A functional graph has exactly one outgoing edge of a graph at every vertex of a graph, representing a function from its vertex of a graph set to itself. In a finite functional graph, each weakly connected component contains one directed cycle, with directed trees feeding into it. Iterating the represented function eventually reaches that directed cycle.

A strongly connected component is a maximal set of vertices in a directed graph such that each can reach every other by a directed path.

This abelian group has eight elements $c^r{a}^s$, with $r$ taken modulo four and $s$ modulo two. Its group operation adds these exponent pairs separately. The elements of order two are $c^2,a,c^{2}a$, and the other four nonidentity elements $c,c^3,ca,c^{3}a$ have order four. It is a direct product of groups rather than a cyclic group, since no element has order eight.

The number of alpha decay events per material volume accumulated over time. For constant alpha-decay event rate density $R$, $D=Rt$; otherwise $D(t)={\int }_0^{t}R(s)ds$. Multiplying by a damaged volume per event gives a dimensionless overlap parameter.

The effective volume converted to an amorphous state by one independent radioactive decay event in a direct-impact model. The subscript alpha can index the event without identifying the emitted alpha particle as the dominant source of ballistic radiation damage.

For four consecutively bonded atoms A–B–C–D, the torsion angle is the signed dihedral between planes A–B–C and B–C–D. State the viewing and sign convention when comparing projections.

Developmental assignment of position-associated anatomical and molecular character to a digit. A change in digit number or growth is not automatically a homeotic change of identity.

Binary digit dynamic programming counts or optimizes integers by processing their base-two digits while recording a finite state. A usual state includes the current digit position, whether the prefix still equals the upper bound's prefix, and whatever automaton or arithmetic information controls future digits.

To count $0\le x\le N$ without consecutive one bits, retain the previous bit and reject a new one after a one. A tight prefix may use only digits no larger than the corresponding bit of $N$; a smaller prefix may use either bit thereafter. Leading zeros allow equal-length representations, but properties involving the first nonzero digit may require an additional started flag.

For two dilute spherical species with absorbing contact and negligible interaction barriers, relative diffusion has coefficient $D=D_1+D_2$ and contact radius $R=a_1+a_2$. The steady concentration outside one target is $c(r)=c_{\infty }(1-R/r)$, solving the radial Laplace equation with zero contact concentration. Its inward diffusive flux is $4\pi DR{c}_{\infty }$, proving the displayed bimolecular rate coefficient. For identical particles, converting this encounter coefficient into a rate of loss of cluster number additionally needs the factor that prevents counting the same pair twice.

Migration through successive exchanges of atoms or ions with vacant lattice sites. The diffusivity depends on both defect abundance and jump kinetics; mobile-defect diffusivity and atomic tracer diffusivity are different quantities.

diffusion by hopping of an interstitial particle between interstitial sites without displacing a host lattice particle. Direction sums over neighbour vectors determine the conductivity tensor in a dilute independent-jump model.

The factor $f_D$ correcting an independent-jump mean-square-displacement estimate. If ${\tau }_j$ is the total waiting time between jumps of length $\ell$, then $D=f_D{\ell }^2/(6{\tau }_j)$ in three dimensions.

diffusion inferred by tracking labelled particles in an equilibrium material. Local jump correlations and carrier concentration must be accounted for when relating it to electrical conductivity.

Mass transport is movement of material through space, whether as whole-fluid motion, relative motion of constituents or migration driven by an external field. Bulk advection carries a constituent with the local velocity field; diffusion moves it relative to that mean motion because of microscopic fluctuations and spatial variations in its thermodynamic driving quantities.

For a dilute passive constituent with concentration $c$ and constant diffusion coefficient $D$, a simple model writes total flux $\mathbf{J}=c\mathbf{v}-D\nabla c$. The first term transports concentration with the flow, while the second smooths concentration differences. Charged constituents can also migrate under an electric field, requiring an additional flux term and appropriate electrochemical constraints. Combining the flux with a local conservation equation predicts how concentration evolves. Measuring only total material crossing a boundary does not by itself distinguish which transport mechanism supplied it.

A transport closure in which a current is proportional to the negative gradient of a scalar field. It discards detailed angular dependence and becomes unreliable where a nearly isotropic local distribution cannot be assumed. Effective coefficients and boundary modelling must match the physical application.

For translational drag coefficient $\zeta$ and mobility $1/\zeta$, thermal equilibrium requires $D=k_{B}T/\zeta$. This connects random displacement to dissipative response and is a form of the fluctuation-dissipation theorem. Combining it with Stokes drag law gives the Stokes–Einstein relation.

The scale $\sqrt{Dt}$ characterizes diffusion over time $t$. For unrestricted isotropic diffusion, the rms displacement in one coordinate is $\sqrt{2Dt}$ and the total three-dimensional rms displacement is $\sqrt{6Dt}$. Stating the convention avoids spurious disagreement over a numerical prefactor.

Random displacement of a tagged constituent, even in a material of uniform composition. The displayed long-time relation defines ordinary isotropic diffusion in $d$ dimensions.

Relaxation of a density or composition fluctuation involving many constituents. Its coefficient can depend on osmotic compressibility and collective mobility, and need not equal the tagged-particle self-diffusion coefficient at nonzero concentration.

For $N$ equally spaced, coherently illuminated narrow slits at normal incidence, the phasor sum is proportional to $\sin (N\delta /2)/\sin (\delta /2)$, where $\delta =2\pi d\sin \theta /\lambda$. The first zero around the central maximum has $d\sin \theta =\lambda /N$. Principal maxima are removable zero-over-zero limits, not intensity zeros.