous, completely positive, trace-preserving quantum dynamical semigroup in a finite-dimensional system. In units $\hslash =1$, the Hamiltonian $H$ supplies coherent evolution and the Lindblad operators $L_{\alpha }$ specify dissipative contributions.

A Markovian continuous-time quantum channel obeys

The Lindblad dissipator combines its gain term and two anticommutator terms so that the trace remains one. For $L=\sqrt{\gamma /2}Z$ and $H=0$, diagonal entries in the $Z$ basis stay fixed while an off-diagonal element obeys ${\dot{\rho }}_{01}=-\gamma {\rho }_{01}$: dephasing without population decay. The semigroup assumption is a specific model of environmental memory loss. ^[ref]

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