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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.

The simplest Pease polynya model balances offshore ice export at speed $U$ against production of new ice at thickness rate $F$ within an open strip of width $L$. Newly formed ice is collected into an edge of prescribed thickness $H$, with negligible residence time. Per unit coastline length, conservation gives $H(U-\dot{L})=FL$. For constant positive parameters,Thus the equilibrium width is $HU/F$ and the relaxation time is $H/F$. If warming reduces $F$ while $U,H$ stay fixed, both increase. With $F\le 0$ the positive-freezing equilibrium ceases to apply. Finite frazil ice residence, variable atmospheric forcing and alongshore transport require extensions of the model.

This reduced polynya model describes a wind-driven, nearly uniform surface layer containing dilute frazil ice. Let $x$ be distance from the opening, $b(x)$ the layer depth, $u(x)>0$ its horizontal speed, and $\varphi (x)\ll 1$ its ice volume fraction. Use reference liquid mass density $\rho$ except in buoyancy, where $r=\rho -{\rho }_s>0$ is the liquid-minus-ice mass density difference. Let $S>0$ be the uniform wind shear stress and $f=Q_a/\rho >0$ the atmospheric cooling heat flux divided by $\rho$.

For a linear liquidus $T=-mC$, far-field salinity $C_{\infty }$, specific heat capacity $c_p$ and specific latent heat $L$, define $H=L+c_{p}m{C}_{\infty }$, $K=f/H$ and $J=3S/(4\rho )$. The extra sensible term in $H$ accounts for the cooling associated with salt enrichment. The chosen depth-averaged momentum, ice-production and buoyancy balances areTheir numerical coefficients belong to this particular layer closure. The first two give $b=Jx/u^2$ and $\varphi =(K/J)u$. Seeking a similarity solution with $u\propto x^{1/3}$ gives $b^{\prime }=b/(3x)$; substitution into the third balance yieldsThus layer depth, speed and retained ice fraction all grow as $x^{1/3}$. The increasing $\varphi$ eventually violates the dilute approximation. If crystals are exported out of the layer, their enthalpy and ice-volume flux must be included through the frazil export enthalpy balance; retained fraction is not automatically total production.

There is a sequence of complex polynomials converging pointwise on the whole plane to the sign of the imaginary part, with value zero on the real axis. For each $n$, take the compact union of the upper rectangle $[-n,n]+i[1/n,n]$, its lower reflection and the real segment $[-n,n]$. Its complement is connected. The function equal to one, minus one and zero on disjoint neighborhoods of the three components is holomorphic. The polynomial Runge theorem provides a polynomial within $1/n$ of it on the union. Every fixed point lies in its corresponding component for all large $n$, proving pointwise convergence. The limit is discontinuous, so the convergence cannot be locally uniform near the real axis. This illustrates how pointwise approximation by holomorphic functions differs from locally uniform approximation.

For the unit disk with the nonpositive real radius removed, compact sets $K_m=\{r{e}^{i\theta }:1/m\le r\le 1-1/m,|\theta |\le \pi -1/m\}$, $m\ge 3$, exhaust the domain. Each has connected complement: its inner and outer complementary regions join through the missing angular sector. The polynomial Runge theorem gives a polynomial within $1/m$ of a given holomorphic function on $K_m$. Every compact subset lies in some $K_m$, so these polynomials converge uniformly on compact subsets.

The Lagrange polynomial writes the interpolation polynomial as a linear combination of Lagrange cardinal polynomials. Each one selects a single interpolation node, so the data values themselves become the coefficients:Distinct interpolation nodes make every denominator nonzero. At $x_j$, all terms vanish except $y_j{\ell }_j(x_j)=y_j$. This gives an explicit construction without solving for monomial coefficients. The elegance of the Lagrange polynomial is that it builds all constraints into the basis, leaving no equations to solve for the data values.

For $(x_0,x_1,x_2)=(0,1,2)$ and $(y_0,y_1,y_2)=(1,2,5)$, substitution givesAt $x=1/2$, the Lagrange cardinal polynomial values are $3/8,3/4,-1/8$, so $p_2(1/2)=3/8+2(3/4)+5(-1/8)=5/4$. A negative Lagrange cardinal polynomial value is allowed: these weights sum to one but need not be probabilities.

The product form is especially useful for deriving differentiation weights from nodal evaluations and quadrature rules. For repeated numerical evaluation, the barycentric interpolation formula reorganizes the same Lagrange polynomial into precomputed weights and sums; it avoids evaluating every full product at every argument.

The Runge phenomenon is the failure of increasing-degree polynomial interpolation at equally spaced interpolation nodes to converge to some smooth target functions, with large oscillations near the ends of the interval. It is a property of the approximation problem, and can occur even with exact arithmetic. A numerically stable evaluation method cannot remove it. Endpoint clustering, as in Chebyshev-Lobatto nodes, changes the approximation problem and can greatly improve it.^[ref]

Take $f(x)=1/(1+25x^2)$ on $[-1,1]$ and polynomial degree at most $10$. Use either $x_k=-1+k/5$ or $x_k=\cos (k\pi /10)$ for $k=0,\dots ,10$. Each interpolation polynomial passes through all eleven of its own samples, yet the first has severe endpoint oscillations and the second stays much closer to the target function. Evaluating both by the barycentric interpolation formula on $20,001$ equally spaced points, including the endpoints, gives sampled maximum absolute errors of approximately $1.915659$ and $0.132197$, respectively: about a $14.49$-fold reduction. These are maxima over the stated grid, not certified maxima over the continuous interval. The generator separately checks exact reproduction of all stored nodal values.

The polynomial interpolation error formula explains why interpolation nodes enter the analysis. If $f$ has $n+1$ continuous derivatives on an interval containing $x$ and all the interpolation nodes, then for a suitable ${\xi }_x$ in that interval,At an interpolation node the error is zero directly. Away from the interpolation nodes, both the product and the high-order derivative matter. A short proof applies Rolle's theorem repeatedly to $f(t)-p_n(t)-[f(x)-p_n(x)]{\prod }_i(t-x_i)/{\prod }_i(x-x_i)$, which vanishes at $n+2$ distinct points.^[ref]

The successful comparison here does not assert that Chebyshev-Lobatto nodes solve every approximation problem for every function. The derivatives, the polynomial degree, and measurement errors still matter. The Lebesgue constant of interpolation measures worst-case amplification of nodal errors, while the polynomial reproduction error bound relates interpolation to the best uniform approximation available at that polynomial degree.

For a real continuous function on a compact interval, choose its best uniform approximation $p_n^{\ast }$ of polynomial degree at most $n$. If its error is nonzero, the Chebyshev alternation theorem supplies $n+2$ ordered alternating extrema. The intermediate value theorem gives a zero of $f-p_n^{\ast }$ in each intervening interval. At these $n+1$ distinct points, the polynomial interpolant is exactly $p_n^{\ast }$. If the best error is zero, arbitrary distinct nodes suffice. The Weierstrass approximation theorem shows that these function-dependent interpolants converge uniformly. The nodes depend on the target function; the conclusion does not provide one universal node sequence for every continuous target.

An interpolation polynomial for distinct nodes $x_0,\dots ,x_n$ and prescribed values $y_0,\dots ,y_n$ is a polynomial $p$ satisfying $p(x_i)=y_i$. There is a unique such polynomial of degree at most $n$: the Lagrange interpolation polynomial constructs it explicitly, and uniqueness follows because a nonzero degree-at-most-$n$ difference cannot have $n+1$ distinct roots. Newton interpolation polynomials express the same object using divided differences.

A poly-infinite cyclic group has a finite subnormal series all of whose nontrivial factors are infinite cyclic groups. Intersecting its series with a subgroup again gives trivial or infinite cyclic factors; repetitions may be deleted. Splicing a kernel series with the preimage of a quotient series proves closure under extensions. Such groups are finitely presented groups: induct on the series, split the last extension by an infinite cyclic quotient, and use finite presentation of a semidirect product.