A language is computably enumerable when a machine recognizes its members, possibly running forever on nonmembers.

A finite-state machine processes inputs using a finite set of automaton states and input-dependent automaton transitions. An automaton state records everything the machine remembers; an automaton transition says how that memory changes on the next symbol. A finite-state machine may also emit outputs. Here the emphasis is on recognizing a formal language: start in a designated automaton state, read a word from an alphabet, and accept exactly when the final automaton state is designated accepting.

Finite memory does not impose a maximum input length. A finite-state machine can read arbitrarily long words, revisiting its automaton states indefinitely. The restriction is that the part already read must be compressed into one of finitely many summaries. For divisibility by three, a word of binary digits can describe an enormous integer, yet its congruence class modulo three is all that the machine needs to retain. Two already-read words giving the same automaton state behave identically under every subsequent continuation in a DFA.

This makes finite-state machines useful whenever past details can be forgotten safely: checking whether a sequence of commands follows an allowed order, or recognizing a pattern with a regular expression. The beautiful design question is which distinctions about the past matter to the future. DFA minimization makes that question precise by merging indistinguishable states of a deterministic finite automaton. Both DFAs and NFAs recognize exactly the regular languages.

A word over an alphabet $\Sigma$ is a finite sequence of symbols from $\Sigma$. The empty word has length zero, and ${\Sigma }^{\ast }$ denotes the set of all finite words.

Newton's laws of motion consist of inertia in an inertial frame, the relation between net force and change of momentum, and equal-and-opposite pair forces. The following three children state the individual laws.

The Van der Waals force is a distance-dependent nonbonded interaction between atoms or molecules, including dispersion and dipolar contributions. In the attractive nonretarded pair model, an interaction energy $U(r)=-\beta /r^6$ gives force magnitude $6\beta /r^7$. Macroscopic integration introduces a Hamaker constant; it does not replace the sign change and molecular cutoff needed at very short distances.

Tension is a pulling force transmitted along a taut string, rope, or similar body. For an ideal uniform string with linear mass density $\mu$, constant tension $T$ sets the small transverse wave speed $c=\sqrt{T/\mu }$.

A restoring force acts toward an equilibrium when a body is displaced from it. In one dimension, taking displacement $x=0$ at equilibrium, this means $xF(x)<0$ for sufficiently small nonzero $x$. An ideal linear spring has $F(x)=-kx$ with positive spring constant $k$, producing simple harmonic motion for a constant mass without damping. More general restoring forces need not be linear.

Here gravitational weight means the force on a body due to gravity. In the Newtonian model, a body of mass $m$ in a uniform gravitational acceleration $\mathbf{g}$ has gravitational weight $\mathbf{W}=m\mathbf{g}$, with magnitude $W=mg$. Its direction follows $\mathbf{g}$; a buoyant force is a separate contribution to the body's net force. ^[ref].

A food chain orders species so that each consumer feeds on the level below and may be eaten by the level above. Basal resource growth, consumer mortality, conversion efficiency and density regulation determine whether all levels can persist.

Two consumers with different break-even requirements cannot both have bounded persistent abundance on a single well-mixed limiting resource when their per-capita growth is linear in that resource. For $\dot{Y}/Y=cX-d$ and $\dot{Z}/Z=eX-f$, differentiation of $\log Y/c-\log Z/e$ gives the nonzero constant $-d/c+f/e$ when their break-even levels differ. This excludes simultaneous bounded persistence without assuming convergence to an equilibrium. Additional resources, environmental variation or different interactions can invalidate the simple one-resource argument.

For a cooriented integrable codimension-one distribution, choose a nonvanishing defining form $\zeta$ and an auxiliary form with $d\zeta =\eta \wedge \zeta$. Changing the auxiliary form to $\eta +a\zeta$ changes $\eta \wedge d\eta$ by $-d(ad\zeta )$. Replacing $\zeta$ by $h\zeta$ changes the auxiliary form by $d\log |h|$, changing the three-form by the exact differential $d(-d\log |h|\wedge \eta )$. Stokes' theorem proves invariance on a closed oriented three-manifold. For the weak stable foliation of a curvature-minus-one surface geodesic flow, take $\zeta =\theta +\omega$, $\eta =\alpha$ to obtain $\mathrm{gv}=4{\pi }^2\chi (M)$ in the positive base/fibre orientation.

A diffusive flux transports a density $u$ down its gradient according to Fick's first law. For scalar diffusion coefficient $D>0$, $\mathbf{j}=-D\nabla u$ and the conservation law $u_t+\mathrm{div}\mathbf{j}=0$ gives a diffusion equation. If $D$ depends on $u$, this becomes a nonlinear diffusion equation.

Fick's laws describe transport down a concentration gradient and the resulting change of concentration with time. Combining Fick's first law with a conservation law gives the diffusion equation.

In a closed tank of cross-sectional area $S$, entrainment into a uniform upper layer conserves the density deficit $({\rho }_l-{\rho }_u)h$. Thus $B=g^{\prime }h$ is constant. With depth $z$ increasing downward, gravitational potential energy is $-gS{\int }_0^H\rho (z)zdz=E_{\ast }+{\rho }_{l}SBh/2$, so its rate of increase is proportional to the rate of deepening. This includes homogenization throughout the upper layer, not merely lifting a thin parcel across the interface.

An interfacial stress $\tau =c_D{\rho }_l{u}_I^2$ supplies local stress power $P_I=S\tau u_I$. Assuming a fixed fraction $C_W$ becomes global potential energy gives ${\dot{E}}_P=C_W{P}_I$. Together with potential energy of two-layer entrainment, this gives $\dot{h}=2C_W{c}_D{u}_I^3/(g^{\prime }h)$. This is an additional local power-transfer closure. The shaft power of a rotating lid is its torque times its prescribed angular speed and need not equal $P_I$. Geometric factors and a constant transfer fraction may be absorbed into the empirical coefficients only when explicitly assumed time-independent.

The mean speed, measured normal to a chosen plume or jet boundary, at which ambient fluid enters the turbulent region. The boundary and averaging convention must be specified: this speed is not the plume's axial velocity. In the Batchelor entrainment hypothesis for a rising Boussinesq plume, $u_e=\alpha w$, where $\alpha$ is the dimensionless entrainment coefficient and $w$ the characteristic axial speed. Multiplying by the boundary perimeter gives the entrained volume flux per axial length.

For a constant real square matrix $A$, the autonomous differential equation $\dot{x}=Ax$ has the linear flow ${\phi }_t(x)=e^{tA}x$, where $e^{tA}$ is the matrix exponential. The identity $e^{(t+s)A}=e^{tA}{e}^{sA}$ gives the flow map composition law, and $e^{-tA}$ reverses the evolution. Its determinant is $e^{t\mathrm{tr}A}$, so the matrix trace gives the instantaneous fractional rate of volume change, as illustrated in trace and local volume change. A zero trace preserves volume even when the flow stretches one direction and contracts another.

For a photon in the flat-background scalar-force theory of gravity, $e^{\Phi }{p}^i$ is constant along its path. In a static field, energies measured by stationary laboratories therefore satisfy the displayed ratio. At first order, $\Delta \nu /\nu =-\Delta \Phi$ in units $c=1$, matching the weak-field Pound-Rebka experiment. This agreement does not imply light bending: a momentum change parallel to the original null tangent changes its scale, not its direction.

Every continuous self-map of a nonempty compact convex subset of a locally convex space has a fixed point. Compactness is in the locally convex topology, which may be weaker than a norm topology. This version permits fixed-point arguments on a weak-star topology compact set of linear functionals. See Hartman's fixed-point chapter, Theorem 0.2.

A nonempty compact convex set $K$ in a finite-dimensional space and an upper hemicontinuous set-valued mapping $F:K\rightrightarrows K$ with nonempty closed convex values admit $x\in F(x)$. A closed graph suffices when the whole output space is compact.