An internal vertex capacity bounds the total flow through that vertex, equivalently its total inflow or outflow by flow conservation. Replace the vertex by an input and output vertex, direct incoming edges to the input and outgoing edges from the output, and connect input to output with an edge of the vertex capacity. This converts the constraint into an ordinary flow network edge capacity.

The conformal Friedmann equation is ${\mathcal{H}}^2=H_0^2[{\Omega }_m/a+1-{\Omega }_m]$. Differentiating and using $a^{\prime }=a\mathcal{H}$ eliminates the matter term and yields a first-order Riccati equation. The expanding Big-Bang branch for $0<{\Omega }_m<1$ has $\mathcal{H}=\alpha \coth (\alpha \tau /2)$.

With constant interface thickness and the local entrainment exponent $3/2$, the fixed-energy turbulent mixed layer satisfies $\dot{h}=C{\Omega }^4{R}^6/(N^3{\delta }^{3/2}{h}^{7/2})$. Integration gives a linear growth law for $h^{9/2}$. Once an initial-depth offset is negligible, $h/R$ scales as $[{\Omega }^{2}R/(N^2\delta ){]}^{1/3}(\Omega t{)}^{2/9}$. The law applies before the mixed layer reaches the tank bottom and after the initial forcing transient.

For $f=1-s/r^2+q/r^4$ and the spacetime orientation $-dt\wedge dr\wedge d{\Omega }_3$, the Hodge star operator gives ${\int }_{S^3}\star d{k}^{♭}=2{\pi }^2{r}^3{f}^{\prime }\to 4{\pi }^{2}s$. A prefactor $1/(16\pi )$ produces $\pi s/4$, whereas the dimension-adjusted ADM mass normalization produces $3\pi s/8$. The normalization and orientation must be specified rather than imported from four dimensions.

For distinct horizon radii $a>b>0$, integrating $d{r}_{\star }/dr=f^{-1}$ gives $r_{\star }=r+\frac{a^3}{2(a^2-b^2)}\log |(r-a)/(r+a)|-\frac{b^3}{2(a^2-b^2)}\log |(r-b)/(r+b)|+C$. The tortoise coordinate defines constant retarded or advanced coordinates on radial null geodesics.

In the regular metric $d{s}^2=-fd{v}^2+2dvdr+r^{2}d{\Omega }^2$, a future causal tangent has $\dot{v}\ge 0$. For $\dot{v}>0$, causality gives $\dot{r}\le f\dot{v}/2-r^2|\dot{\Omega }{|}^2/(2\dot{v})$. If $f<0$ between two horizons, the areal radius must decrease. An inner event cannot reach the original exterior future null infinity without traversing that band in the forbidden direction.

Coherent logic uses atomic formulas and equality, finite logical conjunctions and logical disjunctions, truth, falsity and existential quantification. Sequents are universally interpreted in a finite variable context. Its categorical semantics is provided by coherent categories, where conjunction is intersection, disjunction is finite union and existential quantification is image.

For the first-order Cauchy-Riemann operator, the equation ${\varphi }_t-i{\varphi }_x=0$ admits the entire functions ${\varphi }_n(x,t)=e^{-\sqrt{n}}\cos (n(x+it))$. Every fixed-order derivative of the initial data converges uniformly to zero, since $n^r{e}^{-\sqrt{n}}\to 0$. Nevertheless ${\sup }_x|{\varphi }_n(x,t)|=e^{-\sqrt{n}}\cosh (n|t|)\to \infty$ for each $t\ne 0$. Thus even convergence of all initial derivatives does not give continuous dependence on initial data away from the initial line. Analytic existence and smooth-data stability are different properties.

A prescribed inhomogeneous Wirtinger derivative equation. The distributional identity ${\partial }_{\bar{z}}(1/(\pi z))={\delta }_0$ gives its whole-plane right inverse through the Cauchy-Green operator. Boundary or decay conditions select a solution from the freedom to add a holomorphic function.

A first-class constraint generates an infinitesimal canonical transformation $\delta F=\{F,{ϵ}^i{\phi }_i\}$. For the phase-space action $I=\int (p\dot{q}-{\lambda }^i{\phi }_i)dt$, the multiplier transformation $\delta {\lambda }^k={\dot{ϵ}}^k-{\lambda }^i{ϵ}^j{C}_{ij}{}^k$ cancels the bulk variation when $\{{\phi }_i,{\phi }_j\}=C_{ij}{}^k{\phi }_k$. The remaining temporal boundary term vanishes for parameters that vanish at the endpoints.

Let $A$ be a commutative algebra over a field $k$. Suppose $f_1,\dots ,f_r\in A$ generate the unit ideal, and every localization of a ring $A_{f_i}$ is a finitely generated algebra over $k$. Then $A$ is finitely generated over $k$.

Choose $g_i\in A$ with ${\sum }_i{g}_i{f}_i=1$. Let $B\subseteq A$ be the $k$-subalgebra generated by the $f_i,g_i$ and the numerators of finite generating sets of all $A_{f_i}$. Then $B$ is a finitely generated algebra and $B_{f_i}=A_{f_i}$. For $a\in A$, clear the finitely many denominators to choose one $N$ with $f_i^{N}a\in B$ for every $i$. The unit-ideal identity for powers supplies $1={\sum }_i{c}_i{f}_i^N$ with $c_i\in B$, giving $a={\sum }_i{c}_i{f}_i^{N}a\in B$. Thus $A=B$.

A commutative algebra over a field $k$ has a finite presentation if finitely many generators and finitely many polynomial relations describe it:The polynomial ring supplies freely chosen commuting generators, and the ideal of relations identifies expressions representing the same element. For example, $k[X,Y]/(XY-1)$ is the Laurent polynomial ring in one variable: $Y$ represents the multiplicative inverse of $X$.^[ref]

Every commutative finitely generated algebra over a field admits a finite presentation. Indeed, a field is a Noetherian ring; the Hilbert basis theorem makes its polynomial ring Noetherian, so the kernel of a chosen surjective algebra homomorphism is a finitely generated ideal. This is where commutativity enters the assertion. For a noncommutative associative algebra, a finite presentation instead uses a free associative algebra and finitely many generators for its two-sided ideal of relations; the commutative Hilbert basis theorem does not establish finite presentation there.^[ref]

A finite presentation also organizes differentiation: the module of Kähler differentials is generated by $d{X}_1,\dots ,d{X}_n$, with relations $d{f}_j={\sum }_i(\partial f_j/\partial X_i)d{X}_i$. The Jacobian matrix records those relations; this connects an algebraic description by equations to its infinitesimal geometry.

A dead state of a finite automaton has no path to an accepting state. In a complete deterministic finite automaton, all dead states may be merged into one rejecting sink without changing the accepted formal language.

A priority restraint protects finitely many oracle answers by requiring lower-priority enumerations to avoid positions below a numerical bound. Higher-priority action can cancel the priority restraint by initializing its owner. After the last such injury, the protected computation is permanent.

A regular tetrahedron has a proper rotational symmetry group $T$ of order 12, isomorphic to the alternating group $A_4$ acting on its four vertices. Its full spatial symmetry group $T_d$ has order 24 and is isomorphic to the symmetric group $S_4$; the additional operations reverse orientation. Three-fold vertex/face-axis rotations and half-turns about opposite-edge midpoints generate the proper group. The distinction matters when a field symmetry uses only proper rotations, but a scalar density also admits reflections.

Matching the monomials $1,x,x^2$ gives $f^{\prime \prime }(2)\approx (2/3)f(0)-f(1)+(1/3)f(3)$. With error defined as the true derivative minus this approximation, its Peano kernel is $K(\theta )=21_{\{\theta <2\}}+(1-\theta {)}_{+}^2-(3-\theta {)}^2/3$ on $[0,3]$. The derivative evaluation causes a jump at two. Applying the Peano kernel theorem gives $\lambda (f)=\frac{1}{2}{\int }_0^{3}K{f}^{(3)}$.

For a proper filter on a set, write ${\forall }_{\mathcal{F}}xp(x)$ when the truth set $\{x:p(x)\}$ belongs to the filter. This quantifier preserves finite conjunctions. For an ultrafilter it also preserves finite disjunctions and obeys classical negation; a general proper filter need not have those latter properties.

For proper filters on positive integers, define $A\in \mathcal{F}+\mathcal{G}$ when $\{x:A-x\in \mathcal{G}\}\in \mathcal{F}$. This is again a proper filter. The inner truth-set map preserves the whole set, the empty set, inclusions and finite intersections; applying the outer filter gives all filter axioms. The order of the two filter quantifiers is part of the definition and is not silently interchangeable.

In a filling box model, the first front separates fluid already modified by plume discharge from the initially unmodified fluid below. If the plume occupies a negligible fraction of horizontal area $A$, its volume flux across the front is $P(h)$, and the ceiling exhaust is $F$, volume conservation gives $A\dot{h}=F-P(h)$. For a closed box with a negligible-volume buoyancy source, $F=0$. For a finite injected source flux $Q$ with ceiling overflow $Q$, $F=Q$. The first front need not delimit a uniformly mixed upper layer throughout the unventilated transient.

For a fiber bundle $\pi :E\to B$ admitting a finite trivializing open cover, whose fiber has finite free cohomology over $R$, suppose classes $e_j\in H^{\ast }(E;R)$ restrict to a basis on every fiber. Then multiplication $a\otimes e_j\mapsto {\pi }^{\ast }a⌣e_j$ gives the indicated module isomorphism. A finite trivializing open cover suffices, by the local Künneth theorem and the Mayer–Vietoris sequence. This determines an additive module structure; multiplicative relations must still be computed. Powers of the hyperplane class on a complex projective bundle provide an important application.