A shallow buoyant porous gravity current in a deep aquifer with uniform imposed Darcy velocity $U$ obeys $\varphi h_t+(Uh-K_{b}h{h}_x{)}_x=0$ away from sources, where $K_b=k\Delta \rho g/\mu$. Here $k$ is permeability of a porous medium, $\varphi$ is porosity, and $\mu$ is dynamic viscosity. The pore-fluid drift speed is $u=U/\varphi$, and the gravity diffusion coefficient multiplying $(h{h}_x{)}_x$ is $a=K_b/\varphi$. The sharp-interface model neglects capillary trapping and dissolution.

A Laurent polynomial ring is the localization of a ring $k[X_1,\dots ,X_r]$ obtained by inverting all variables. Its elements are finite sums of Laurent polynomials with exponent vectors in ${\mathbb{Z}}^r$. It is a Noetherian ring by the Hilbert basis theorem and Localization of a Noetherian ring.

The dilogarithm is the order-two polylogarithm, defined for $|z|<1$ by ${\mathrm{Li}}_2(z)={\sum }_{n\ge 1}{z}^n/n^2$. Its derivative is $-\mathrm{Log}(1-z)/z$, with a removable value at zero; a chosen complex logarithm branch gives its analytic continuation.

For positive lapse, eliminating momentum gives $(\dot{X}-u{X}^{\prime }{)}^2/(2e)-e{T}^2{X}^{\prime 2}/2$. This is the Polyakov action for ${\gamma }_{ab}=\left(\begin{array}{cc}{u}^2-e^2{T}^2& u\\ u& 1\end{array}\right)$, up to a Weyl transformation. Thus $e=1/T,u=0$ is conformal gauge.

Varying the independent worldsheet metric gives $T_{ab}=T[{\partial }_{a}X\cdot {\partial }_{b}X-\frac{1}{2}{\gamma }_{ab}{\gamma }^{cd}{\partial }_{c}X\cdot {\partial }_{d}X]$. Its classical vanishing is the metric equation and produces the Virasoro constraints.

Two orthonormal transverse polarization vectors span the plane perpendicular to a nonzero momentum. Summing their dyadic products gives the spatial transverse projector. It has trace and rank two and acts as the identity on physical photon polarizations.

A physical polarization of a massive spin-one gauge boson whose spatial polarization points along its spatial momentum. Together with two transverse polarizations it supplies the three physical spin states. In the Higgs mechanism, a broken-direction Goldstone boson supplies this extra state as the gauge boson acquires mass.

On a smooth bounded connected domain, the homogeneous Neumann problem seeks $u\in H^1(U)$ such that $\int \nabla u\cdot \nabla v=\int fv$ for every $v\in H^1(U)$. For $f\in L^2(U)$ it is solvable exactly when $\int f=0$, and the weak solution is unique up to constants. The Poincare-Wirtinger inequality and the Riesz representation theorem give existence on the mean-zero Sobolev space. The test function one gives necessity of the compatibility condition.

An even Poisson bracket on a space with commuting and anticommuting variables obeys $\{F,G\}=-(-1{)}^{|F||G|}\{G,F\}$ and the graded Leibniz rule. In particular the bracket of two odd functions is symmetric. Quantization turns the odd-odd bracket into an anticommutator.

Second-class constraints ${\chi }_a$ with invertible bracket matrix can be eliminated with a modified bracket. For an ordinary even constraint set it is $\{F,G{\}}_D=\{F,G\}-\{F,{\chi }_a\}(C^{-1}{)}^{ab}\{{\chi }_b,G\}$, $C_{ab}=\{{\chi }_a,{\chi }_b\}$. Graded constraints require the matching graded sign convention.

Localization of a finite fraction of oppositely charged ions near a charged object even as the confining volume grows without bound. In the salt-free infinite-cylinder mean-field model, the Manning parameter must exceed one for a nonzero limiting local density. The condensed charge does not include ions that escape to infinity; finite-cell charge neutrality and the local infinite-volume limit must not be interchanged.

For $u=\log (r/a)$ and $\Phi =\beta q\varphi -2u$, a counterion-only Poisson-Boltzmann equation reduces to ${\Phi }^{\prime \prime }=-C{e}^{-\Phi }$. Conditions $\Phi (0)=0$, $\Phi \to \infty$ and ${\Phi }^{\prime }\to 0$ give $\Phi =2\log (1+bu)$, $b=\sqrt{C/2}>0$. Matching surface Gauss's law gives $b=\eta -1$, so this branch requires the Manning parameter $\eta >1$. Its number density is $n(r)=(\eta -1{)}^2(a/r{)}^2/[2\pi {\ell }_B{a}^2(1+(\eta -1)\log (r/a){)}^2]$. It leaves residual line-charge magnitude $q/{\ell }_B$.

Confine the counterions of a uniformly charged infinite cylinder within a coaxial cylinder of radius $R$, with total mobile line charge $\lambda$. The electric field vanishes at $R$, while its surface value at $a$ is fixed by the bare line charge. This gives $a{\varphi }^{\prime }(a)=2\lambda /ϵ$ and $R{\varphi }^{\prime }(R)=0$ in Gaussian units. Sending $R/a\to \infty$ defines the infinite-dilution limit of cylindrical counterions.

For $|a|<1$, the Möbius transformation $f_a(z)=(z-a)/(1-\overline{a}z)$ is an isometry of the Poincare disc model sending $a$ to zero. Rotations combined with these transformations describe its orientation-preserving isometries.

Planetary migration changes a planet's orbital semimajor axis through torques from a disc or exchanges of energy and angular momentum with other bodies, often followed by tidal dissipation. Short-period giant planets, eccentric progenitors, resonant architectures, and stellar obliquity provide complementary constraints rather than one unique migration diagnostic.

A passing star perturbs a planetary or cometary system through its differential Newtonian gravitational field. A distant encounter is predominantly tidal; a sufficiently close one can substantially change specific orbital energy and angular momentum, eject a comet, or shift its periapsis into or out of the planetary region.

Anisotropic thermal radiation produces recoil that changes the semi-major axis of a rotating solid body. Finite thermal inertia provides the temperature lag. Large bodies have small acceleration per unit mass, while very small bodies become nearly isothermal. The Yarkovsky effect can supply bodies to dynamical escape regions.

Accretion releases gravitational energy. Efficient radiation at an accretion shock leaves a lower-entropy cold start, while greater retained energy gives a higher-entropy hot start; intermediate outcomes are possible. Initial entropy changes a young planet's radius, luminosity and spectrum. Consequently an exoplanet direct imaging luminosity and age need not uniquely fix mass without a formation/evolution model.

The minimum-mass solar nebula estimates a primordial gas-and-solid protoplanetary disk by spreading planetary material into annuli and restoring a solar mixture. A common idealized surface density is proportional to $r^{-3/2}$. This reconstruction is a model of minimum material requirements, not a unique history of the Solar System. Extrapolating it far beyond the planetary region need not give a physically consistent disk.

Core accretion forms a planet by building a solid core through growth and accretion of solid material. A sufficiently massive core can then bind a gaseous envelope. It is distinct from direct fragmentation through gravitational instability of an astrophysical disk.