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Put $C=ϵ+1-c_s^2$. For constant $H,ϵ,c_s>0$ and cubic interaction $H_{\mathrm{int}}=-M_{\mathrm{Pl}}^2\int d^{3}x{a}^3ϵC{\dot{\zeta }}^3/(H{c}_s^2)$, the Bunch-Davies vacuum modes $u_k=H(1+ik{c}_s\tau )e^{-ik{c}_s\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{c}_s{k}^3}$ giveTo derive this primordial bispectrum, transform $dt=ad\tau$, $\dot{\zeta }={\zeta }^{\prime }/a$, use the six connected Wick contractions, and integrate ${\int }_{-\infty (1-i0)}^0{\tau }^2{e}^{i{c}_s(k_1+k_2+k_3)\tau }d\tau =2i/[c_s(k_1+k_2+k_3){]}^3$ in the in-in formalism. The normalized equilateral amplitude is $B(k,k,k)/P(k{)}^2=2C/9$; raw inverse powers of $c_s$ must not be confused with normalized primordial non-Gaussianity.

For a scale-invariant primordial bispectrum, the displayed shape function is invariant under common rescaling of all three wavenumbers. It isolates triangle dependence from the overall power spectrum amplitude. It indicates intrinsic shape preference but alone does not determine observational signal-to-noise, which also depends on transfer functions, covariance and survey weights.

An equilateral primordial bispectrum configuration has equal external wavenumbers. For a cubic time-derivative curvature bispectrum, $S\propto k_1{k}_2{k}_3/(k_1+k_2+k_3{)}^3$ is maximized here at fixed perimeter by the arithmetic-geometric mean inequality.

A flattened primordial bispectrum configuration lies at the collinear boundary of the allowed momentum-triangle domain. For a cubic time-derivative curvature bispectrum, it is finite rather than singular. Its equal-short-side value $k_1=k_2=k_3/2$ gives $k_1{k}_2{k}_3/(k_1+k_2+k_3{)}^3=1/32$, compared with $1/27$ at an equilateral bispectrum configuration.

A constant primordial bispectrum means a constant bispectrum shape function, not a bispectrum independent of wavenumber. With $P(k)=2{\pi }^2{\Delta }_{\zeta }^2/k^3$ and scale-independent ${\Delta }_{\zeta }^2$, one normalization is $B=f_{\mathrm{NL}}[P(k_1)P(k_2)P(k_3){]}^{2/3}=4{\pi }^4{f}_{\mathrm{NL}}{\Delta }_{\zeta }^4/(k_1{k}_2{k}_3{)}^2$.

For $H_{\mathrm{int}}=-M_{\mathrm{Pl}}^2\int d^{3}x{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$ and the standard Gaussian de Sitter modes, the tree in-in formalism gives $B=H^4{\sum }_{\mathrm{cyc}}{k}_2^2{k}_3^2(K^{-1}+k_1{K}^{-2})/[16ϵM_{\mathrm{Pl}}^4(k_1{k}_2{k}_3{)}^3]$, where $K={\sum }_i{k}_i$. The result includes six connected Wick contractions and a vacuum contour.

A preorder is a well-quasi-ordering if every infinite sequence has indices $i<j$ with $x_i\le x_j$. An infinite sequence with no such pair is a bad sequence. This formulation handles combinatorial embedding relations that need not be antisymmetric on syntactic codes.

For subsets of a preorder, Hoare domination means every source element has some greater or equal target element: $S{\le }_{H}T⟺\forall s\in S\exists t\in T(s\le t)$. It is reflexive and transitive. If the base is a well-quasi-ordering, its finite subsets are well-quasi-ordered by this relation; arbitrary subsets need not be, as the Rado order shows.

A stationary point in a non-monotone velocity profile can support a quarter-power critical layer whose phase-speed correction has positive imaginary part. The resulting normal mode grows at a rate proportional to the square root of the wavenumber. Such unbounded high-frequency amplification demonstrates linear instability of the frozen layer; nonlinear evolution and precise well-posedness consequences require separate analysis.

The unsteady form of the Prandtl boundary-layer equation evolves the tangential boundary-layer velocity under advection, outer pressure forcing and normal viscous diffusion. Incompressibility supplies the normal velocity. A stationary wall imposes the no-slip boundary condition, and tangential velocity matches the prescribed outer slip velocity far from the wall. The normal far-field disturbance need not vanish.

For a blackbody grain with geometric cross-section $\sigma$, mass $m$ and stellar-frame velocity $\mathbf{v}$, define the radiation-pressure coefficient $\beta =L_{\star }\sigma /(4\pi cG{M}_{\star }m)$. To first order in $v/c$, steady absorption and isotropic rest-frame reradiation giveThe relative incoming flux provides $1-\dot{r}/c$; photon reradiation carries stellar-frame momentum proportional to $\mathbf{v}/c$. Subtracting static radiation pressure gives radial drag $-2m\beta G{M}_{\star }\dot{r}/(c{r}^2)$ and transverse drag $-m\beta G{M}_{\star }{v}_{\theta }/(c{r}^2)$. Non-blackbody grains require their actual radiation-pressure efficiency.

With $A=\beta G{M}_{\star }/c$ constant, average the drag work $\dot{\epsilon }=-A(2{\dot{r}}^2+v_{\theta }^2)/r^2$ over a bound elliptic orbit. Using $dt=r^{2}df/h$ gives the integral $\pi (2+3e^2){\mu }^2/h^2$, where $\mu$ is the conservative central parameter. Combining with $\dot{a}=2a^2\dot{\epsilon }/\mu$ givesThis is a weak-drag secular average. It also holds when the conservative parameter is $(1-\beta )G{M}_{\star }$, provided a bound orbit exists; the conservative parameter cancels but the drag amplitude remains $A$. For a circular Kepler orbit, $a^2(t)=a_0^2-4At$.

Consider a circular circumplanetary orbit well inside the Hill sphere, with dust speed $u^2=G{M}_p/a$ and unit plane normal $\hat{\mathbf{k}}$. During one dust orbital period, take the stellar direction $\hat{\mathbf{R}}$ and flux at the planet's stellar radius $a_p$ as constant. The velocity-dependent Poynting–Robertson drag has relative workThe constant terms have zero work over the closed circular orbit. Since $⟨u_i{u}_j⟩=(u^2/2)({\delta }_{ij}-k_i{k}_j)$,The coefficient is three for coplanar stellar and dust orbits; it is two for a dust plane normal to the instantaneous stellar direction. Averaging over a circular planetary year at fixed dust orbital inclination $I$ gives $3-{\sin }^{2}I/2$. Conservative perturbations must remain small enough for this circular-orbit average.

A power series is lacunary when only a sparse sequence of powers has nonzero coefficients. For ${\sum }_j{c}_j{z}^{m_j}$ with strictly increasing integer exponents, the radius of convergence satisfies $R^{-1}={\mathrm{limsup}}_j|c_j{|}^{1/m_j}$ by the Cauchy-Hadamard theorem. The exponent $m_j$, rather than the term index $j$, controls the coefficient root.

A potential vorticity budget has the formThe material derivative follows the fluid motion and $S_q$ collects forcing or dissipation. With $S_q=0$ it becomes potential-vorticity conservation. For a quasi-geostrophic streamfunction, horizontal advection is $J(\psi ,q)={\psi }_x{q}_y-{\psi }_y{q}_x$.

The spatial gradient of potential vorticity controls the anomaly created when a parcel is displaced. The linear potential-vorticity equation contains ${\mathbf{u}}^{\prime }\cdot \nabla \bar{q}$. A gradient of the Coriolis parameter contributes planetary vorticity, but gradients of relative vorticity and layer stretching can be equally important.

The set of real positive semidefinite matrices with unit diagonal. Equivalently, its members are Gram matrices of unit vectors, by the real spectral theorem. Every two-by-two principal minor gives $|X_{ij}|\le 1$, so the set is bounded; it is also closed, hence compact in finite dimension. This gives attainment of linear objectives used in semidefinite programming.

A real-linear contraction on $C(X)$ that preserves $1$ is positive: for $0\le f\le 1$, $\Vert 1-Tf{\Vert }_{\infty }\le 1$ forces $Tf\ge 0$, then scale. In the complex case, a unital norm-one functional is real on real functions, since $|1+it\varphi (g)|\le \Vert 1+itg{\Vert }_{\infty }=1+O(t^2)$ for both signs of $t$. The same positivity argument applies pointwise to $\varphi (f)=Tf(x)$. This turns a unital contraction semigroup into a Feller semigroup.

For a function $f$ and one-form $\alpha$, the codifferential obeys $\delta (f\alpha )=f\delta \alpha -⟨df,\alpha {⟩}_g$. This follows from the graded Leibniz rule and the definition of the Hodge star operator. Applying it to $d(fh)=fdh+hdf$ gives the product formula. In particular ${\Delta }_{+}(f^2)=2f{\Delta }_{+}f-2|df{|}_g^2$. The cross-term sign reverses for the opposite Laplace-Beltrami operator convention.

Constant actual volume input $Q$ into a porous gravity current with background flow gives early symmetric scales $h\sim [(Q/\varphi {)}^{2}t/a{]}^{1/3}$ and $x\sim [a(Q/\varphi )t^2{]}^{1/3}$. For $U>0$, drift dominates after $t_c=\varphi K_{b}Q/U^3$. The late interior depth is $Q/U$ and the steady upstream edge is $x_{-}=-K_{b}Q/U^2$, with upstream linear profile $h=U(x-x_{-})/K_b$. Upstream volume flux is zero because background drift balances gravity spreading.