The Knudsen number is the dimensionless ratio of an intermolecular mean free path ${\ell }_{\mathrm{mfp}}$ to a specified characteristic length $L$ of the apparatus or flow. The mean free path measures the distance travelled between gas–gas collisions; $L$ measures the distance over which boundaries or macroscopic conditions matter. Wall encounters must not be counted as gas–gas collisions in this comparison: doing so would hide the very wall-dominated behavior we want to detect. The same gas can therefore have a small Knudsen number in a large vessel and a large Knudsen number in a tiny opening.

For a dilute hard-sphere gas of identical molecular diameter $d$ in thermal equilibrium, use the conventionHere $n$ is the number density, $p$ the pressure, $T$ the absolute temperature, and $k_B$ the Boltzmann constant; the second equality uses the ideal gas law. The collision cross-section is $\pi d^2$, and the $\sqrt{2}$ accounts for both partners moving with independent equilibrium velocities. Mixtures and velocity-dependent collision cross-sections require a stated molecular model and mean free path convention; the displayed expression is not an exact formula for every real gas. ^{[ref] [ref]}

As a numerical toy example, choose $T=300\mathrm{K}$ and $d=0.37\mathrm{nm}$, without claiming that this is a measured molecular diameter for a particular gas. The formula gives ${\ell }_{\mathrm{mfp}}\simeq 68.1\mathrm{nm}$ at $p=10^5\mathrm{Pa}$. It yields $\mathrm{Kn}\simeq 6.81\times 10^{-6}$ for a $1\mathrm{cm}$ vessel but $\mathrm{Kn}\simeq 0.681$ for a $100\mathrm{nm}$ gap. At $p=1\mathrm{Pa}$ the same toy gas has ${\ell }_{\mathrm{mfp}}\simeq 6.81\mathrm{mm}$, comparable to the centimetre vessel. Reducing either pressure or apparatus size can thus expose molecular transport.

Choose $L$ for the question being asked: a tube diameter compares the mean free path with wall separation, while tube length describes a different geometric scale. For a positive macroscopic field $q$, a local gradient scale can instead be $L_q=|q|/|\nabla q|$, where its gradient is nonzero. A large vessel can contain a thin region with a large local Knudsen number. Small geometric Knudsen number alone does not guarantee a slowly varying near-equilibrium flow. The time, velocity and space scalings must also be stated when the Knudsen number appears as a coefficient in a Boltzmann equation. ^[ref]

For $N$ hard particles in a fixed macroscopic volume, shrink their diameter $a$ while keeping $N{a}^{d-1}$ of order one. The occupied volume fraction tends to zero, but the mean free path stays macroscopic. Deriving the Boltzmann equation in this dilute limit requires control of recollisions and propagation of an incoming independence hypothesis related to molecular chaos for the Boltzmann equation. Its validity is not an automatic all-time consequence of small particle size.

Map each real into $(0,1)$ by a fixed bijection, choose its canonical binary expansion, and alternate the two bit sequences as ternary digits $0$ and $2$. If two such outputs first differ at position $k$, the leading difference has size $2\cdot 3^{-k}$ and the remaining tail has size at most $3^{-k}$, so they are unequal. This gives an explicit injection pairing two reals. Iteration encodes every fixed finite real tuple without invoking ambiguous decimal expansions.

A bounded subset of ${\mathbb{R}}^n$ is a Kakeya set if it contains a unit line segment in every direction. Segment positions may vary with direction. Such a set need not have positive Lebesgue measure; the dimension problem concerns its Minkowski dimension or other notions of dimension.

For distinct real affine lines in ${\mathbb{R}}^n$, $n\ge 2$, a joint of a line collection is a point lying on $n$ lines with linearly independent direction vectors. In three dimensions the directions must not all lie in one plane. Extra incident lines are allowed; existence of one spanning choice is enough. This counts joint of a line collection points, rather than tuples of lines or incidence multiplicities.

Gaussian blur applies convolution with $G_{\sigma }(x)=(2\pi {\sigma }^2{)}^{-n/2}{e}^{-|x{|}^2/(2{\sigma }^2)}$ to an image signal. It averages over a spatial scale $\sigma$, suppressing rapid variation while also blurring image edges. The heat-kernel convolution realizes this filter at time $t={\sigma }^2/2$.

A median filter replaces an image signal value by the median of nearby values. Unlike a linear average, it can reject isolated extreme observations without averaging both sides of an image edge. The small-radius response depends on neighbourhood geometry and measure; a uniform disk-area median and a circle-boundary median have different time scales.

A six-sided polygon in the hyperbolic plane whose sides are geodesic segments. The especially useful right-angled case is a right-angled hyperbolic hexagon.

This stable hydrogen isotope has one proton and one neutron. In the Proton–proton chain it is produced by the slow weak proton-pair reaction and is rapidly consumed by proton capture to make helium-3. Its short nuclear lifetime inside a stellar core allows a deuterium quasi-equilibrium in proton-proton burning approximation.

A perturbation preserving both values and slopes has $\eta =\eta \prime =0$ at the endpoints. Apply the Poincaré inequality to $\eta$ and then to $\eta \prime$. The resulting inequality makes a quadratic second variation proportional to $\int ({\eta }^2-(\eta \prime \prime {)}^2)$ nonpositive. If a functional is quadratic, its exact expansion about a stationary function then proves a global maximum.

This stable helium isotope contains two protons and one neutron. In solar proton-proton burning it is an intermediate: two such nuclei can form helium-4 and return two protons, or one can capture helium-4 to enter the second chain branch. The balance between production and destruction determines the helium-3 equilibrium abundance and its helium-3 relaxation time.

The uncentered maximal function takes the supremum over all open Euclidean balls containing $x$, not only balls centred at $x$. It is an extended nonnegative lower semicontinuous function: each strict superlevel set is the union of the balls whose average density exceeds that level. For finite positive measures it satisfies the uncentered maximal weak-type inequality.

In a first-order light-deflection calculation, evaluate the Weyl lensing potential and its gradients on the unperturbed null geodesic. The position correction is first order, so inserting it into an already first-order potential produces a second-order correction. This approximation omits repeated deflection and lens-lens coupling.

When gravitational lensing gives small angular distortions without generic multiple imaging, its deflection field changes source positions, magnification and shear. Angular derivatives of the lensing potential determine these distortions. Ensemble galaxy-shape distortions and cosmic microwave background remapping probe the foreground gravitational field.

A choice of direction for each edge of an undirected graph, producing a directed graph.

An orientation of a graph keeps the same vertex set and the same individual edges. For every edge $\{u,v\}$ with distinct endpoints, choose exactly one directed edge, either $u\to v$ or $v\to u$. Replacing it by both arrows is a different construction: it doubles the edges and permits travel both ways along that connection. In an orientation of a graph, the return journey must use another route.^[ref]

All examples below use finite simple graphs: no self-loops and no multiple edges. For a multigraph, each individual nonloop edge receives its own direction, so two multiple edges can point in opposite directions. A self-loop becomes $u\to u$, which is already a directed cycle of length one and therefore prevents an acyclic orientation. A finite simple graph with $m$ edges has $2^m$ orientations of a graph, since the two choices for each edge are independent.

Think of vertices as junctions and edges as streets. An orientation of a graph makes each street one-way; reachability asks whether a legal journey exists from one specified junction to another. Two useful aims pull in different directions: an acyclic orientation rules out circular dependencies, while a strong orientation permits journeys between every ordered pair of junctions.

A graph property is a class of finite graphs closed under isomorphism. Its labelled order-$n$ slice consists of members on vertex set $[n]$. Closure under taking arbitrary subgraphs and closure under taking only induced subgraphs lead respectively to subgraph-closed graph properties and hereditary graph properties.

A nonsmooth gradient flow driven by the subdifferential of a convex function. Explicit discretization gives the forward subgradient step; implicit discretization gives the proximal operator. Monotonicity of a convex subdifferential underlies uniqueness and contraction properties.

The completed-square form gives global minima at $(1,-1)$ and $(-1,1)$, both with value $-1$, and a saddle point at $(0,0)$ with value zero. Negative levels split into two closed components; the zero contour joins their lobes at the saddle. The gradient flow $\dot{x}=-V_x,\dot{y}=-V_y$ decreases $V$ by the gradient-flow dissipation identity. A negative initial level traps each trajectory on its original side of $x=0$, because $V(0,y)\ge 0$.

The Burger number compares the square of a relevant deformation radius with the square of the horizontal flow scale: $\mathrm{Bu}=R_d^2/l^2$. In a two-layer geostrophic model $R_d^2=g\prime H_1{H}_2/[f_0^2(H_1+H_2)]$. Its inverse measures the scale ratio of baroclinic available potential energy to kinetic energy. Keeping it of order unity retains both interface and relative-vorticity contributions in QG scaling.

A horizontal scale comparing gravitational or buoyancy restoring forces with rotation. For a shallow-water gravity mode with propagation speed $c$, the deformation radius is $c/|f|$. The barotropic deformation radius uses $c=\sqrt{gH}$; internal layered modes use a speed set by reduced gravity and layer depths.