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An instantaneous point release of actual volume $V$ per unit width in a porous gravity current with background flow has the Barenblatt solution $h=[R(t{)}^2-(x-ut{)}^2{]}_{+}/(6at)$, where $R^3=9aVt/(2\varphi )$. Its centre translates at pore speed $u=U/\varphi$ and its radius grows as $t^{1/3}$. The conserved physical volume is $\varphi \int hdx=V$.

An algorithm runs in polynomial time if its worst-case running time is bounded by a fixed polynomial in the input length. Arithmetic with fixed-degree algebraic numbers also needs polynomial bit cost when used in such a guarantee. The Edmonds–Karp algorithm and the method of conditional probabilities with efficiently computable clause expectations are examples.

Let the uniform closure of polynomials on a compact set be an algebra. A reciprocal at a sufficiently distant pole belongs to it by a geometric series. A path of poles avoiding the compact set can be divided into short steps; expanding each reciprocal in powers of the previous one transfers polynomial approximability along the path.

A harmonic function on all of ${\mathbb{R}}^n$ with a finite global Hölder seminorm of exponent $0<\mu <1$ is constant. Indeed $|w(x)-w(0)|\le C|x{|}^{\mu }$, and the interior derivative estimate for a harmonic function on balls of radius $R$ bounds $|\nabla w(0)|$ by $C{R}^{\mu -1}$. Letting $R\to \infty$ makes this gradient zero, and translating the centre proves the assertion everywhere.

For finite $S\subset \mathbb{Z}$, use vertices $\mathbb{Z}$ and hyperedges $S+t$. Independent uniform $k$-coloring gives missing-color probability at most $k{e}^{-|S|/k}$, and the dependency graph of events has degree at most $|S|(|S|-1)$. For $k\ge 20$ and $|S|\ge 6k\log k$, the symmetric Lovász local lemma condition holds. The compactness extension of the Lovász local lemma gives a coloring in which every translate contains every color.

For the Polonyi model with canonical Kähler potential and $W=m^2(z+\beta )$, $\beta >0$, a constant supersymmetric vacuum with $m\ne 0$ requires real $z=(-\beta \pm \sqrt{{\beta }^2-4})/2$ and exists exactly for $\beta \ge 2$. Its supergravity F-term potential is negative because $W\ne 0$. For $0<\beta <2$, the supergravity auxiliary field cannot vanish. If $m=0$, the model is flat with unbroken supersymmetry. This parameter-dependent statement is sharper than asserting that every linear superpotential breaks supersymmetry in supergravity.

For $m\ne 0$ and positive $\beta$, the Polonyi model has a unique zero-energy global minimum at $\beta =2-\sqrt{3}$, $⟨z⟩=\sqrt{3}-1$. Writing $u=\mathrm{Re}z-(\sqrt{3}-1)$ and $y=\mathrm{Im}z$, its potential bracket is $(u^2+y^2+\sqrt{3}u{)}^2+(2\sqrt{3}-3)u^2+(4-2\sqrt{3})y^2$, a sum of nonnegative terms. The real and imaginary curvatures are $4\sqrt{3}$ and $8-4\sqrt{3}$. The other zero-energy stationary point with $\beta =2+\sqrt{3}$ and $z=-\sqrt{3}-1$ is a saddle point. The stable vacuum has nonzero supergravity auxiliary field, so it has supersymmetry breaking even with zero cosmological constant. The $m=0$ branch does not fix these values.

A wave is linearly polarized when its oscillating electric field always lies along one fixed line. For a transverse plane wave, that line is perpendicular to the wavevector. Unlike circular polarization, the field oscillates along the line rather than rotating through directions.

The Stokes parameters $I,Q,U,V$ describe intensity, two linear-polarization components and circular polarization. $Q$ and $U$ depend on the chosen transverse basis. On the celestial sphere, $Q\pm iU$ transform as spin-two fields rather than as ordinary scalars.

For $z=r{e}^{i\theta }\ne 0$ and a positive integer $n$, all solutions of $w^n=z$ are $r^{1/n}{e}^{i(\theta +2\pi k)/n}$ for $k=0,\dots ,n-1$. They differ by multiplication by roots of unity and are distinct. If $z=0$, the sole root is zero.

Mark each member of an independent Poisson infection cohort by its symptom-onset interval, using independent delays. Poisson thinning gives independent onset-bin counts, and sums over independent cohorts remain independent Poisson counts. Fixed cohort totals instead induce multinomial dependence. The marking assumptions are essential.

For a Poisson random measure on $E$ with finite intensity $\nu (E)$, its integral of a finite-valued measurable mark function is the finite sum of the marks at its random atoms. Its characteristic function is $\mathbb{E}\exp (iu\int fdN)=\exp (\int (e^{iuf}-1)d\nu )$. On a time-mark space with intensity $ds\nu (dz)$ and finite mark intensity, restriction to each bounded time interval has finite intensity and defines a Compound Poisson process. Zero-valued marks do not affect the integral and need not appear in its Lévy measure.

A smooth antisymmetric contravariant two-tensor defines a biderivation $\{F,G\}=\omega (dF,dG)$. It is a Poisson bivector when this bracket satisfies the Jacobi identity. Its rank may vary; where it is nondegenerate it inverts to a symplectic form. The coordinate Jacobi condition for a Poisson bivector makes the integrability constraint explicit.

The dual space ${\mathfrak{g}}^{\ast }$ of a finite-dimensional Lie algebra has this canonical Poisson bracket. For a basis $v^i$, linear coordinates $x^i(\ell )=\ell (v^i)$ give $\{x^i,x^j\}=c_k^{ij}{x}^k$. The Lie algebra's Jacobi identity is precisely the coordinate Jacobi condition for a Poisson bivector. This construction is intrinsic on the dual space; writing the same formula in an arbitrary manifold chart does not by itself provide a compatible global atlas.

A pointwise limit assigns to each input the limit of a sequence of function values there, on the specified domain of a function. It is unique wherever it exists, by uniqueness of the numerical limit of a sequence. Its existence supplies no common approximation rate over the domain of a function. For $f_n(x)=x^n$ on $[0,1]$, it equals zero below one and one at one; restricting to $[0,a]$ with $a<1$ gives the zero function.

For $n\ge 2$, a round annulus $A_R=B_{2R}\setminus B_R$ is connected and obeys ${\int }_{A_R}|v-v_{A_R}{|}^2\le C(n)R^2{\int }_{A_R}|\nabla v{|}^2$. Scaling reduces the claim to a fixed annulus. In dimension one the annulus has two components, so the inequality fails with a single average; one needs a separate average on each component.

A Platonic solid with six square faces, twelve edges and eight vertices. Its rotational symmetry group of a cube has order 24; including orientation-reversing isometries gives the order-48 symmetry group of a cube.

Downward recycling of a lithospheric plate into the mantle. A sufficiently cold, dense slab can provide slab pull, while bending and friction resist motion. The resulting material and volatile recycling connects plate tectonics with volcanism and long-term climate.

Gravitational sliding of lithospheric material away from elevated spreading regions. It supplies a driving contribution to plate tectonics together with slab pull and mantle stresses.

For a planet on a circular stellar orbit, a common convention is $\Theta =\frac{1}{2}(v_{\mathrm{esc}}/v_k{)}^2$. The escape-to-orbital-speed ratio measures the size of a strong gravitational assist relative to the stellar binding energy of a nearby comet. Large values tend to favour ejection over collision; small values favour repeated weak encounters with more collision opportunities. This is a statistical comparison, dependent on the initial orbit and encounter geometry, rather than a universal outcome threshold.