On the square lattice, a self-avoiding walk using north, east and west steps but no south steps. At each horizontal height, self-avoidance forces one monotone horizontal run. Conversely, such runs separated by north steps always give self-avoiding walks, because every new run is at a previously unvisited height.

An ice floe is a connected floating piece of sea ice. Its diameter relative to wavelength and its thickness affect the reflection, transmission and dissipation of surface gravity waves.

The marginal ice zone is the transition between open ocean and compact sea ice, where interacting ice floes and surface gravity waves can dominate the mechanical evolution.

Frazil ice consists of small suspended ice crystals formed in supercooled turbulent water. A layer of crystals can damp surface gravity waves through viscous dissipation, even when individual crystals scatter very little wave energy.

Pancake ice consists of approximately disk-shaped young ice floes, often with raised edges formed by collisions. Wave-driven collisions and motion in the surrounding frazil ice can dissipate energy.

A polynya is a persistent or recurrent area of open water or relatively thin ice within surrounding sea ice. Wind can maintain it by transporting ice away; an oceanic heat flux can prevent freezing. Open-water width supplies an open-water fetch over which wind waves can grow.

Wave attenuation is a reduction of a chosen wave component as it travels through sea ice. An amplitude coefficient $\alpha$ corresponds to an energy coefficient $2\alpha$. Wave scattering at a planar interface can remove energy from the forward wave without dissipating total energy; viscous dissipation, collisions and internal ice losses irreversibly convert mechanical energy to heat.

A pressure ridge is a pile of fractured sea ice formed by horizontal convergence. Ice displaced upward forms a sail; ice displaced downward forms a keel. Newly formed ridges contain angular blocks and pores, and freezing, melting and further deformation alter their strength and shape.

A decline in Arctic sea ice can mean reduced extent, concentration, thickness, volume or survival of older ice. These measures differ: extent counts areas above a concentration threshold, whereas area weights actual coverage. Comparisons require consistent season, geographical domain and measurement method.

Sea-ice draft is the depth of the ice underside below the local water surface. Total thickness also includes the part above the water. A sea-ice pressure ridge has a spatially variable draft, and its peak draft differs from its mean along a track.

First-year sea ice is ice that has formed since the preceding summer melt season and has not yet survived a summer. It is usually thinner than older ice, although deformation can create thick sea-ice pressure ridges in a young cover.

Multi-year sea ice has survived at least one summer melt season. Its age, thickness, salinity and deformation history affect how it responds to further melting and mechanical forcing. Ice age does not uniquely determine ridge shape.

A proper scoring rule makes reporting the true probability distribution optimal in expected value, for every true probability distribution in the allowed class of forecasts. In the reward convention, the inequality in the title holds for every report $P$ and true law $Q$. The expected value is always taken under $Q$, because the report does not change how the outcome is generated. In the loss convention the same definition readsThis requirement makes a proper scoring rule useful for eliciting probabilities and comparing predictive distributions: deliberately distorting a known forecast cannot improve its expected reward or expected loss. The reward convention and its strict version are set out in Strictly Proper Scoring Rules, Prediction, and Estimation by Tilmann Gneiting and others (Section 1).

For the Bernoulli distribution with $q=0.7$, the Brier score has expected loss $(p-0.7{)}^2+0.21$, and the negative logarithmic scoring rule has expected loss $-0.7\ln p-0.3\ln (1-p)$. Both are minimized at the honest report $p=0.7$. The linear probability score, by contrast, maximizes its expected reward at $p=1$. These are statements about averaging over both possible outcomes with their actual probabilities.

The left panel displays each excess expected loss divided by that rule's expected loss at $p=q$: $0.21$ for the Brier score and approximately $0.610864$ for the logarithmic scoring rule. This normalization permits comparison of curve shapes and preserves the minimizing report, since it subtracts a constant and divides by a positive constant independent of $p$. Zero means no excess expected loss. The right panel shows the unmodified linear probability score reward, with its maximizing direction labeled explicitly.

The linear probability score pays the reported probability of whichever binary outcome occurs. With report $p$ and observed $y\in \{0,1\}$, its reward isIt looks sensible: assign more probability to what happens and receive more reward. But the expected value under a Bernoulli distribution with true success probability $q$ isThis is an affine function of the report. When $q>1/2$, the maximum over $[0,1]$ is at $p=1$; when $q<1/2$, it is at $p=0$. At $q=1/2$, every report ties. Thus the linear probability score encourages reporting certainty about the more likely outcome and is not a proper scoring rule.

For the stipulated $q=0.7$, the expected value of the reward is $0.3+0.4p$. Honest reporting $p=0.7$ earns $0.58$ in expectation, whereas $p=1$ earns $0.7$. The forecaster improves expected reward by replacing a 70% forecast with a 100% forecast. The Brier score and logarithmic scoring rule avoid this incentive, as the expected-score diagram under proper scoring rule shows.

For two equal-mass inward timelike geodesics of unit specific Killing energy, their invariant center-of-mass energy has this finite horizon limit. If both are captured from infinity, Schwarzschild marginally bound capture gives $E_{\mathrm{com}}<m\sqrt{20}$, with $m\sqrt{20}$ a supremum. No such universal bound follows for particles prepared inside the angular-momentum barrier.

These seminorms measure all polynomially weighted derivatives and generate the Fréchet topology of the Schwartz space. Finitely many such bounds control each weighted derivative's integrable norm by adding a spatial decay factor stronger than $⟨x{⟩}^{-n}$. Continuity of the Fourier transform follows by applying those bounds after differentiation and integration by parts. Equivalent families use powers of $⟨x⟩$ instead of individual monomials.

For a Hermitian matrix, let $x_i$ be its $i$th column without the diagonal entry and let $G^{(i)}=(X^{(i)}-zI{)}^{-1}$. Solve the two block equations of the inverse against the $i$th coordinate vector to obtain this formula. Both full and minor inverses must exist; nonreal $z$ guarantees that. In the real symmetric case $x_i^{\ast }$ may be replaced by $x_i^T$.

An accidental massless scalar has zero quadratic mass without being required by the stated broken continuous symmetry. A singlet with potential $\kappa z^4$ around $z=0$ is an example: its Hessian matrix vanishes although no continuous symmetry of that single real field protects a mass term. Such a zero mass can require tuning once quantum corrections are included.

The average of the two conformal Newtonian gauge potentials controls scalar gravitational lensing. A conformal rescaling makes the spatial optical metric flat and its time component $1+4\Psi$ to first order. Vanishing scalar anisotropic stress makes $\varphi =\psi$ in general relativity, but lensing depends on their sum even when they differ.

For equal independent binomial samples, the displayed normal-approximation size per group targets a two-sided significance level $\alpha$ and power $1-\beta$ at specified alternative rates, with $\bar{p}=(p_1+p_0)/2$. Its first variance term sets the null critical value and its second the alternative distribution. A numerical sample size is not determined by an effect ratio alone; power, sidedness, allocation and baseline rates must also be specified. Rare-event discreteness and clustered designs can materially change the achieved power.