For independent site percolation or bond percolation on the cubic lattice ${\mathbb{Z}}^d$, there is almost surely at most one infinite percolation cluster at every fixed parameter. Above the percolation critical probability, there is exactly one almost surely. This theorem is called the Burton-Keane theorem. Existence uses translation ergodicity together with positive percolation probability; uniqueness is a separate conclusion. At $p=1$ uniqueness is deterministic.

For a future-directed causal four-momentum $p$ crossing a future Killing horizon with generator $\xi =t+{\Omega }_H\phi$, the Lorentzian inner product obeys $p\cdot \xi \le 0$. With $E=-p\cdot t$ and $L=p\cdot \phi$, this is $E\ge {\Omega }_{H}L$. A massive particle has strict inequality. The same bound follows in the nonextremal quasistationary regime from the Physical-process first law for a rotating black hole and the second law of black-hole mechanics. Negative Killing energy in a Penrose process is allowed, but its angular momentum must satisfy this bound.

A parton distribution function specifies a hadron's partonic content as a function of longitudinal momentum fraction and a stated parton factorization scale. In the leading collinear parton model, $f_i(x)dx$ has a number-density interpretation for partons of species $i$ with fractions in $dx$. It is not generally normalized to one.

For a proton, the integrals of the valence differences count two net up quarks and one net down quark. A momentum-weighted sum over all quark, antiquark and gluon species gives total momentum fraction one. These are different sum rules. Beyond the simplest approximation, the definition depends on factorization scheme and renormalized field correlations; a naive positive classical probability interpretation is not universal. The functions enter factorized cross sections and evolve with resolution, rather than being predictions of a free-quark model.^[ref]

The leading high-energy approximation neglects quark and target masses and transverse constituent momentum. A lightlike reference momentum is used for the hadron; this does not describe a massless particle at rest.

For massless spin-one-half partons at leading order, the unpolarized electromagnetic deep-inelastic structure functions obey this relation. Radiative quantum chromodynamics corrections and target-mass effects can modify it.

At leading order in the naive parton model, dimensionless deep-inelastic structure functions depend only on Bjorken x. Quantum chromodynamics evolution introduces scale dependence.

The time-independent particle Lagrangian in a shearing sheet has conserved rotating-frame energyThe velocity-linear Coriolis acceleration term cancels from this expression. Up to the reference-orbit constant, it is the second-order expansion of inertial specific orbital energy minus ${\Omega }_0$ times inertial specific angular momentum. Its negative radial tidal term allows inelastic collisions to lower the total energy while increasing the radial extent of a ring.

A finite-volume gravity-current box model in a widening triangular channel has volume proportional to $h^2{L}^2$, so $hL=K$ is constant. A gravity-current front condition and absorbing-bed particle deposition flux giveEliminating time makes $\sqrt{\varphi }$ affine in $L^{5/2}$, producing a finite limiting runout length of a gravity current despite an infinite idealized stopping time.

In a channel with width $2\lambda xz$ at height $z$, a well-mixed particle-laden gravity current of depth $h$ occupies area $\lambda x{h}^2$. Its volume conservation, momentum and particle deposition flux balances givewhere the reduced gravity is $g^{\prime }=d\varphi$. The $h/3$ coefficient is the cross-section-weighted mean of $h-z$ in the hydrostatic pressure force. The characteristic curves have speeds $u$ and $u\pm \sqrt{g^{\prime }h/2}$.

A heated particle-laden layer released along a bed forms a gravity current while $g^{\prime }=g(\gamma \varphi -\theta )>0$. In a gravity-current box model, heating can make $g^{\prime }$ vanish in finite time, whereas settling velocity alone can give finite runout length of a gravity current approached asymptotically.

A partial type $p(\mathbf{x})$ is principal over $T$ if some formula $\theta (\mathbf{x})$ with $T\cup \{\exists \mathbf{x}\theta \}$ consistent implies each formula of $p$ modulo $T$. For complete types this is the usual notion of an isolated type. For an incomplete type, the isolating formula need not belong to the type itself.

A partial type is nonprincipal when it is not a principal partial type. Equivalently, for every formula $\theta (\mathbf{x})$ consistent with $T$, some $\psi \in p$ makes $\theta \wedge \neg \psi$ consistent with $T$. This is the exact extension property needed in the Henkin construction proving the omitting types theorem.

A time derivative is the partial derivative of an evolving field with respect to time while holding its spatial coordinates fixed. For a quantity depending only on time, it is an ordinary derivative.

After clearing denominators in a matrix, for every disjoint pair of index sets $I,B$ with $B$ nonempty and ${\sum }_{i\in B}{a}_i$ outside the linear span of the $I$-columns, choose an integer linear functional vanishing on those columns but not on the indicated sum. Choose a prime number dividing none of the finitely many nonzero evaluations. The last nonzero digit coloring for this prime, applied to a monochromatic positive solution, groups coordinates into blocks by p-adic valuation. Applying the appropriate functional and reducing modulo the prime shows that each block sum lies in the linear span of all earlier columns. This proves necessity in Rado's theorem without an asymptotic limiting argument.

For smooth non-increasing $V$ and an ordered subsolution and supersolution, begin with $u_0={\phi }^{-}$ and solve successive Poisson equations with forcing $V(u_k)$ and the fixed boundary data. The weak maximum principle for elliptic operators applied to the negatives of consecutive differences givesThe global Schauder estimate and the Hölder interpolation inequality bound the iterates uniformly in $C^{2,\alpha }$. Their bounded monotone sequence converges, and compact embedding of Hölder spaces permits passing to the equation. The uniform Hölder seminorm bounds retain $C^{2,\alpha }$ regularity of the limit.

This representation of an ordered pair by sets satisfies $⟨a,b⟩=⟨c,d⟩$ exactly when $a=c$ and $b=d$. Its intersection recovers $\{a\}$, and its union recovers $\{a,b\}$; if the latter is a singleton then $b=a$. Its rank of a set is $\max (\mathrm{rank}(a),\mathrm{rank}(b))+2$, which bounds the ranks of function graphs in absoluteness of cardinalhood in limit ranks.

The invariable plane is perpendicular to a system's total angular momentum. For an isolated planetary system, it is fixed while individual orbital planes precess.

If $e^{-\theta N_{t\wedge {\sigma }_K}}$ is a nonnegative supermartingale, with ${\sigma }_K$ the first time $N_t\le K$, the optional stopping theorem yields ${\mathbb{P}}_N({\sigma }_K<\infty )\le e^{-\theta (N-K)}$. Such a bound can prove escape of a backlog independently of heuristic fluid models.

The negative hydrogen ion supplies important continuum opacity in cool hydrogen-rich stellar atmospheres. A useful local approximate law is $\kappa ={\kappa }_0{\rho }^{1/2}{T}^9$; it is not valid over arbitrary temperatures and compositions. In an ideal gas power-law opacity radiative envelope matched to ${\nabla }_{\mathrm{phot}}=1/8$, it gives ${\nabla }_{\mathrm{rad}}=-1/3+(11/24)(T/T_{\mathrm{eff}}{)}^{9/2}$, rising rapidly toward convective instability.

For $R=4{\pi }^2+{ϵ}^2$, fixed roll phase, $T={ϵ}^{2}t$ and normalization $\psi =ϵA(T)\cos (\pi x)\sin (\pi z)+\cdots$, the Landau amplitude equation isThe mean-temperature correction in weakly nonlinear Darcy convection supplies the cubic negative feedback. Projection of the third-order equations onto the critical eigenfunction via the Fredholm solvability condition for a self-adjoint operator fixes both coefficients. The stable nonzero amplitudes are $\pm 2/\pi$ within the chosen phase, giving a supercritical pitchfork bifurcation in this real-amplitude reduction.