A line $r=b+sm$ intersects a plane $(r-a)\cdot n=0$ exactly when the scalar equation $s(m\cdot n)=(a-b)\cdot n$ can be solved. If $m\cdot n\ne 0$, there is one point. If that product vanishes, the line is either contained in the plane or disjoint from it. Direction and normal vectors must be nonzero.

Planes with common unit normal $n$ have distance $|(a_1-a_2)\cdot n|$. Every joining displacement has this normal component, giving a lower bound by its absolute value; a displacement parallel to the normal attains the bound.

For a common nonzero normal vector $\mathbf{n}$, not necessarily of unit Euclidean norm, the planes $\mathbf{n}\cdot \mathbf{x}=d_1$ and $\mathbf{n}\cdot \mathbf{x}=d_2$ have distance $|d_2-d_1|/\Vert \mathbf{n}\Vert$. Thus $2x-y+2z=3$ and $2x-y+2z=9$ are distance $6/3=2$ apart. First put both plane equations into the same normal-vector scale: subtracting offsets from independently rescaled equations need not give a geometric distance.

The Felsenstein pruning algorithm computes a site likelihood function by sum-product belief propagation from leaves to root. For a finite-state Markov kernel $K$, the subtree likelihood function obeys $L_v(a)={\prod }_{w\text{ child of }v}{\sum }_{b}K(a,b)L_w(b)$, with observed-state indicators at leaves. Summing against the root probability distribution gives the site likelihood function. An outward pass gives edge posterior probabilities and expected transition counts for the expectation-maximization algorithm.

Photon emission creates a photon, for example in bremsstrahlung or the decay of an excited material state. It produces a positive photon-number collision source when not balanced by inverse processes. The photon number current is conserved only if its integrated creation and destruction rates cancel.

Photon absorption removes a photon and transfers its energy and momentum to matter. It contributes a negative photon-number collision source. It differs from Compton scattering, which redistributes a photon's energy and direction while preserving the count. The integrated source can vanish if photon emission and absorption balance.

Two first-class constraints in Maxwell theory remove four phase-space variables from the eight variables of a four-component potential and its momenta. The remaining four phase-space variables form two canonical pairs. In radiation gauge these are transverse polarization modes; circular combinations have helicity $+1$ and $-1$. Auxiliary covariant-gauge components do not add physical photon states.

The thermal binding scale defined by equating heated gas sound speed to circular Keplerian orbit speed. It is $r_g=GM/c_s^2$. The circular-orbit binding energy is $GM/(2r)$ per unit mass, while the gas specific enthalpy is of order $c_s^2$. A heated disk surface at radii comparable to or larger than $r_g$ can therefore drive photoevaporation. The actual launching radius can differ by a factor of order unity; $c_s$ is not the ballistic escape speed.

For a sink $W=W_0(r/r_g{)}^{-5/2}$ outside $r_g$, mass conservation gives $d{F}_{\mathrm{acc}}/dr=2\pi rW$ in steady state. Remote feeding gives $F_{\mathrm{acc}}={\dot{M}}_d-{\dot{M}}_w$ inside $r_g$, and ${\dot{M}}_d-{\dot{M}}_w\sqrt{r_g/r}$ outside, where ${\dot{M}}_w=4\pi W_0{r}_g^2$. The inward flux decreases towards the star because mass is diverted into the wind. A finite feeding radius or outer truncation supplies an additional boundary condition.

When a protoplanetary disk's inward viscous supply approaches its wind-loss rate, photoevaporation can prevent replenishment of the inner disk. That region drains on a viscous timescale, leaving a gap or hole. A negative density in a formal steady state calculation signals the breakdown of that approximation, not a physical continuation. An exposed outer edge can subsequently disperse rapidly.

A radiative correction is a quantum correction beyond the classical or tree approximation, represented in perturbative quantum field theory by Feynman diagrams of positive loop order. Such corrections can alter a scalar potential and its derivatives; symmetries and renormalization constrain which changes are allowed.

A Hermitian matrix model replaces a spacetime field by a finite-dimensional Hermitian matrix. Its normalized integral supplies correlation functions and a zero-dimensional perturbative quantum field theory expansion. For an action built from traces of matrix powers, unitary matrix conjugation is a symmetry and the eigenvalues describe the invariant degrees of freedom. A quartic real integral needs a nonnegative highest-order coupling for convergence; a formal power series can be studied independently of this convergence issue.

A generalised permutation is a finite multiset of ordered pairs, displayed as a two-row array or encoded by a finite-support matrix of nonnegative integer multiplicities $m_{a,b}$. The no-repeated-column condition means every multiplicity is zero or one; entries within either individual row may still repeat.

With a time-slice regulator, inserting field-coordinate resolutions of identity in evolution kernels produces a bosonic phase-space path integral; integrating the canonical momentum gives the configuration-space action. For fermions, resolutions in fermionic coherent states instead produce a Berezin integral and the first-order kinetic term. Vacuum endpoint projection supplies the Feynman i-epsilon prescription. Both constructions reproduce the same free oscillator correlators and their Wick theorem expansions, and thus agree order by order in a regulated perturbation series.

A path integral with weight $e^{-S_E}$ uses the Euclidean action obtained by Wick rotation. Bosonic convergence requires appropriate positivity and boundary conditions.

A phase-space path integral represents a quantum transition kernel by integrating both coordinates and conjugate momenta with action $\int (p\dot{q}-H)dt$. It follows from inserting position and momentum completeness relations between short-time evolution operators. For a quadratic momentum dependence the momentum variables can be integrated by a Gaussian integral.

A configuration-space path integral integrates histories of the coordinates with weight $e^{iS}$. Fixed endpoint conditions specify a transition amplitude. For a quadratic kinetic energy it follows by Gaussian momentum integration from the phase-space path integral. Time-slicing fixes the normalization, and a regulator gives meaning to the otherwise formal functional measure.

The scalar field path integral generalizes the configuration-space path integral by replacing finitely many coordinates with field values. Its regulated measure integrates one field variable at every spacetime lattice point. Fixed boundary fields give transition kernels; vacuum boundary conditions and sources give the normalized vacuum generating functional.

Insert the coherent-state resolution of identity between short-time evolution operators. For a normally ordered Hamiltonian, the kernel uses the next bra label and current ket label: $⟨{\psi }_{j+1}|e^{-ϵH}|{\psi }_j⟩=\exp ({\bar{\psi }}_{j+1}{\psi }_j-ϵH({\bar{\psi }}_{j+1},{\psi }_j)+O({ϵ}^2))$. Together with the Gaussian measure this yields the first-order derivative term $\bar{\psi }{\partial }_{\tau }\psi$. The adjacent-label prescription retains operator ordering information that a formal continuum expression alone can hide.

More suspended heavy particles above fewer particles puts denser fluid above lighter fluid. Such a particle suspension may develop Rayleigh-Taylor instability and overturning in addition to one-dimensional kinematic sedimentation. A rarefaction solution alone does not establish hydrodynamic stability.

A horizon-length patch on the matter-radiation equality surface has physical size $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$ in a flat universe with conformal time measured from the Big Bang. Dividing by its angular diameter distance $a_{\mathrm{eq}}({\tau }_0-{\tau }_{\mathrm{eq}})$ gives its small angular size. For the radiation-matter scale factor in conformal time, this is $(\sqrt{2}-1)\sqrt{{\Omega }_{r,0}}/[1-\sqrt{2{\Omega }_{r,0}}]$. A patch diameter is twice the angle associated with one horizon radius. The particle horizon is an integral over past evolution and must not be silently replaced by the instantaneous Hubble radius.