The shallow-water quasi-geostrophic potential vorticity relation is a modified Helmholtz equation for the quasi-geostrophic streamfunction. Its inverse with specified boundary conditions recovers horizontal geostrophic flow and surface elevation $\eta =f_0\psi /g$. The Rossby deformation radius $L_D=\sqrt{gH}/|f_0|$ is the range of the balanced response: for a horizontal Fourier component, $\hat{\psi }=-\hat{q}/(|{\mathbf{k}}_h{|}^2+L_D^{-2})$, after subtracting background potential vorticity.

In two-dimensional vortex dynamics, solve the Poisson equation for the streamfunction with the prescribed boundary conditions. On the plane its fundamental solution is $(2\pi {)}^{-1}\log |\mathbf{x}|$, so differentiating the displayed inverse gives the planar Biot-Savart kernel. The harmonic function ${\psi }_h$ represents boundary and global-flow data. An additive constant changes no velocity; circulation around holes and a uniform flow are additional degrees of freedom if their boundary or far-field values are not prescribed. The same inverse acts independently on each level in nonrotating layerwise-two-dimensional vortex dynamics.

Despite its notation, ${\xi }_V^2$ need not be nonnegative. It is the signed dimensionless derivative combination entering the leading running of the scalar spectral index.

At horizon exit during single-field slow-roll inflation, $d/d\log k\simeq -M_{\mathrm{Pl}}^2(V^{\prime }/V)d/d\Phi$. Applying this operator to the potential slow-roll parameter and the second potential slow-roll parameter yields the displayed leading-order flow equations. Corrections to the horizon-exit derivative contribute only at higher slow-roll order here.

Here $I_t={\sum }_{j=1}^t{1}_{\{{\xi }_j^{(1)}={\xi }_j^{(2)}\}}$ counts simultaneous collisions of two independent walks. A shared environment sign contributes $\mathbb{E}(1+\epsilon h{)}^2=1+{\epsilon }^2$; distinct signs contribute one. If the difference walk returns to zero with probability $\rho <1$, its total number of positive-time returns has law $\mathbb{P}(I_{\infty }=k)=(1-\rho ){\rho }^k$. Therefore the second moments are bounded by $(1-\rho )/(1-\rho (1+{\epsilon }^2))$ whenever $\rho (1+{\epsilon }^2)<1$. Convergence in L2 then preserves the limit's mean one.

Delete the first $m$ environment factors from the path-weight expectation and call the result $Z_T^{(m)}$. Pointwise, $(1-\epsilon {)}^m{Z}_T^{(m)}\le Z_T\le (1+\epsilon {)}^m{Z}_T^{(m)}$. Thus $\{{\mathrm{liminf}}_T{Z}_T=0\}$ equals $\{{\mathrm{liminf}}_T{Z}_T^{(m)}=0\}$ and belongs to every future-time environment sigma-algebra. The Kolmogorov zero-one law makes its probability zero or one. If the nonnegative martingale limit has mean one, the probability cannot be one, so its limit is strictly positive almost surely. The pointwise comparison establishes a genuine tail event, rather than merely invariance outside an unspecified exceptional set.

Two positive rank-one effects summing to $I$ in dimension two must be complementary orthogonal projections. Write the first as $\alpha |\psi ⟩⟨\psi |$ with normalized $\psi$. The second has eigenvalues $1-\alpha$ and $1$, so its rank-one condition forces $\alpha =1$. The two unit Bloch vectors are antipodal, and this POVM is a projective measurement.

A POVM is informationally complete if its outcome probabilities determine every input density operator uniquely. Its effects must span the real space of Hermitian operators. For a qubit, normalization leaves at most $m-1$ independent probabilities from $m$ outcomes; an arbitrary Bloch vector has three real parameters, so one fixed informationally complete measurement requires at least four outcomes. Using several differently oriented binary measurements is a different experimental setting and can also perform quantum state tomography.

Choose three pure qubit states whose unit Bloch vectors are coplanar, sum to zero and have pairwise angle $120$ degrees. The three effects $E_j=\frac{2}{3}|{\varphi }_j⟩⟨{\varphi }_j|$ sum to the identity, giving a POVM. If an input is the state orthogonal to ${\varphi }_1$ or to ${\varphi }_2$, outcome one excludes the first input and outcome two excludes the second. Each conclusive outcome has probability one half on its corresponding input; the third outcome is inconclusive with probability one half on either input. This gives unambiguous quantum state discrimination, rather than identification on every trial.

A POVM whose nonzero effects have rank one, $E_i=w_i|{\psi }_i⟩⟨{\psi }_i|$, with normalized pure states and $w_i>0$. Completeness is ${\sum }_i{w}_i|{\psi }_i⟩⟨{\psi }_i|=I$. This use of pure concerns effect rank, and does not mean an extremal point in the convex set of all POVMs.

A POVM $\{E_i\}$ in dimension $d$ corresponds to a quantum state ensemble with $p_i=\mathrm{Tr}(E_i)/d$ and ${\rho }_i=E_i/\mathrm{Tr}(E_i)$, omitting zero effects. Its average is the maximally mixed state $I/d$. Conversely any ensemble with that average defines a POVM by $E_i=d{p}_i{\rho }_i$. These transformations are inverse. A pure POVM corresponds exactly to a pure-state ensemble decomposition of $I/d$.

For a bounded real linear functional on the space of continuous functions on a compact space, define ${\varphi }^{+}(f)={\sup }_{0\le g\le f}\varphi (g)$ for $f\ge 0$. The decomposition $g=\min (g,f)+(g-f{)}^{+}$ for $0\le g\le f+h$ proves additivity on the positive cone; positive homogeneity is immediate. Extend by differences, and put ${\varphi }^{-}={\varphi }^{+}-\varphi$. Both are positive linear functionals. Their norms sum to $2{\varphi }^{+}(1)-\varphi (1)={\sup }_{\Vert g{\Vert }_{\infty }\le 1}\varphi (g)=\Vert \varphi \Vert$, using the substitution $g=2h-1$.

For a slender porous thermal plume, Darcy law gives $w=\beta (T-T_0)$, with $\beta =Kg\alpha /\nu$. If $C_h={\rho }_0{c}_p$ and $F_h=C_h\int w(T-T_0)dx$ is the conserved excess convective heat flux per span, then the kinematic momentum flux is $M=\int w^{2}dx=\beta F_h/C_h$, and the buoyancy flux is $B=\int g\alpha (T-T_0)wdx=g\alpha F_h/C_h$. The hyperbolic-secant porous plume profile has $Q=(36\kappa Mz{)}^{1/3}$. Multiplication of $Q$ and $M$ by ${\rho }_0$ gives physical mass flux and the density-weighted Darcy second moment. Intrinsic pore-velocity momentum flux instead has an additional inverse-porosity factor. Momentum conservation here follows from the heat/velocity proportionality, not from an inertia balance: porous drag balances buoyancy locally.

For the planar diffusion equation $n_t=D_0\nabla \cdot [(n/n_0{)}^2\nabla n]$, a mass-preserving radial similarity solution has $n=n_0{\lambda }^{-2}F(r/(r_0\lambda ))$. Substitution reduces the equation to $(\xi F^2{F}^{\prime }{)}^{\prime }=-({\xi }^{2}F{)}^{\prime }$ when ${\lambda }^5\dot{\lambda }=D_0/r_0^2$. Zero central flux and $F(0)=1$ give the compact profile shown above, with ${\lambda }^6=6D_{0}t/r_0^2$. Its height decays as $t^{-1/3}$ and its radius grows as $t^{1/6}$; its total mass is $2\pi n_0{r}_0^2/3$.

For $C_t=k\nabla \cdot (C\nabla C)$ in three dimensions and total mass $4\pi M$, the nonnegative radial source solution is $C=M^{2/5}(kt{)}^{-3/5}f(r/(Mkt{)}^{1/5})$, with $f(\xi )=(75^{2/5}-{\xi }^2{)}_{+}/10$. Integrating the similarity equation once gives $f{f}^{\prime }=-\xi f/5$, and normalization ${\int }_0^{\infty }{\xi }^{2}fd\xi =1$ fixes the compact-support radius. The flux vanishes at the moving front, so the profile solves the equation weakly and approaches the prescribed point mass as time tends to zero.

In a plane-parallel polytropic atmosphere of index $m$, let $z$ measure depth from a zero-pressure boundary and take constant gravitational acceleration $g$. The stellar hydrostatic equation and $p=K{\rho }^{1+1/m}$ integrate to $(m+1)p/\rho =gz$. Consequently the adiabatic sound speed satisfies $c_s^2=\gamma p/\rho =\gamma gz/(m+1)$, so $c_s\propto z^{1/2}$. This local relation does not justify extrapolating a surface model to the stellar core.

A latent-heat polynya remains relatively open because winds or currents remove newly formed sea ice. Sustained cooling can produce large amounts of ice even while local ice concentration remains low. Opening is maintained by export, not by latent heat warming the ocean enough to prevent freezing.

A sensible-heat polynya is maintained by ocean heat delivered through currents, mixing or upwelling, sufficient to inhibit local sea ice growth. Mechanical export and sensible heating may act together, so this classification identifies a dominant mechanism rather than an exclusive geographical class.

The Northeast Water Polynya is a recurrent open-water region off northeast Greenland. Ice barriers, local currents and winds organize ice divergence; summer opening and winter new-ice export need not have the same dominant forcing. Its regional northward coastal flow must not be confused with assuming that every current along east Greenland flows southward.

The North Water Polynya, also called Pikialasorsuaq, lies between northern Greenland and the Canadian Arctic in the Smith Sound and northern Baffin Bay region. An upstream ice arch can restrict replenishment of drifting Arctic ice while winds and currents export newly formed ice. Ocean heat contributes in some parts of the polynya.