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A superbubble is an expanding cavity of hot gas driven by the combined energy input of stellar winds and multiple supernova explosions. Constant mechanical power and negligible radiative losses lead to a radius proportional to $t^{3/5}$ in a uniform ambient medium.

For a smooth projective surface and an ample divisor $A$, the intersection pairing on the Picard group of a surface is negative definite on the orthogonal complement of $[A]$ in $N^1(X{)}_{\mathbb{R}}$. Its signature is $(1,\rho -1)$. This is an algebraic statement valid in arbitrary characteristic, distinct from its analytic formulation for compact Kähler surfaces.

For a two-dimensional internal gravity wave with phase $kx+\int m(z)dz-\omega t$, a constant-phase line at fixed time has slope $dz/dx=-k/m(z)$. Its normal spacing from the next crest is $2\pi /(k^2+m^2{)}^{1/2}$; the wavevector $(k,m)$ is normal to the line. The intrinsic energy ray is tangent to these phase lines, while mean-flow advection changes the laboratory ray direction.

For a propagating, non-cutoff two-dimensional internal gravity wave proportional to $e^{i(kx+mz-\omega t)}$, incompressibility gives $U=-mW/k$. Thus the oscillatory velocity field is perpendicular to the wavevector and parallel to the group velocity, whereas the phase velocity is parallel to the wavevector.

An integral extension $A\subset B$ need not give an integral extension $A_{P\cap A}\subset B_P$. For example, take $\mathbb{Z}\subset \mathbb{Z}[i]$ and $P=(2+i)$, whose contraction is $(5)$. The element $1/(2-i)$ belongs to $\mathbb{Z}[i{]}_P$ but is not integral over ${\mathbb{Z}}_{(5)}$: multiplying a hypothetical monic equation by a power of $2-i$ and reducing modulo the other Gaussian prime ideal $(2-i)$ gives $1=0$ in ${\mathbb{F}}_5$. Localizing both rings by the same multiplicative subset does preserve integrality.

An integral representation expresses a function through an integral of specified data and an integral kernel. Green-function representations solve differential equations this way, while the Bromwich inversion formula reconstructs a function from its Laplace transform.

An integral manifold of a rank-$r$ smooth distribution $D$ is an immersed $r$-dimensional submanifold with $T_{p}N=D_p$ at every point. For an integrable distribution, maximal connected integral manifolds form its leaves.

For $n\ge 0$, define $B_0(X)=1$ and $B_n(X)=X(X-1)\cdots (X-n+1)/n!$. The binomial coefficient formula shows $B_n(m)\in \mathbb{Z}$ for every nonnegative integer $m$. Since these integers are dense in the p-adic integers, continuity of the polynomial implies $B_n({\mathbb{Z}}_p)\subseteq {\mathbb{Z}}_p$, so it is an integer-valued polynomial there and $\Vert B_n{\Vert }_{\infty }=1$. The Pascal's identity gives $\Delta B_{n+1}=B_n$ for the forward difference operator.

The Cepheid instability strip is the region of the Hertzsprung-Russell diagram where Cepheid variables are unstable to pulsation driven by the opacity mechanism.

For concentration $C$ per unit surface area, conservation on a moving material interface givesThe material derivative follows the surface particles. The two dilution terms are tangential expansion and curvature-induced area change from normal velocity $u_n$; the last term is surface diffusion.

Surface diffusion transports particles along an interface down a concentration gradient. On an isotropic smooth interface with constant diffusivity $D_s$, its flux is $-D_s{\nabla }_{s}C$ and its concentration source is $D_s{\Delta }_{s}C$, using the surface gradient and surface Laplacian.

An initial object $0$ is strict when every morphism into $0$ is invertible. Adjoining a new strict initial object to a category means adding one map from it to every object and no map to it from any old object. Its only endomorphism is the identity.

For an event time with an exponential distribution of mean $\mu$ and an independent censoring time, let $X$ be their minimum. Memorylessness of the exponential distribution makes $X+\mu$ the conditional mean of the latent event time when it is censored. Keeping $X$ when the event is observed gives an unbiased imputed event time by the tower property of conditional expectation.

For a nonzero finitely generated indecomposable module $P$ that is a projective module over a right Artinian ring $A$, its top $P/PJ(A)$ is a simple module. The Jacobson radical $J(A)$ is a nilpotent ideal, so the top is nonzero; the quotient is a semisimple module over the semisimple ring $A/J(A)$. If it decomposed, a nontrivial idempotent of its endomorphism ring would lift by projectivity to an endomorphism of $P$. The Hopkins-Levitzki theorem gives finite composition length, and the Fitting lemma makes the lift either invertible or nilpotent. Its induced map on the top would have the same property, impossible for that nontrivial idempotent.

For zero source volume flux, constant Batchelor entrainment, and a horizontal source, the near-source trajectory is cubic: $z\sim \alpha \sqrt{\pi }{f}_0{x}^3/(3m_0^{3/2})$. Far from the source the centreline approaches a vertical asymptote, with its horizontal distance from that asymptote proportional to $z^{-1/3}$.

Adiabatic shielding suppresses the heating of an orbit when an external tidal tensor varies slowly compared with its orbital frequency. In a nonresonant slow encounter, adiabatic invariance of an orbital action lets the orbit adjust approximately reversibly, so a frozen-position impulse approximation overestimates the energy transfer. An orbital resonance can invalidate the simple slow-variation argument.

For a vector field uniformly Lipschitz continuous in the state with constant $L$, the stage fixed-point map of an implicit Runge-Kutta method is a contraction in the maximum stage norm if $hL{\max }_i{\sum }_j|a_{ij}|<1$. The contraction mapping theorem then gives unique stages. This is a sufficient small-step condition; a dissipative vector field may permit solvability for larger steps. An algebraic stability or B-stability estimate compares existing stage solutions and should not be mistaken for an unqualified stage-existence theorem.

For a charge $q$ at $\mathbf{p}$, $R=|\mathbf{p}|>a$, outside a grounded sphere of radius $a$, the displayed image gives zero potential on the sphere and decay at infinity. On $|\mathbf{x}|=a$, one has $|\mathbf{x}-{\mathbf{p}}^{\prime }|=(a/R)|\mathbf{x}-\mathbf{p}|$, proving boundary cancellation. The induced force is inward, of magnitude $q^{2}aR/[4\pi {\epsilon }_0(R^2-a^2{)}^2]$, from Coulomb's law applied to the image field. The self-field is excluded.

A category is Cauchy-complete when every idempotent morphism splits: for $e:A\to A$ with $e^2=e$, there are $r:A\to B$ and $i:B\to A$ with $ir=e$ and $ri=1_B$. Its Karoubi envelope freely supplies these splittings. For a small category, this condition makes every retract of a representable functor representable.

A proper ideal $I$ is irreducible if $I=J\cap K$ implies $I=J$ or $I=K$. In a Noetherian ring, every irreducible ideal is a primary ideal; every proper ideal is a finite intersection of irreducible ideals.