The Lyman-alpha line is the hydrogen $2p\leftrightarrow 1s$ electric-dipole transition, with photon energy about $10.2\mathrm{eV}$ and vacuum wavelength about $1216$ angstroms. Its emission cools collisionally excited gas, while absorption traces neutral hydrogen along a background-source sightline.

Ordinary baryons have three valence quarks, and ordinary mesons have a quark and an antiquark. Additive quantum numbers such as electric charge, baryon number and strangeness are sums of constituent values. This valence description organizes flavor symmetry multiplets while allowing sea constituents and gluons.

The quark model makes a large catalogue manageable: the same few quark flavors generate many different hadrons. It predicts relations between electric charge, strangeness and isospin, rather than assigning these independently to each new particle. It is especially useful for asking which combinations are possible and which additional measurements could distinguish them. A constituent list alone does not determine a mass, a decay rate or whether a proposed combination forms a bound state.

The study of convex sets, their convex hulls, supporting closed half-spaces, and inequalities for their sizes.

The ingoing Vaidya metric describes a spherically symmetric spacetime sourced by radial null matter, using advanced time $v$ and areal radius $r$. In geometrized units it is $d{s}^2=-(1-2M(v)/r)d{v}^2+2dvdr+r^{2}d{\Omega }^2$. A constant mass function reduces it to the Schwarzschild metric in ingoing null coordinates; a varying mass function changes the curvature and the radial null geodesics.

Geometrized units set $G=c=1$, allowing mass, length, and time to be expressed in common units. Restore $GM/c^2$ for a mass parameter in a radial metric tensor.

In a spherically symmetric spacetime, the areal radius $r$ is defined by the area $A=4\pi r^2$ of each symmetry sphere. It is independent of which radial coordinate chart is used.

A coordinate singularity is a failure of a coordinate description which disappears in a regular chart. The horizon singularity of the Schwarzschild metric in $(t,r)$ coordinates is removable; its $r=0$ curvature singularity is not.

A timelike geodesic has timelike tangent and vanishing covariant acceleration in an affine parameter. Choosing proper time gives $g(\dot{\gamma },\dot{\gamma })=-1$.

Physically, an ideal massive test particle in free fall follows a timelike geodesic when nongravitational forces and appreciable self-gravity can be neglected. The normalization above uses $c=1$; in SI units it is $g(u,u)=-c^2$. Zero covariant acceleration does not mean constant coordinate velocity: a dropped object accelerates relative to the ground. Conversely, a person standing on the ground is pushed away from free fall by the supporting surface. A photon follows a null geodesic, whose parameter cannot be proper time. See Tong's geodesics in spacetime.

Transforming a metric tensor and all matter fields together by the pullback of a diffeomorphism preserves the Einstein field equations and covariant matter equations. Passive coordinate changes only change the description of the fields; active pullbacks produce another representative of the geometry. The treatment of transformations acting on a boundary depends on the boundary conditions and possible charges. A diffeomorphism is not necessarily an isometry of a fixed metric.

A binary string is a finite word over the alphabet $\{0,1\}$. There are $2^n$ strings of length $n$. Finite binary strings have an effective bijection with natural numbers, so computable enumerability and immunity apply to sets of strings.

A fluid flow is motion described by a velocity field $\mathbf{u}(\mathbf{x},t)$. The trajectories of fluid elements satisfy $d\mathbf{x}/dt=\mathbf{u}(\mathbf{x},t)$; the mass density and other fields obey their conservation laws and constitutive equations. This physical motion differs from a flow on a flow network.

Scalar transport follows a material quantity through advection and diffusion equation terms, for example ${\Theta }_t+\mathbf{U}\cdot \nabla \Theta =\kappa \Delta \Theta$. A passive scalar does not affect the velocity field; an active scalar feeds back through forces such as buoyancy. With incompressible flow bounded by impermeable walls and homogeneous Neumann boundary conditions, the spatial mean is conserved.

A material curve consists of the same fluid elements at all times. Differentiating their flow map gives $D\mathbf{\ell }/Dt=(\mathbf{\ell }\cdot \nabla )\mathbf{u}$ for its tangent. In ideal magnetohydrodynamics a magnetic-field tangent obeys this same transport law after division by mass density.

A material surface is transported with its fluid elements, including its boundary when open. Its oriented material surface element evolves by tangent transport. Magnetic flux freezing conserves the flux through such an open surface.

A fluid free boundary is an initially unknown material interface separating a fluid from its exterior. For a polytropic elliptical patch in a shearing sheet bounded by vacuum, it is the level set $p=\rho =Q=0$, and the velocity is tangent to it in a steady solution.

The Navier slip boundary condition allows a fluid to slide tangentially past a solid wall while resisting that motion through wall friction in the Navier slip boundary condition. For parallel flow past a flat, impermeable wall translating uniformly with tangential velocity $U_w$, write the fluid's tangential velocity as $u$ and its relative tangential velocity as $u_{\mathrm{rel}}=u-U_w$. Thenwhere $b$ is the slip length, also written ${\ell }_s$. The unit normal $\mathbf{n}$ points from the wall into the fluid, and ${\partial }_{n}u=\mathbf{n}\cdot \nabla u$ is the normal derivative. The separate no-penetration boundary condition requires $(\mathbf{u}-{\mathbf{U}}_w)\cdot \mathbf{n}=0$: tangential sliding does not allow flow through the wall.

The sign convention matters. In a channel $0<z<H$, the lower wall has inward unit normal $+{\mathbf{e}}_z$ and the upper wall has inward unit normal $-{\mathbf{e}}_z$. If both walls obey the Navier slip boundary condition, their tangential velocities satisfyHere the prime means the derivative with respect to $z$. Using $+b_H{u}^{\prime }(H)$ at the upper wall would reverse the friction law. The examples Couette flow with a slipping lower wall and planar Poiseuille flow with equal wall slip use different upper-wall boundary conditions.

This flat-wall formula is a local linear model. On a curved wall the Newtonian fluid stress tensor must supply the tangential traction; the normal derivative alone generally misses curvature terms. The author-written review by Lauga, Brenner and Stone states the full strain-based condition with the unit normal directed into the liquid. ^[ref]

The same review credits Navier's 1823 treatise with the linear slip law. Its history is a reminder that the no-slip boundary condition is an experimentally successful model, whose validity depends on the interface and scale. The history here follows that review rather than a new reading of Navier's original memoir. ^[ref]

A control volume is a chosen region over which fluid mass conservation and momentum balance are integrated. It may be fixed or move with a prescribed velocity. Boundary fluxes must use velocity relative to the moving boundary. In Stokes flow, negligible inertia reduces the momentum balance to the sum of body forces and boundary tractions, making a carefully chosen control volume useful for finding drag without integrating directly over a curved body.

Phase locking maintains a fixed relation between the phases of oscillations, or between an oscillation and periodic forcing. For a one-to-one locked pair, ${\theta }_2-{\theta }_1$ is constant; more general locking can fix an integer combination $m{\theta }_1-n{\theta }_2$. Locking concerns the relative phases, so the individual oscillations can continue indefinitely.

For two equally coupled phase oscillators in the Kuramoto model, the difference satisfies $\dot{\delta }=\Delta \omega -K\sin \delta$. A stable locked value has $\sin \delta =\Delta \omega /K$ and $\cos \delta >0$ when $|\Delta \omega |<K$. The uncoupled frequencies need not coincide: coupling shifts both observed frequencies to the same value. Outside the locking range the difference keeps winding.

Spatial forcing can likewise pin the phase of a pattern. For the real-coefficient conjugately forced amplitude equationputting $A=\rho e^{i\varphi }$ with $\rho >0$ gives $\dot{\rho }=(\mu +\beta \cos 2\varphi -\lambda {\rho }^2)\rho$ and $\dot{\varphi }=-\beta \sin 2\varphi$. The phase-stable values are $0,\pi$ for $\beta >0$ and $\pi /2,3\pi /2$ for $\beta <0$. A physical nonzero branch also needs a positive squared amplitude $(\mu +\beta \cos 2\varphi )/\lambda$ when $\lambda \ne 0$; radial stability is a separate check. At $\beta =0$ the phase is neutral and this pinning disappears.

A heteroclinic orbit is a trajectory tending to one invariant equilibrium or orbit in backward time and a different one in forward time. Coupled convection amplitudes can have such trajectories describing replacement of one convection roll orientation by another.

The Routh-Hurwitz criterion tests whether all roots of a real polynomial have negative real part, by sign conditions on determinants formed from its coefficients. Applied to a characteristic polynomial, it gives an algebraic test for strict linear asymptotic stability.