A point measure is a countable sum of Dirac measures. It is simple if all its atoms have mass one, and locally finite if every bounded set has finite mass. Simplicity lets fine enough cells count individual points, whereas occupation indicators would lose multiplicities in a nonsimple measure.

The void probability functional assigns to a measurable set the probability that it contains no atom of a random point measure. For a Poisson random measure of intensity $\mu$, it is $e^{-\mu (A)}$. The exponential void probabilities characterize a Poisson random measure result uses simplicity; doubling every atom of a Poisson measure preserves void probabilities but changes count distributions.

The intensity measure of a point process $\Pi$ is its expected counting measure:It may take the value $+\infty$.^[ref] Countable additivity follows by applying the monotone convergence theorem to the counts of finite unions of pairwise disjoint observation windows. If $\mu$ is locally finite and absolutely continuous with respect to Lebesgue measure, its Radon-Nikodym derivative is the intensity function of a point process $\lambda$:This is an expected number per unit volume, not a probability density function for a randomly selected point. For example, a single deterministic point has $\mu ={\delta }_0$, a Dirac measure, and no Lebesgue measure density. The density representation requires its own assumption.^[ref]

For a bounded Borel set $W$, $\mathbb{E}[N_W]=\mu (W)$, but local finiteness alone does not ensure that this expected value is finite. Conversely, $\mu (W)<\infty$ implies $N_W<\infty$ almost surely, even when $W$ is unbounded. A constant finite intensity function of a point process on all of ${\mathbb{R}}^d$ has infinite total intensity measure, although every bounded window has finite expected value.

A Neyman-Scott process places independent offspring clusters around points of a Poisson parent process, with an independently drawn offspring count and displacements. Poisson offspring counts make the offspring process conditionally Poisson given the parents, and therefore give a Cox process. Arbitrary offspring counts need not have that property.

A point process on ${\mathbb{R}}^d$ is stationary if every translation preserves its probability law. If its mean count in a unit cube is finite, its intensity measure is $\mu (A)=\rho |A|$ for some $\rho \ge 0$, where $|A|$ denotes Lebesgue measure. This follows because stationarity makes the mean measure invariant under translations. A constant intensity function of a point process describes only these mean counts, and by itself asserts neither stationarity of the entire law nor independence.^[ref]

For a bounded Borel set $W\subset {\mathbb{R}}^d$, the window count $N_W$ of a locally finite point process is a finite nonnegative integer random variable. It counts multiplicities, and the convention for window boundaries matters. On the time half-line, $N(t)=\Pi ((0,t])$ is a counting process and $N(t)-N(s)=\Pi ((s,t])$. For pairwise disjoint observation windows $W_1,\dots ,W_m$, the measure identityis valid for every realization. Additivity of counts supplies no independence assumption.

A point process is simple when, on a single almost sure event, every location has multiplicity at most one. Its realizations are simple locally finite point measures. For instance, ${\delta }_{0.4}+2{\delta }_{1.1}+{\delta }_{1.8}$ is locally finite but not simple: it has four points counted with multiplicity at three distinct locations. Simplicity is a statement about each entire realization. Checking $\mathbb{P}(\Pi (\{x\})>1)=0$ separately for every fixed $x$ is insufficient: $2{\delta }_U$ with $U$ having the uniform distribution on $(0,1)$ passes every such fixed-location check, yet always has a double point.^[ref]

A rotation preserving a rank-two lattice has order $1,2,3,4$ or $6$. In a lattice basis its matrix is integral, while its trace is $2\cos \theta$. The only integral values in $[-2,2]$ are $-2,-1,0,1,2$, which give precisely those rotation orders. Finiteness of a symmetry group alone would not impose this restriction.

The boundary of a chosen Poincare coordinates on anti-de Sitter spacetime patch inside the full spacetime is a null coordinate horizon. It is not the timelike conformal boundary, and not a curvature singularity. Timelike geodesics can cross it in finite proper time while the patch time diverges. In a signed-diameter conformal strip, the patch containing $T=\rho =0$ satisfies $|T|<\pi /2+\rho$.

For unit-radius metric $d{y}^2+e^{-2y}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$, translations conserve $p^{\mu }=e^{-2y}{\dot{x}}^{\mu }$. Timelike normalization gives $K^2=-{\eta }_{\mu \nu }{p}^{\mu }{p}^{\nu }>0$ and ${\dot{y}}^2=K^2{e}^{2y}-1$. Setting $w=e^{-y}$ yields ${\dot{w}}^2+w^2=K^2$, hence the displayed proper-time solution where the cosine is positive. Its two patch-horizon crossings are separated by proper time $\pi$, although the globally extended geodesic is complete.

An astronomical conjunction occurs when two orbiting bodies have the same apparent longitude as viewed from their primary or from an observer.

A planetary dynamo maintains an intrinsic magnetic field by motion of an electrically conducting fluid. The resistive induction equation balances stretching and transport against magnetic diffusion. A sufficiently large magnetic Reynolds number is necessary but does not ensure that the flow regenerates every required field component. Thermal or compositional convection supplies mechanical power; rotation organizes the flow through the Coriolis force, and the Lorentz force density eventually modifies the flow and limits growth. A fluid core can fail to operate a dynamo if it is stably stratified or inadequately driven.

The innermost planet has a large metallic core and a weak but global intrinsic magnetic field. Its observed field supports an active planetary dynamo in a fluid conducting part of that core. Slow rotation does not by itself exclude dynamo action.

Two viscous layers occupy $-a<y<0$ and $0<y<a$, with viscosities ${\mu }_{-}$ and ${\mu }_{+}$. A uniform driving pressure gradient $G=-p_x>0$, a stationary flat interface and no-slip outer walls give the displayed plane Poiseuille flow, where $U_0=G{a}^2/({\mu }_{-}+{\mu }_{+})$ and $C=Ga({\mu }_{-}-{\mu }_{+})/[2({\mu }_{-}+{\mu }_{+})]$. These constants follow from continuity of velocity and shear stress at the interface and zero wall velocities. The upper-to-lower volume-flux ratio is $(1+7r)/[r(7+r)]$, with $r={\mu }_{+}/{\mu }_{-}$. As $r\to \infty$, the interface becomes effectively stationary, the upper flux vanishes, and the lower layer approaches flow between two stationary plates a distance $a$ apart. The upper wall stress remains finite despite its vanishing velocity gradient.

At large Reynolds number $\mathcal{R}$, a long indentation of relative height $ϵ$ has nonlinear near-wall interaction when the streamwise scale ratio satisfies $\lambda /\mathcal{R}={ϵ}^3$. The regime ${\mathcal{R}}^{1/7}\ll \lambda \ll \mathcal{R}$ gives a thin viscous layer and an effectively common pressure across the core. The core displacement is $u_1=A(x)U_0^{\prime }(y)$ and $v_1=-A^{\prime }(x)U_0(y)$. A lower wall displaced to $ϵF(x)$ produces layer displacement $F+A$, while the upper wall produces $-A$. Uniqueness for their common pressure forcing and upstream data makes the two layer displacements equal, so $A=-F/2$.

A steady planar viscous jet whose conserved momentum flux and convection-diffusion balance give the displayed scaling. Choosing $U{\delta }^2=\nu x$ and $U$ as centre-line speed gives streamfunction profile $f^{\prime \prime \prime }+(f{f}^{\prime \prime }+(f^{\prime }{)}^2)/3=0$, with $f(0)=f^{\prime \prime }(0)=0$, $f^{\prime }(0)=1$ and $f^{\prime }(\pm \infty )=0$. Integration yields $f=\sqrt{6}\tanh (\eta /\sqrt{6})$ and velocity profile $f^{\prime }={\mathrm{sech}}^2(\eta /\sqrt{6})$. Its momentum integral is $4\sqrt{6}/3$, fixing the dimensional prefactors. Transverse entrainment does not require $f(\infty )=0$.

For $d{s}^2=-2dudv+d{x}^2+d{y}^2-2H(u,x,y)d{u}^2$, the nonzero connections are ${\Gamma }_{uu}^v=H_u$, ${\Gamma }_{ux}^v=H_x$, ${\Gamma }_{uy}^v=H_y$, ${\Gamma }_{uu}^x=H_x$, and ${\Gamma }_{uu}^y=H_y$, with symmetric lower indices. Their traces and quadratic Ricci products vanish, leaving only the displayed Ricci component. Hence the vacuum equations require a harmonic transverse profile. The sign is tied to the metric's $-2Hd{u}^2$ convention.

Across an interface with unit normal $\mathbf{n}$ from medium one to medium two, Maxwell equations imply $\mathbf{n}\times ({\mathbf{E}}_2-{\mathbf{E}}_1)=0$, $\mathbf{n}\cdot ({\mathbf{B}}_2-{\mathbf{B}}_1)=0$, $\mathbf{n}\cdot ({\mathbf{D}}_2-{\mathbf{D}}_1)={\sigma }_f$, and $\mathbf{n}\times ({\mathbf{H}}_2-{\mathbf{H}}_1)={\mathbf{K}}_f$. Here ${\sigma }_f$ and ${\mathbf{K}}_f$ are the free surface electric charge and electric current densities.

A figure-eight plane curve with two congruent lobes and a crossing at the origin. With scale $a>0$, its polar equation is $r^2=a^2\cos 2\theta$. One quarter has coordinates $x=at\sqrt{(1+t^2)/2}$, $y=at\sqrt{(1-t^2)/2}$, $0\le t\le 1$. Differentiation gives arc element $adt/\sqrt{1-t^4}$. The total arc length is therefore $4a{\int }_0^{1}dt/\sqrt{1-t^4}$.

Reduced Planck units choose $\hslash =c=8\pi G=1$. With the reduced Planck mass defined by $M_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}$ after setting $\hslash =c=1$, this convention gives $M_{\mathrm{Pl}}=1$ and the Einstein-Hilbert action coefficient $M_{\mathrm{Pl}}^2/2=1/2$.

For a spatially flat Friedmann-Lemaître-Robertson-Walker metric, the Friedmann equation is then $3H^2=\rho$, where $\rho$ is the total energy density, including any vacuum contribution. With spatial-curvature parameter $k$, it is instead $3(H^2+k/a^2)=\rho$. Setting the unreduced Planck mass to one would set $G=1$, leaving the factor $8\pi$ in the gravitational equation. Keeping these conventions distinct prevents spurious factors in gravitational and inflationary calculations.