Sea ice anchored to a coast, grounded feature or other fixed support is fast ice. It can be smooth or deformed, young or old. Currents can transport an oil spill under sea ice beneath it even though the ice itself remains stationary.

A melt pond is liquid water collected in a depression on melting ice. Its lower albedo than snow-covered ice can strengthen ice-albedo feedback. Pond drainage and connected pores can redistribute an oil spill under sea ice during warming.

The Odden ice tongue is an intermittent extension of young sea ice into the Greenland Sea. Repeated formation and export of frazil ice and pancake ice can produce brine rejection without a steadily thickening local ice cover, increasing the salinity of the water left behind.

The sea-ice crystal fabric describes grain shapes and crystallographic orientations. Calm-water congelation ice tends to develop vertical columns with predominantly horizontal crystal c-axes, whereas accumulated frazil ice produces a granular texture with much less systematic orientation.

// Target: geophysics.bigb

Nilas is a thin, flexible, relatively continuous initial sea-ice sheet. Further freezing at its underside in quiet water produces congelation ice.

// Target: geophysics.bigb

Snow ice forms when flooded snow or a seawater-snow mixture freezes. Snow loading of a thin floe can depress the ice surface below water level and initiate flooding; wave overwash is another route.

// Target: geophysics.bigb

A sea-ice thickness distribution assigns fractional horizontal area to ice thickness intervals: $g(h)dh$ is the fraction between $h$ and $h+dh$. Include open water as a mass at $h=0$ when normalizing over the entire ocean area. The first moment is ice volume per ocean area. With growth rate $f$ and velocity $v$, its conservative evolution is ${\partial }_{t}g+\nabla \cdot (vg)+{\partial }_h(fg)=\psi$, with mechanical redistribution of sea ice represented by $\psi$.

// Target: geophysics.bigb

A sea-ice lead is a fracture or elongated opening in a sea-ice cover, containing open water or young thin ice. Divergence and shear create leads, while freezing and convergence close them.

// Target: geophysics.bigb

The horizontal sea-ice drift momentum balance per unit area includes air and water drag forces, the Coriolis force, depth-integrated internal ice stress and the sea-surface pressure gradient. Here $m$ is ice-plus-snow mass per area and $\Sigma$ is a depth-integrated stress tensor.

// Target: geophysics.bigb

The tortoise coordinate is $r_{\ast }=r+(r_h/2)\log |(r-r_h)/(r+r_h)|$. Set $u=t-r_{\ast }$ and $v=t+r_{\ast }$, then $U=-e^{-u/r_h}$ and $V=e^{v/r_h}$ in the right exterior. The extended radial metric is $-r_h^2(r+r_h{)}^2{e}^{-2r/r_h}dUdV/r^2$, regular at $r=r_h$. The horizons are $UV=0$, while $r=0$ gives $UV=1$ and is a spacelike curvature singularity. Compactification by $\arctan U,\arctan V$ displays two exteriors, a future black hole and a past white hole.

A freely falling source emits optical pulses at proper times $\delta +nP$, for integers $n\ge 0$. If its horizon-crossing proper time is finite, only pulses beginning before $s_H$ can escape. Their number is the displayed ceiling, for $0\le \delta <P<s_H$. Infinite Schwarzschild time at crossing does not give infinitely many emitted pulses. The pulse phase $\delta$ fixes both the count and the redshift of the last pulse. If a finite pulse straddles the horizon, its exterior portion has an arbitrarily late, increasingly redshifted received tail.

Let $L>0$ and consider holomorphic functions $f$ from the unit disc into $e^{-L}<|w|<e^L$, normalized by $f(0)=1$. Their largest possible derivative modulus is $4L/\pi$. Lift $f$ to its holomorphic logarithm $h$ with $h(0)=0$, so $|\mathrm{Re}h|<L$. The function $\tan (\pi h/(4L))$ maps the disc into itself and fixes zero. The Schwarz lemma gives $|h^{\prime }(0)|\le 4L/\pi$. Equality holds precisely forHere $\arctan$ is its holomorphic branch vanishing at zero; $(4L/\pi )\arctan z$ is a conformal map from the disc onto the vertical strip. Thus the maximal derivative modulus is attained, with a circle of distinct extremizers rather than a unique map.

A holomorphic function between hyperbolic Riemann surfaces lifts to a disk map. If the lift is not an automorphism, strict Schwarz-Pick lemma derivative inequality makes every nonconstant geodesic shorter. Equality of distance for one distinct pair therefore forces the map to be a covering map and a local isometry. Covering maps attain equality for sufficiently close pairs in a single lift of a small convex metric ball; they need not preserve all global distances. A punctured disk covers a thrice-punctured sphere by including a cyclic parabolic subgroup into $\Gamma (2)$, so equality is possible in that example.

// Destination: analysis.bigb

For holomorphic functions $f:\mathbb{D}\to \mathbb{D}$ with $f(z_0)=w_0$, put $r=|(z-z_0)/(1-{\overline{z}}_{0}z)|$. Their possible values at $z$ form the closed disc with centre $(1-r^2)w_0/(1-r^2|w_0{|}^2)$ and radius $r(1-|w_0{|}^2)/(1-r^2|w_0{|}^2)$. Indeed ${\psi }_{w_0}\circ f={\psi }_{z_0}h$ with holomorphic $|h|\le 1$; all constant choices of $h$ in the closed unit disc are admissible. Completing the square in the displayed pseudohyperbolic inequality gives the centre and radius.

If a holomorphic function $f$ on the unit disc satisfies $|f|\le M$ and $f(a)=0$, the Schwarz-Pick lemma applied to $f/M$ gives $|f(0)|/M\le |a|$. Equality for a nonzero central value forces a scaled automorphism of the unit disk. If $f$ extends past the unit circle and $|f(0)|$ is smaller than its minimum boundary modulus, the maximum modulus principle applied to $1/f$ proves that a zero occurs inside the circle.

For $0<k<1$, principal upper-half-plane branches give a conformal map onto the upper half-plane cut along $[0,\tau ]$, where $\tau =2k^k(1-k{)}^{1-k}{e}^{i\pi k}$. Its derivative is $(z-1+2k)(z-1{)}^{k-1}(z+1{)}^{-k}$. The two prevertices $-1,1$ map to the slit base, and $1-2k$ maps to its tip. Boundary traversal goes out along the slit and back, while $g(z)=z+1-2k+O(1/z)$ at infinity. The argument principle gives one preimage of every point off the slit.

A boundary point of the source half-plane that maps to a polygon vertex. If that corner has angle $\pi \alpha$, the Schwarz-Christoffel mapping has local behavior $F(z)-F(a)\sim C(z-a{)}^{\alpha }$ at its prevertex, so the derivative has exponent $\alpha -1$. Infinity may also be a prevertex, expressed in the local coordinate $1/z$.

Sum the weight monomials of all Semistandard Young tableaux on the skew Young diagram $\lambda /\mu$, with weak rows and strict columns. A Bender-Knuth involution interchanges the counts of adjacent letters without changing the shape, so the sum is a symmetric function. Taking $\mu$ empty gives the Schur function.

Let $a_{\gamma }=\det [x_j^{{\gamma }_i}]$ and $\delta =(N-1,\dots ,0)$. Multiply the Jacobi–Trudi identity matrix by $B_{jk}=(-1{)}^{N-j}{e}_{N-j}(X\setminus x_k)$. The generating-function identity $H(t){\prod }_{i\ne k}(1-x_{i}t)=(1-x_{k}t{)}^{-1}$ makes the product matrix $[x_k^{{\lambda }_i+N-i}]$. Setting $\lambda =0$ shows $\det B=a_{\delta }$, proving the formula as a polynomial identity.

A pure-state source emits independent states from a fixed quantum state ensemble. Its average block density operator is the tensor power of the one-use average. Unassisted reliable quantum source compression preserves the emitted quantum states, rather than only their classical labels. Schumacher compression gives the optimum asymptotic qubit rate $S(\rho )$ for such a source, with every larger rate achievable. Nonorthogonal equally likely pure states need not have entropy one: their classical labels have entropy one but their average density operator can have lower Von Neumann entropy.