For finite sets of points $P$ and geometric curves $\mathcal{C}$, a point-curve incidence is a pair $(p,\gamma )$ with $p\in \gamma$. The incidence count is $I(P,\mathcal{C})={\sum }_{\gamma \in \mathcal{C}}|P\cap \gamma |$. Bounds in incidence geometry exploit restrictions on how curves can meet points, rather than treating the incidence relation as an arbitrary bipartite graph.

For a finite planar point set $P$, its distinct-distance set is $\Delta (P)=\{|p-q|:p,q\in P\}$. It includes zero for nonempty $P$. Its cardinality measures how many different distances the configuration determines. Ordered point pairs can be grouped according to their distance, which relates the problem to incidences between points and curves.

Relative to a finite point set, an ordinary line is a line containing exactly two points of that set. It is an exact-richness condition, not a count of unordered pairs alone: a line with three points contains three point pairs but is not ordinary. Point-line duality turns ordinary lines into intersections incident to exactly two dual lines.

The Gaussian Bunch-Davies vacuum is selected by tilting the early endpoint to $-\infty (1-i\epsilon )$ for integrands $e^{iK\tau }$ with $K>0$. They then decay at the early boundary. The complex-conjugate integrand requires the opposite tilt. Ordered contractions and their contour must be conjugated together.

With ${\zeta }_k(\tau )=u_k(\tau )a_k+u_k^{\ast }(\tau )a_{-k}^{†}$, the operator order external field then vertex gives $⟨{\zeta }_k(0){\zeta }_p^{\prime }(\tau )⟩=(2\pi {)}^3\delta (k+p)u_k(0)u_p^{\prime }{}^{\ast }(\tau )$. For positive-frequency $u\propto e^{-ik\tau }$, these vertex contractions contain $e^{+iK\tau }$ and converge on ${\tau }_i=-\infty (1-i0)$. Conjugating the contraction representation also conjugates the outside in-in prefactor. Mixing the two changes signs and defeats the vacuum prescription.

Subtract a smoothed image signal and add a scaled residual: $u=g+\lambda (g-G_t\ast g)$. The Fourier multiplier $1+\lambda (1-e^{-t|\xi {|}^2})$ is bounded by $1+\lambda$. This enhances apparent contrast but also amplifies noise; it is not a stable exact inverse of heat image smoothing.

The formal shock-filter equation $u_t=-\mathrm{sign}(\Delta u)|\nabla u|$ steepens transitions around inflection boundaries. It is a Hamilton–Jacobi-type transport mechanism rather than positive diffusion. Combining it with controlled forward image smoothing limits noise amplification.

The Beltrami-Klein model realizes the hyperbolic plane inside an open Euclidean disk, with geodesics given by straight chords. Its metric differs from the Euclidean metric, and angles are generally distorted. The straight chords make geodesic convexity especially easy to see.

An absolute hydrodynamic instability makes the response to a localized impulse grow at a fixed spatial point in a specified frame. This differs from convective hydrodynamic instability, which amplifies a moving wave packet while its response decays at fixed positions. For analytic dispersion relations, the relevant absolute frequency is associated with an accessible zero-group velocity saddle, with the spatial-branch selection checked rather than assumed for an arbitrary dispersion relation.

A convective hydrodynamic instability amplifies an advected localized disturbance while its amplitude decays at any fixed position. The distinction from absolute hydrodynamic instability depends on the observation frame.

On a ray $x/t=V$, a localized wave packet is governed by a physically selected saddle point satisfying $d\omega /dk=V$ in its analytic dispersion relation. The exponent's real growth is $\mathrm{Im}({\omega }_{\star }-V{k}_{\star })$. The fixed-frame case $V=0$ gives the absolute growth rate. This ray formulation makes the frame dependence of convective hydrodynamic instability explicit.

For the linear complex Ginzburg-Landau equation with real $U,c_d,\mu$ and positive real diffusion coefficient scaled to one, the maximum temporal growth rate is $\mu$ and the absolute growth rate is $\mu -U^2/[4(1+c_d^2)]$. For $U>0$, $\mu <0$ is stable, $0<\mu <{\mu }_a$ has convective hydrodynamic instability, and $\mu >{\mu }_a$ has absolute hydrodynamic instability. Both equality boundaries are marginal in exponential rate.

For a constant reference density, fluid perturbation kinetic energy is proportional to $\Vert u{\Vert }^2/2$ under the displayed $L^2$ inner product. An adjoint operator for perturbation evolution must be defined relative to this inner product, including boundary conditions and the projection onto incompressible flow.

Relative to the kinetic-energy inner product, integration by parts reverses incompressible base transport, transposes the velocity-gradient coupling, and preserves viscous diffusion. With reverse time $\tau =-t$, the adjoint base state is evaluated at $-\tau$. The Leray-Helmholtz projection or adjoint pressure enforces divergence-free adjoint velocity. The curl of a cross product and vorticity cross-product identity give the equivalent form ${\partial }_{\tau }v=\Omega \times v-\nabla \times (U\times v)-\nabla \pi +{\mathrm{Re}}^{-1}\Delta v$.

For an incompressible parallel flow with base velocity $U(y)$ and perturbation proportional to $e^{i\alpha (x-ct)}$, divergence of the linearized momentum equations gives $p\prime \prime -{\alpha }^{2}p=-2i\alpha U\prime v$. At a no-slip boundary condition, streamwise momentum gives $p(0)=u\prime \prime (0)/(i\alpha R)$. The equation encodes how base shear generates perturbation fluid pressure.

A high-Reynolds number long-wave mode near a wall has a viscous layer of thickness $\delta =(\alpha R{)}^{-1/3}$, an order-one region following the base profile, and a pressure-decay region of thickness $1/\alpha$. The lower Airy function solution fixes velocity displacement and fluid pressure. The middle Rayleigh equation for inviscid shear flow transports these to the outer constant-velocity region. Matching its decaying pressure produces an Airy wall-layer matching determinant. For order-one amplitudes, constant leading middle pressure requires ${\alpha }^2\ll \delta$; the distinguished balance $\alpha \sim \delta$ gives $\alpha \sim R^{-1/4}$.

Baroclinic instability converts potential energy of a stratified sheared basic flow into growing disturbances. In the Eady model, coupled Boundary Rossby waves provide a useful mechanism: their induced velocities displace boundary buoyancy anomalies, and suitable wavelengths permit phase locking. A single neutral boundary wave does not by itself imply exponential instability.

For a wave with zonal phase velocity $c$ in a basic shear $U(z)$, a critical level is a height $z_c$ at which $U(z_c)=c$. The Doppler-shifted material frequency then vanishes. This means equality of the laboratory velocities; the $U$ and $-c$ terms cancel in the wave frame. It does not mean $U=-c$. A semi-infinite Eady edge wave with zero interior potential vorticity anomaly can remain regular through its steering height.

A critical layer is a region where the leading wave phase speed coincides with the local shear-flow velocity and an outer approximation degenerates. Small viscosity or other corrections must then be retained on a distinguished transverse scale. The width depends on the order of vanishing of $U-c_0$ and on the governing mode equation; simple-shear and stationary-profile layers have different balances.

The astronomical horizon is the great circle where the horizontal plane through the observer intersects the celestial sphere. It is perpendicular to the local vertical, so every direction on the astronomical horizon has altitude $0^{\circ }$ and zenith distance $90^{\circ }$. In the local horizontal coordinate system, it consists of the unit vectors satisfying $\mathbf{n}\cdot {\mathbf{e}}_U=0$. This geometric reference remains defined when buildings or mountains hide it; a visible skyline is not a calibration of zero altitude.

Three boundaries must be distinguished. The astronomical horizon fixes the angular reference. Terrain supplies a site-dependent boundary with apparent altitude $t(\eta )$ that changes with azimuth. Atmospheric refraction changes the apparent direction of incoming light, and can also change the apparent distant boundary. A point source clears an opaque terrain boundary only if ${\gamma }_{\mathrm{app}}>t(\eta )$ when both are expressed in the same apparent horizontal coordinate system; this clearance condition does not guarantee visibility through clouds or bright sky. In the synthetic diagram, a source with geometric altitude $3^{\circ }$ and an imposed illustrative upward correction $R=0.2^{\circ }$ remains behind a $4^{\circ }$ boundary. The chosen $R$ is an algebraic illustration, not a fitted atmospheric prediction.

Even an unobstructed sea boundary need not coincide with the astronomical horizon. On a spherical Earth of radius $r$, an observer at height $q$ sees the unrefracted tangent to the surface at a downward angle $D$ satisfying $\cos D=r/(r+q)$. The right-angle tangent triangle supplies this relation; the visible boundary then has altitude $-D$. With illustrative $r=6371\mathrm{km}$ and $q=10\mathrm{m}$, $D\simeq 0.1015^{\circ }$, before atmospheric refraction. Raising the eye therefore changes the visible boundary without redefining the horizontal plane.

The US Naval Observatory's rise/set definitions explicitly distinguish the geometric reference from apparent contact with a visible disk: the standard Sun calculation puts its centre $50^{\prime }$ below the horizontal plane, combining an average $16^{\prime }$ radius with $34^{\prime }$ of atmospheric refraction. An angular minute is $1^{\prime }=1/60^{\circ }$ in the minute and second of arc convention. This is an operational average under stated conditions, rather than a universal zero of altitude. The same source identifies weather, local topography and observer height as limits on predicted rise/set time. An observation log should therefore retain the level reference, site and eye height, terrain profile, observation time, and whether each reported altitude is geometric or apparent. ^[ref].