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Double Compton scattering is $e^{-}+\gamma \to e^{-}+\gamma +\gamma$, together with its inverse. The forward reaction has one incoming and two outgoing photons, so it changes their number and can source the divergence of the photon number current. Ordinary Compton scattering has one incoming and one outgoing photon and conserves photon number. Forward and inverse rates can balance without either reaction individually conserving it.

For steady spherical isentropic flow with $p=K{\rho }^{3/2}$ around a point mass, put $j=r^2\rho v>0$ and $\mathcal{E}=v^2/2+2c^2-GM/r>0$. Eliminating the mass density gives $c^2=[j(3K/2{)}^2{]}^{2/5}{r}^{-4/5}{\mathcal{M}}^{-2/5}$. With $r_s=GM/(4\mathcal{E})$, $x=(r/r_s{)}^{1/5}$, $y={\mathcal{M}}^{2/5}$ and $\lambda =[j(3K/2{)}^2{]}^{2/5}/(2\mathcal{E}{r}_s^{4/5})$, the Bernoulli equation reduces to $\lambda F(y)=F(x)$, where $F(t)=t^4+4/t$. Since $F$ has its unique minimum at $t=1$, a regular sonic point requires $\lambda =1$ and $r=r_s$. One transonic branch is $y=x$, which has $v=\sqrt{2\mathcal{E}}$, $\rho \propto r^{-2}$ and $p\propto r^{-3}$. The decreasing branch has inner free fall and an outer nearly static reservoir, giving Bondi accretion for inward fluid flow. A nonradiative stationary shock wave keeps $j,\mathcal{E},r_s$ fixed but raises $K$, so $\lambda$ increases by the factor $(K_2/K_1{)}^{4/5}$. The inward transonic solution is the spherical $n=2$, $\gamma =3/2$ specialization of transonic accretion in a power-law tube.

For an inviscid fluid with specific internal energy $e$, let $E=\rho (e+|\mathbf{u}{|}^2/2)$. In a prescribed potential $\Phi$, ${\partial }_{t}E+\nabla \cdot [(E+p)\mathbf{u}]=-\rho \mathbf{u}\cdot \nabla \Phi +Q$, where $Q$ is external heat supply. A static gravitational potential can instead be included in the conserved energy.

An isothermal shock joins two states at the same temperature and isothermal sound speed. Mass and momentum conservation imply $u_1{u}_2=c_s^2$ for a nontrivial planar jump, so the compression ratio is the upstream isothermal Mach number squared. Heat exchange replaces adiabatic energy conservation across the gas alone.

A smooth radial compressible flow with a fixed adiabatic exponent conserves $r^2\rho u$ and $p/{\rho }^{\gamma }$. In a prescribed Newtonian gravitational potential, its Bernoulli function is $u^2/2+c^2/(\gamma -1)+\Phi$, with $c$ the adiabatic sound speed. Eliminating the density derivative yields $(u-c^2/u)u^{\prime }=2c^2/r-{\Phi }^{\prime }$.

A polytropic flow uses a polytropic equation of state. When $K$ and the adiabatic exponent are constant, a smooth streamline has specific enthalpy $h=\gamma K{\rho }^{\gamma -1}/(\gamma -1)$ and adiabatic sound speed $c_s^2=\gamma K{\rho }^{\gamma -1}$. Spatially varying entropy can make $K$ depend on streamline; the constant-$K$ spherical wind is a special case.

A semiprime is a composite number that is a product of exactly two prime numbers, counted with multiplicity. For example, $6=2\cdot 3$ and $9=3^2$ are semiprimes. Distinct factors give $\phi (pq)=(p-1)(q-1)$ for the Euler totient function, whereas equal factors give $\phi (p^2)=p(p-1)$. This distinction matters when constructing a uniform coprime state for a semiprime.

A logarithmic singularity behaves locally like $a\log (z-z_0)$ with $a\ne 0$, on a chosen branch. Although unbounded at the endpoint, it is integrable along a finite ray. In a rapidly oscillating endpoint integral its contour rotation produces a logarithm of the large parameter and a branch-angle contribution; this differs from the ordinary smooth-amplitude Watson lemma.

A compactification with quantized background form-field strengths on internal cycles. Fluxes can generate potentials for moduli of a string compactification and affect supersymmetry and vacuum geometry. They must satisfy field equations, Bianchi identities and global charge constraints; choosing flux integers alone is not a complete consistency proof.

A parameter of an internal geometry, gauge background or dilaton configuration that becomes a scalar field in the lower-dimensional theory. A continuously undetermined parameter gives a flat direction and often a massless field. Fluxes and other effects can generate a potential, leading to moduli stabilization. Shape, size and coupling moduli affect low-energy observables.

For a commutative algebra $R$ over $k$, put $D_k^{-1}(R)=0$ and define $D_k^j(R)$ to be the $k$-linear endomorphisms $\theta$ with $[\theta ,m_a]\in D_k^{j-1}(R)$ for every multiplication map $m_a$. The union is a ring under composition, with $D_k^0(R)=R$ and $D_k^r{D}_k^s\subseteq D_k^{r+s}$. This commutator definition applies to singular rings and fields of arbitrary characteristic, unlike a definition restricted to polynomial coefficients and partial derivatives.

In collisional ionization equilibrium, collision-driven ionization and recombination balance for the populated ion stages, determining their relative abundances at a given temperature. The resulting cooling curve includes the loss of line-emitting bound states as temperature increases. Rapidly cooling gas or external photoionization can invalidate this equilibrium assumption.

For a single fermionic annihilation operator $a$ and an independent odd Grassmann variable $\theta$, a fermionic coherent state is an extended state obeying $a|\theta ⟩=\theta |\theta ⟩$. With $\{a,\theta \}=0$ one may take $|\theta ⟩=\exp (-\theta a^{†})|0⟩$. Its overlap is exponential in the Grassmann labels. A resolution of identity with a Berezin integral gives the time-slice construction of a fermionic path integral. These formal eigenstates use Grassmann-valued coefficients.

An unnormalized bosonic coherent state is $|\psi ⟩=e^{\psi a^{†}}|0⟩$ for a complex label $\psi$. The canonical commutation relation gives $a|\psi ⟩=\psi |\psi ⟩$ and overlap $⟨\psi |{\psi }^{\prime }⟩=e^{\bar{\psi }{\psi }^{\prime }}$. Multiplication by $e^{-|\psi {|}^2/2}$ produces a normalized coherent state. The unnormalized form makes the coherent-state resolution of identity and thermal time slicing particularly simple.

For an unnormalized bosonic coherent state, the measure is $d^2\psi /\pi$ over the complex plane. Gaussian moments $\int d^2\psi e^{-|\psi {|}^2}{\psi }^n{\bar{\psi }}^m/\pi =n!{\delta }_{nm}$ recover the identity in the Fock state basis. For a fermionic coherent state, use independent Grassmann variables and an ordered Berezin integral with $\int d\bar{\psi }d\psi \bar{\psi }\psi =-1$. The coefficient of $-\bar{\psi }\psi$ in the weighted projector is the identity. Products of these one-mode measures give the finite-mode result.

The quadratic coherent correction weights the power spectrum of surface height by $\beta (q)$, so equal root mean square heights can give different mean fields. Only an additional long-spatial-scale approximation, with spectrum concentrated at $|q|\ll k$, reduces it to $2k^2{\sigma }^2$. Second-order surface scattering therefore contains information absent from first-order coherent reflection.

If a labeled soliton approaches $x=vt+a_{-}$ before a collision and $x=vt+a_{+}$ afterwards, its spatial shift is $\Delta x=a_{+}-a_{-}$. For $v\ne 0$, define its time delay at a fixed spatial point by $\Delta t=-\Delta x/v$. Thus a positive forward shift of a right-moving soliton is a negative time delay. For a stationary soliton, the spatial shift is well defined but this time-delay definition is not.

The rest mass of a stable soliton is the lowest energy in its topological sector minus the vacuum energy, in the infinite-volume limit. If Euclidean kernels $K_Q(\tau )$ have nonzero overlap with the sector's lowest-energy states, then $M=-{\lim }_{\tau \to \infty }{\tau }^{-1}\log [K_1(\tau )/K_0(\tau )]$, followed by the infinite-volume limit. A Euclidean path integral represents these kernels, with boundary wavefunctionals included if necessary.

A weak-coupling expansion of a vacuum-subtracted soliton mass begins with the classical energy and adds the Gaussian fluctuation approximation. For Sine-Gordon theory, $M=8m/{\beta }^2+\Delta M_{1\mathrm{loop}}+O(m{\beta }^2)$. The term of order $m$ includes both the vacuum-subtracted fluctuation frequencies and the appropriate mass counterterm.

A breather is a spatially localized nonlinear-wave solution periodic in time. It does not separate into independently traveling lumps while it breathes. A Sine-Gordon breather arises from a complex-conjugate soliton pair. In the quantum sine-Gordon model, the associated neutral kink-antikink bound particles are also called breathers.