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Airflow is motion of air, described in the continuum approximation by fields of velocity, pressure, mass density and temperature. A fan can sustain a pressure difference, buoyancy can drive a warm plume, and external motion can carry air past a body. The driving mechanism and boundary geometry determine the resulting flow.

For a characteristic speed $U$ much smaller than the speed of sound, an incompressible flow model can often describe mechanical motion when thermal density changes are also small. The Mach number tests the acoustic comparison, while the Reynolds number compares inertial and viscous effects. These are different approximations: low Mach number does not require laminar flow. A fixed probe measures an Eulerian coordinate reading; smoke markers can visualize paths and mixing, with the limitation that their motion must follow the air closely enough for the question being studied.

The time from a triggering clock edge to the corresponding settled output of a flip-flop, under specified loading. The time available between successive active clock edges must accommodate this delay, the combinational critical path, the receiving flip-flop's setup time, and any clock uncertainty.

A hyperbolic source fixed point of a smooth finite-dimensional map has every Jacobian eigenvalue outside the unit circle. Forward iteration repels nearby states. For an ordinary differential equation the analogous test instead uses positive real parts, so a source equilibrium criterion should not be substituted into a discrete map calculation.

A hyperbolic sink fixed point of a smooth finite-dimensional map has every Jacobian eigenvalue inside the unit circle. Nearby iterates converge to it, and zero multipliers are allowed for a noninvertible map. Unit-modulus multipliers require nonlinear or exact analysis rather than this strict linear criterion. A complex pair gives spiral contraction, while negative real multipliers alternate directions.

For $\alpha >\beta >0$ the positive fixed point is $(\alpha -\beta {)}^{1/\gamma }$ with multiplier $1-\gamma (1-\beta /\alpha )$. It is stable for $1<\gamma <{\gamma }_{\ast }$ and crosses through minus one at the displayed threshold. Since ${\gamma }_{\ast }>2$, every positive fixed point is stable when $1<\gamma \le 2$. The increasing power controls overcompensatory density dependence at high abundance. A multiplier crossing identifies the local flip mechanism; a separate nonlinear calculation determines its criticality and any resulting attractor.

For the finite square well $V=-V_0$ on $|x|<a$ and $V=0$ outside, with $a,V_0>0$, there is always an even bound state, however small $V_0$ is. Define $\rho =a\sqrt{2m{V}_0}/\hslash$, the dimensionless decay constant $\alpha =a\sqrt{-2mE}/\hslash$, and interior oscillation constant $\beta =a\sqrt{2m(E+V_0)}/\hslash$. Wavefunction matching at a finite potential step gives $\alpha =\beta \tan \beta$ and ${\alpha }^2+{\beta }^2={\rho }^2$.

On $0<\beta <\pi /2$, these equations reduce to $\beta /\cos \beta =\rho$. The left side is continuous, strictly increasing from zero to infinity, with derivative $\sec \beta (1+\beta \tan \beta )>0$. The intermediate value theorem gives exactly one root on this first branch for every $\rho >0$. It has $0<\beta <\rho$, hence $\alpha >0$ and $-V_0<E<0$. The matched cosine interior and exponential tails therefore give a nonzero even square-integrable function with continuous first derivative, which can be normalized. As the well becomes shallow the binding energy approaches zero but remains negative; see shallow finite-square-well energy.

A finite field has positive characteristic, because its repeated sums of $1$ cannot all be distinct. By the characteristic of an integral domain theorem, this is a prime number $p$. Its multiples of $1$ form a copy of the prime field ${\mathbb{F}}_p$: a nonzero multiple $a1$ has inverse $b1$, where the Bezout identity $ab+cp=1$ supplies $b$.

View $F$ as a vector space over this prime field. Repeatedly extend a linearly independent set until it is maximal; the procedure terminates because $F$ is finite. Maximality makes it a basis. If the dimension is $d$, unique coordinates give exactly $p^d$ elements. Thus the cardinality is a power of a single prime, rather than an arbitrary composite integer.

In a splitting field of $x^{p^n}-x$ over a prime field, the polynomial roots form a field. Frobenius identities close the polynomial roots under addition and multiplication, and nonzero polynomial roots have inverses which are again polynomial roots. The derivative is minus one, so the polynomial has exactly $p^n$ distinct polynomial roots. This constructs a field of every prime-power order.

A process-associated mapping from small nonnegative integers to open file descriptions, with descriptor-specific flags. Several file descriptors can share an open file description; their current file offset is then shared. The traditional model has one table per process, although operating systems can support explicit sharing between processes.

A simple field extension is a field extension generated by one element: $F(\alpha )$ is the smallest subfield of an ambient field containing $F$ and $\alpha$. Its elements are rational expressions in $\alpha$ with coefficients in $F$ and nonzero denominators. If $\alpha$ is algebraic, its minimal polynomial $m_{\alpha }$ identifies $F(\alpha )\cong F[t]/(m_{\alpha })$, so $[F(\alpha ):F]=\mathrm{deg}{m}_{\alpha }$. The classes of $1,t,\dots ,t^{d-1}$ give a basis when $d=\mathrm{deg}{m}_{\alpha }$.

In particular, ${\alpha }^2\in F$ but $\alpha \notin F$ implies $[F(\alpha ):F]=2$. Conversely, in characteristic different from two, any degree-two extension is of this form: for $\beta \notin F$, express ${\beta }^2=a+b\beta$ in the basis $1,\beta$, then $\alpha =\beta -b/2$ has ${\alpha }^2=a+b^2/4\in F$ and generates the same extension. The restriction on the characteristic makes division by two possible.

For a field $K$, the nonzero elements form an abelian group under multiplication, called its multiplicative group. Products of nonzero elements are nonzero because a field has no nonzero zero divisors; multiplication is associative and commutative, the identity is $1$, and every nonzero element has an inverse. Thus every finite subgroup of $K^{\times }$ is a finite abelian group. The finite subgroup of a field multiplicative group is cyclic theorem uses the root bound for a polynomial to prove a stronger restriction on such subgroups.

The field axioms specify an abelian additive group, an abelian multiplicative group of nonzero elements, and distributivity of multiplication over addition. The identity elements are distinct.

A cup-shaped crust characteristic of favus, formed from fungal material and tissue debris. The plural is scutula; its shape is a diagnostic observation rather than proof of a particular fungal strain.

For outer velocity $U_s=-2/x$, the Falkner-Skan equation reduces for the scaled velocity $g=f^{\prime }$ to $g^{\prime \prime }+g^2-4=0$. Multiplication by $g^{\prime }$ gives $\frac{1}{2}(g^{\prime }{)}^2+g^3/3-4g=16/3$ for profiles tending to $-2$ at infinity. No slip gives $g(0)=0$, so $g^{\prime }(0)=\pm 4\sqrt{2/3}$. The two smooth solutions are the displayed translates with ${\eta }_0=\mathrm{arctanh}\sqrt{2/3}$. One decreases monotonically to the outer velocity; the other rises to $4$ before descending, producing an internal reversed-flow region. The latter has inflection points, motivating concern about shear instability, but the steady profile equation alone does not determine temporal stability.

For the Falkner-Skan equation $f^{(3)}+\beta f{f}^{\prime \prime }+m(1-(f^{\prime }{)}^2)=0$, an attached branch has $f^{\prime }>0$ away from the wall. It is then possible to replace the similarity coordinate by $f$ itself. Define $w(f)=f^{\prime }$. The chain rule gives $f^{\prime \prime }=w{w}_f$ and $f^{(3)}=w(w_f^2+w{w}_{ff})$, proving the displayed second-order equation. The wall and outer conditions become $w(0)=0$ and $w(\infty )=1$ when the stream function is set to zero at the wall.

The reduction is singular at the wall: if $s=f^{\prime \prime }(0)>0$, then $f\sim s{\eta }^2/2$ and $w\sim \sqrt{2sf}$. Hence $w_f$ diverges while $w{w}_f\to s$ remains finite. A finite-slope condition on $w$ at zero would incorrectly exclude the ordinary positive-wall-shear branch. At flow reversal or separation, the invertibility of the coordinate change must be checked again.

A historical name for a crystalline material isolated from human faeces. Early descriptions and formulas are not sufficient to establish that every preparation was one pure modern compound.

A tarsal plate is a dense connective-tissue support within an eyelid. It contains specialized glands and helps maintain lid shape. It is not a plate of bone, despite the stiffness implied by its name.

The medial rectus muscle adducts the eye and is supplied by the oculomotor nerve. Coordinated activity with the opposite eye supports convergence. A local muscle action and the binocular control problem are therefore different explanatory levels.

The lateral rectus muscle abducts the eye. It is supplied by the abducens nerve, unlike most other muscles rotating the eyeball. Its relatively direct horizontal action provides a useful anatomical example connecting one muscle, one nerve and one principal movement.