A categorical covariate can be grouped in an interaction term while retaining its original main effect. This shares selected slopes but keeps separate intercepts. Nested Poisson deviance differences test slope pooling and intercept pooling separately.

The conformal map $w=z^2$ sends $x,y>0$ to the complex upper half-plane. For bounded continuous boundary values $g_2$ on the horizontal axis and $g_1$ on the vertical axis with equal corner values, set $f(s)=g_2(\sqrt{s})$ for $s>0$ and $f(s)=g_1(\sqrt{-s})$ for $s<0$. The bounded harmonic function with those Dirichlet boundary conditions isEquivalently, if $z^2=a+ic$ with $c>0$, this is $\frac{2c}{\pi }{\int }_0^{\infty }r\left[g_2(r)/((r^2-a{)}^2+c^2)+g_1(r)/((r^2+a{)}^2+c^2)\right]dr$. The Poisson kernel is a positive approximate identity, so the prescribed edge values are attained. Boundedness is essential for uniqueness on this unbounded domain.

Given $\Theta$, independent observations $Y_j$ have Poisson distribution with means $m_j\Theta$ for known exposures. A gamma distribution prior of shape $\alpha$ and rate $\beta$ yields a gamma posterior with shape $\alpha +{\sum }_j{Y}_j$ and rate $\beta +{\sum }_j{m}_j$. The likelihood kernel is ${\theta }^{{\sum }_j{Y}_j}{e}^{-\theta {\sum }_j{m}_j}$. Thus the posterior mean depends on total claims and total exposure, not the number of periods alone.

For $y>0$ and $\gamma >0$, minimizing $|x-z{|}^2/2+\gamma [y\log (y/x)+x-y]$ on $x>0$ gives $x^2+(\gamma -z)x-\gamma y=0$. The roots have opposite signs, so only the displayed positive root belongs to the effective domain. Strict positivity of the second derivative proves uniqueness. For an integral Poisson data fidelity, the proximal operator separates pointwise under the appropriate integrability assumptions.

The Jacobi identity for a Poisson bivector reduces to the displayed cyclic identity in every local chart. Necessity follows by applying the bracket to coordinate functions. Sufficiency follows because the second-derivative terms in an arbitrary Jacobiator cancel by antisymmetry, leaving this coefficient multiplying first derivatives of the three functions.

A nondegenerate Poisson bivector inverts to a closed differential form of degree two. Differentiating $S=P^{-1}$ gives ${\partial }_{d}S=-S({\partial }_{d}P)S$. Contracting the coordinate Jacobi condition for a Poisson bivector with three copies of $S$ yields ${\partial }_i{S}_{jk}+{\partial }_j{S}_{ki}+{\partial }_k{S}_{ij}=0$, which is precisely closure of the resulting symplectic form. An overall sign in the inverse convention does not change closure.

For a smooth differential one-form $\alpha ={\alpha }_i(x)d{x}^i$ with $d\alpha =0$, take a manifold chart identifying a neighborhood with a ball centered at zero, and set $\Phi (x)={\int }_0^1{x}^i{\alpha }_i(sx)ds$. Closedness means ${\partial }_j{\alpha }_i={\partial }_i{\alpha }_j$. ConsequentlyThus $d\Phi =\alpha$ in any dimension. This is the one-form case of the Poincaré lemma, and applies to spacetime as well as three-dimensional vector calculus. A global potential need not exist on a domain with nonzero periods.

On a star-shaped planar domain containing zero, let $\eta$ be a smooth closed differential one-form and set $f(x)={\int }_{[0,x]}\eta$. Green theorem on the triangle with vertices $0,x,x+h$ gives $f(x+h)-f(x)={\int }_{[x,x+h]}\eta$. Dividing by a displacement and taking its limit proves $df=\eta$. Thus every such form is an exact differential form, a degree-one version of the Poincaré lemma which uses only the planar integral theorem.

On a star-shaped domain, let ${\rho }_t(x)=tx$ and $R=\sum x_j{\partial }_{x_j}$. On positive-degree differential forms, $H\alpha ={\int }_0^1{\rho }_t^{\ast }({\iota }_R\alpha )dt/t$ satisfies $dH+Hd=\mathrm{id}-{\rho }_0^{\ast }$. It gives a primitive of every closed positive-degree form. On a star-shaped domain in a complex vector space, contraction with the two type components of $R$ shows that $H$ sends type $(p,q)$ into $(p-1,q)\oplus (p,q-1)$. There is no primitive assertion for a nonzero constant function.

Normalize volume, momentum and buoyancy by a positive plume similarity in power-law stratification and use $\xi =\ln (z/z_{\ast })$. The normalized equations are ${\hat{Q}}_{\xi }=q(\sqrt{\hat{M}}-\hat{Q})$, ${\hat{M}}_{\xi }=m(\hat{F}\hat{Q}/\hat{M}-\hat{M})$ and ${\hat{F}}_{\xi }=f(\hat{Q}-\hat{F})$. They form an autonomous differential equation.

The transition above a planetary troposphere, usually near a temperature minimum or the change to a weakly varying or outward-increasing temperature profile. The overlying planetary stratosphere is stable against ordinary thermal convection. Its pressure is planet- and location-dependent.

A low-mean molecular weight hydrogen-helium envelope can add a large radius while adding a small fraction of the mass. The exoplanet transit photometry radius is the altitude where tangential optical depth becomes order unity, so temperature, opacity, and exoplanet clouds can move the measured radius above the deep material radius. Atmospheric loss and cooling change this contribution over time.

A prograde, equatorial circular orbit with orbital period equal to the planet's spin period. In the ideal model its satellite remains above one equatorial ground location. Nonzero orbital inclination or orbital eccentricity destroys exact stationarity even when the periods agree.

The energy of the residual flow and magnetic field in a maintained shear flow has source $S=-a{T}_{xy}$, where $T_{xy}=\rho v_x{v}_y-B_x{B}_y/{\mu }_0$. The kinetic and magnetic stresses can exchange energy with the imposed background. The source vanishes when their off-diagonal contributions cancel.

At $w^2=v_{az}^2$, the transverse steady equations require $w{B}_x{w}^{\prime }=0$ and $w{B}_y{w}^{\prime }=a(w{B}_x-B_z{v}_x)$. These conditions can be lost by division when eliminating transverse magnetic derivatives. A scalar equation displaying the magnetosonic critical speeds need not display every compatibility condition at an Alfvén speed.

Let fluid 1 lie above fluid 2, with mass densities ${\rho }_1<{\rho }_2$ and dynamic viscosities ${\mu }_1<{\mu }_2$. Define $\Delta \rho ={\rho }_2-{\rho }_1>0$, $\Delta \mu ={\mu }_2-{\mu }_1>0$, porosity $\varphi$, permeability of a porous medium $k$, and downward interface speed $W$. With Darcy flux $\varphi W$, no capillarity, and two unbounded layers, Darcy's law givesFor a Fourier mode of interface displacement, the two perturbation pressures decay exponentially into the respective layers. The kinematic boundary condition makes their amplitudes proportional to $\varphi {\mu }_1\sigma /(k|\alpha |)$ and $-\varphi {\mu }_2\sigma /(k|\alpha |)$. Linearized pressure continuity adds the base-gradient difference $\varphi W\Delta \mu /k-g\Delta \rho$. Eliminating the amplitudes proves the formula. The critical interface speed is $kg\Delta \rho /(\varphi \Delta \mu )$; the critical Darcy velocity is $kg\Delta \rho /\Delta \mu$. Without surface tension this model has unbounded short-wave growth rate on its unstable side.

For the lattice rectangle with graph vertices $\{0,\dots ,n\}\times \{0,\dots ,n-1\}$, let $H_n$ be an open left-to-right crossing. Its complementary event is a closed dual top-to-bottom crossing. The dual crossing rectangle has width $n-1$ and height $n$, so rotation and translation identify it with the original rectangle. Boundary edges along the starting and ending sides do not affect the crossing event. At $p=1/2$ the primal and dual edge states have identical laws, henceThe one-unit adjustment in the side lengths makes the symmetry exact rather than an informal assertion about a finite graph vertex square.

For independent coordinate Brownian motions, the stochastic area with the stated orientation is $Z_t={\int }_0^t{Y}_{s}d{X}_s-X_{s}d{Y}_s$. Its bracket is ${\int }_0^t(X_s^2+Y_s^2)ds$. Reversing orientation changes its sign but not its law or quadratic variation.

For a small perturbation of a stationary horizon, the linearized Null Raychaudhuri equation gives $\delta {\theta }^{\prime }=-8\pi \delta T_{kk}$. With vanishing boundary term $[\lambda \delta \theta ]$, integration by parts gives $\delta A=8\pi \int \lambda \delta T_{kk}d\lambda d{A}_0$. Combining this with affine horizon-generator scaling converts area response to horizon Killing-energy flux.

In Newtonian gauge in cosmology, the Free-streaming photon Boltzmann equation is $\dot{\Theta }+\mathbf{e}\cdot \nabla \Theta =\dot{\varphi }-\mathbf{e}\cdot \nabla \psi$. Angular averaging and the photon angular temperature moments give the displayed continuity equation. Its dipole yields the photon Euler equation, with the sign of the anisotropic stress term determined by the stress convention.