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Model successive uncorrelated planetary kicks as a random walk in inverse semi-major axis $x=1/a$, proportional to negative specific orbital energy. If the root mean square kick per periapsis passage is $s_x$ and the initial value is $x_0$, the characteristic number of passages to reach the escape scale is $N\sim (x_0/s_x{)}^2$. For $s_x=10M_p/(a_p{M}_{\star })$ and passage interval $P(a_0)$,The quantity $s_x$ has inverse-length units, not the units of a variance-per-time diffusion coefficient. With $⟨(\Delta x{)}^2⟩=2D_{x}t$, that coefficient is $D_x=s_x^2/(2P)$. This estimate assumes many independent encounters; changing orbital periods and resonant correlations can invalidate the clock. It is not an exact mean first-passage time.

The stellar obliquity is the angle between a star's spin and a planet's orbital angular momentum. The Rossiter-McLaughlin effect constrains the angle $\lambda$ between the sky projections of those two axes, rather than a component of $\psi$. If the orbital and stellar-spin inclinations to the line of sight are $i_{\mathrm{orb}}$ and $i_{\ast }$, then $\cos \psi =\cos i_{\ast }\cos i_{\mathrm{orb}}+\sin i_{\ast }\sin i_{\mathrm{orb}}\cos \lambda$; $\lambda$ alone generally does not determine $\psi$. Large misalignments constrain formation and planetary migration, but can also reflect a tilted birth disc.

For a plane internal gravity wave with vertical displacement $\zeta$, linear density advection gives ${\rho }^{\prime }=-{\bar{\rho }}_z\zeta$. In a constant stable gradient, ${\rho }_z={\bar{\rho }}_z(1-{\zeta }_z)$, so static overturning begins when the maximum of ${\zeta }_z$ exceeds one. A monochromatic displacement has maximum $|m{A}_{\zeta }|$. The criterion predicts the failure of the small-amplitude field, rather than a valid continuation of linear theory beyond overturning.

Blow up the origin and use the chart $x=uy$, giving the strict transform $u^2-y^3=0$. Blow up its remaining singular point and use $u=vy$, giving $v^2-y=0$, which is smooth because its derivative in $y$ is $-1$. The complementary chart at each stage has no point of the strict transform on its exceptional divisor. These two blowups of a smooth algebraic surface resolve the branch, although the smooth transform is still tangent to an exceptional divisor.

For a fixed finite plane graph, the complementary dual configuration opens a dual edge exactly when its primal edge is closed. Its components correspond to faces of the spanning open primal graph. The componentwise Euler formula for a connected planar graph gives the displayed identity. The outer face and isolated primal vertices are included; the identity also handles disconnected graphs, bridges and loops.

A binary string is incompressible relative to a fixed optimal description machine when its plain Kolmogorov complexity is at least its length. A counting argument gives at least one such string at each length, and their set is immune.

The pion decay constant gives $⟨0|J_A^{\mu }|\pi (p)⟩\propto f_{\pi }{p}^{\mu }$. Contracting the momentum with the leptonic weak charged current and applying the external Dirac equations makes the amplitude proportional to the charged-lepton mass. This is an explicit example of helicity suppression.

At relative rapidity $iu$, the analytically continued sum of the constituent four-momentum vectors is on the bound-state mass shell with the displayed mass. For equal constituents, rapidities $\chi \pm iu/2$ sum to $2m_a\cos (u/2)(\cosh \chi ,\sinh \chi )$. A pole at $u=0$ is at threshold rather than a strictly bound state.

Let $⟨F⟩=(4\pi {)}^{-1}\int Fd\Omega$ and $f=\bar{f}(ϵ)-ϵ{\bar{f}}^{\prime }(ϵ)\Theta (\mathbf{e})$ with comoving energy $ϵ=aE$. If ${ϵ}^4\bar{f}$ vanishes at both integration endpoints, integration by parts gives $-\int {ϵ}^4{\bar{f}}^{\prime }dϵ=4\int {ϵ}^3\bar{f}dϵ$. The kinetic stress-energy tensor then gives ${\delta }_{\gamma }=4⟨\Theta ⟩$, $v_{\gamma }^i=3⟨\Theta e^i⟩$, $\delta P=\bar{\rho }{\delta }_{\gamma }/3$, and ${\pi }^{ij}=4\bar{\rho }⟨\Theta (e^i{e}^j-{\delta }^{ij}/3)⟩$. If $\Pi =-\pi$ is used, the last expression changes sign. Frequency independence of $\Theta$ is essential to this common temperature description.

For a fixed angular order $m$, write $\Theta ={\sum }_{\ell }{C}_{\ell }{q}_{\ell }{Y}_{\ell m}$. Streaming by $ik\mu$ contributes $ik(C_{\ell +1}/C_{\ell })a_{\ell +1,m}{q}_{\ell +1}+ik(C_{\ell -1}/C_{\ell })a_{\ell ,m}{q}_{\ell -1}$. Thus rescaling the expansion coefficients also rescales the hierarchy couplings and source amplitudes. Replacing $C_{\ell }=(-i{)}^{\ell }\sqrt{(2\ell +1)/(8\pi )}$ by ${\tilde{C}}_{\ell }=(-i{)}^{\ell }\sqrt{2\pi /(2\ell +1)}$ requires ${\tilde{q}}_{\ell }=(2\ell +1)q_{\ell }/(4\pi )$; the temperature field is unchanged. This prevents mixing incompatible photon temperature multipole conventions.

Time-ordering the canonical transverse photon field gives a propagator with numerator equal to the spatial transverse projector and denominator $k^2+i0$. Its two energy poles encode forward propagation of the positive-frequency term and backward propagation of the negative-frequency term. This is the reduced free-field propagator; it is distinct from the covariant four-potential propagator.

In an orthonormal tetrad, $J^{\hat{0}}=\int f{d}^{3}p$ is the photon number density and $J^{\hat{i}}=\int f{e}^i{d}^{3}p$ is its number flux. Integrating the Boltzmann equation gives ${\nabla }_{\mu }{J}^{\mu }=\int C[f]d^{3}p/E$ when momentum-boundary terms vanish. Collisionless propagation and photon-number-conserving scattering give zero divergence. Photon emission, photon absorption, electron-positron annihilation in cosmology, bremsstrahlung and double Compton scattering generally do not. This is a number current, not an electric charge current: photons are neutral.

The inertia ratio of the photon-baryon fluid is $R=3{\bar{\rho }}_b/(4{\bar{\rho }}_{\gamma })$. Photon momentum is weighted by the radiation enthalpy $4{\bar{\rho }}_{\gamma }/3$, so this ratio controls momentum exchange, the common sound speed and the offset of the acoustic oscillator. Background conservation gives $R\propto a$ and $\dot{R}=\mathcal{H}R$.

For the velocity convention $v_{\gamma }=-{\Theta }_1$ and $\dot{\tau }<0$, momentum-conserving Thomson scattering gives ${\dot{v}}_b+\mathcal{H}{v}_b+k\psi =(\dot{\tau }/R)(v_b-v_{\gamma })$. Here $R$ is the baryon loading parameter. Its collision force is opposite to the photon force when both are weighted by their enthalpies.

With interaction $\lambda {\varphi }^4/4!$ for one real scalar, the three one-loop four-point bubble channels give the four-dimensional renormalization-group beta function $\beta (\lambda )=3{\lambda }^2/(16{\pi }^2)+O({\lambda }^3)$. Its positive sign for $\lambda >0$ excludes asymptotic freedom and produces a perturbative Landau pole on extrapolation.

For $V(\varphi )=\lambda (m^2-{\varphi }^2{)}^2$ with $\lambda ,m>0$, set $\kappa =\sqrt{2\lambda }m$. A well-separated kink–antikink pair at $-d/2,d/2$ is approximated by $\varphi =m\tanh \kappa (x+d/2)-m\tanh \kappa (x-d/2)-m$. At its midpoint, ${\varphi }_x=0$ and $\varphi =m-4m{e}^{-\kappa d}+O(e^{-2\kappa d})$. The static stress-energy tensor component $T^{11}={\varphi }_x^2/2-V$ is therefore negative there. Momentum conservation gives the force on the left half-line as $T^{11}(-\infty )-T^{11}(0)$, which is positive. Its leading magnitude is $F\sim 32m^2{\kappa }^2{e}^{-2\kappa d}$. Thus the pair attracts. This is a controlled leading-tail estimate at large separation, not an exact superposed solution or a theorem about all subsequent collision outcomes.

The two-vertex cubic bubble in $d=6-ϵ$ dimensions has the displayed amputated pole. A Feynman parameter gives mass $m^2+x(1-x)k^2$, and $\Gamma (-1+ϵ/2)$ supplies the pole. The full propagator receives $D_0^{2}I$, while its inverse receives $-I$. This sign distinguishes a two-point insertion from an inverse-propagator correction.

A noise-driven barrier crossing between adjacent phase-locked wells. The Kramers escape rate is proportional to $e^{-\Delta V/T_{\mathrm{eff}}}$ when a mobility-one Langevin equation has white-noise covariance $2T_{\mathrm{eff}}\delta (t-t^{\prime })$. Both the barrier and local relaxation time determine whether an intrawell phase autocorrelation approximation is appropriate.

A pure thermodynamic phase is an extremal equilibrium Gibbs state rather than a probabilistic mixture of distinct macroscopic phases. On an ordered Ising branch, taking the thermodynamic limit before removing a selecting conjugate field gives a nonzero magnetization. A symmetric mixture can instead have zero mean but a nondecaying two-point contribution. The connected correlation function and correlation length describing ordinary critical fluctuations should be evaluated in a chosen pure phase when this distinction matters.

Suppose a finite group acts freely on $S^{n-1}$ and trivially on its integral cohomology. The Borel construction gives a sphere fibration over $BG$ whose total space is homotopy equivalent to the $(n-1)$-dimensional quotient. Its Serre spectral sequence has just two rows with trivial local coefficients. Transgressing the fibre generator gives $\Delta$; the sole differential is multiplication by it up to sign. For $i>0$, both the source and target terms have total degrees above $n-1$, so their surviving kernel and cokernel must vanish. Hence multiplication by $\Delta$ gives period $n$ in positive-degree cohomology over any field.