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For $q\cdot J=q\cdot K=0$, the instantaneous term in the Coulomb-gauge photon propagator combines with its spatial longitudinal subtraction:The identity follows from $\mathbf{q}\cdot \mathbf{J}=q^0{J}^0$ and $q^2=(q^0{)}^2-|\mathbf{q}{|}^2$. It is equality after contraction with conserved currents, not equality of the uncontracted propagator tensors. It is especially useful in showing that tree scattering amplitudes computed in Coulomb gauge are Lorentz invariant.

For a coreflexive pair $f,g:B\rightrightarrows A$ with equalizer $e:E\hookrightarrow B$, direct image along the mono $f$ satisfies $Pf{\exists }_f=1$ and $Pg{\exists }_f={\exists }_{e}Pe$. Thus any map $h:PB\to Z$ equalizing $Pf,Pg$ obeys $h=h{\exists }_{e}Pe$. Since $Pe{\exists }_e=1$, the map $Pe$ is their coequalizer. These identities hold for parameterized subobjects in any elementary topos.

The contravariant power-object functor of an elementary topos is monadic. Its double-power unit ${\eta }_A(a)(S)=(a\in S)$ is monic, which helps prove that $P$ reflects isomorphisms. Finite equalizers and the fact that power objects turn coreflexive equalizers into coequalizers give the remaining hypotheses of the crude monadicity theorem.

Bounded continuous functions on a topological space form a Banach space with the supremum norm. A uniform limit is continuous and bounded, proving completeness. This space differs from the bounded uniformly continuous functions: free-transport composition is jointly continuous in time and position, but need not be continuous in the global sup norm as time varies.

A contact form $\alpha$ defines the hyperplane distribution $\xi =\ker \alpha$. Multiplying $\alpha$ by a nowhere-zero function leaves this distribution unchanged. The contact distribution symplectic form distinguishes it from an integrable hyperplane distribution.

A smooth interval family of contact forms on a compact manifold without boundary is carried back to its initial contact distribution by a smooth isotopy. The pullback of each form is a positive smooth multiple of the initial form. Solve for the horizontal generator of contact stability and integrate the resulting scalar equation for the contact conformal factor along an isotopy.

Every set belongs to some level of the constructible hierarchy. The assertion is $\forall x\exists \alpha (x\in L_{\alpha })$, not that each set is definable without parameters. The constructible universe theorem supplies a relative-consistency model of this assertion from ZFC.

The conserved first moment of a draining porous current fixes $\alpha +2\beta =0$ in a self-similar solution $h=a{t}^{\alpha }f(x/(d{t}^{\beta }))$. The Boussinesq equation for an unconfined aquifer supplies $\alpha =2\beta -1$, giving $\alpha =-1/2$, $\beta =1/4$. Normalize $\kappa a=d^2$ and the nose at $\eta =1$. Then $-f/2-\eta f^{\prime }/4=(f{f}^{\prime }{)}^{\prime }$ is solved byThe square-root outlet has finite drainage flux and the nose has zero flux. This is a particular similarity solution and its scaling family, not an exact description of arbitrary initial data.

A connection on a vector bundle induces a connection on its top exterior power by differentiating each factor in turn. Its local connection one-form is $\mathrm{Tr}A$ and its curvature form of a connection is $\mathrm{Tr}{\Theta }_D$.

A local trivialization of a smooth vector bundle gives a local connection by differentiating its component functions. Weight these connections by a subordinate locally finite partition of unity and extend the weighted terms by zero. The weights sum to one, so the connection Leibniz rule survives. This proves existence on a paracompact smooth manifold. Averaging connections is valid with weights summing to one because the collection is the affine space of vector-bundle connections.

The difference of two connections on a vector bundle is a smooth one-form with values in the endomorphism bundle: the Leibniz derivative terms cancel, leaving pointwise linearity in both the tangent vector and section value. Conversely every such form added to a connection gives another connection. Thus the collection is an affine space with this model vector space. Choosing ${\nabla }^0$ supplies a noncanonical origin; there is no distinguished zero connection.

With local frame written as a row $e$ and $D(eu)=e(du+Au)$, a change $e^{\prime }=eg$ gives $A^{\prime }=g^{-1}Ag+g^{-1}dg$. The vector-bundle curvature transforms by conjugation. This convention makes the inhomogeneous derivative term and every subsequent vector-bundle curvature sign explicit.

The graded Leibniz rule terms of two connections cancel, making their difference $C^{\infty }$-linear in the section. It is therefore an element of $A^1(\mathrm{End}E)$. Conversely adding any such form to a connection gives another connection, making the space of connections an affine space.

The largest graph component is one with the most vertices. Ties may be broken arbitrarily; $L_1$ records the order, which is unaffected by the tie. For a sequence of random graphs, the size of its giant component is a central observable.

A tree component is a graph component whose induced graph is a tree. A tree component with $j$ vertices has exactly $j-1$ edges.

For open-edge sets $S,T$ in a fixed finite graph, including isolated graph vertices in $k$, one has $k(S\cap T)+k(S\cup T)\ge k(S)+k(T)$. Incidence-vector spans have rank $r(S)=|V|-k(S)$; their union span is the sum of the two spans, while the intersection-edge span is contained in the intersection of spans. The vector-space dimension formula for a sum of subspaces proves submodularity of $r$, equivalently supermodularity of $k$.

A Pareto distribution prior for the upper limit of independent uniform distributions remains Pareto after observing data. The likelihood adds the sample size to the shape and replaces the lower bound by the larger of the prior bound and sample maximum.

Conditionally independent gamma distributions with known shape $r$ and unknown rate $\Theta$ have likelihood proportional to ${\Theta }^{nr}{e}^{-\Theta {\sum }_i{x}_i}$. A gamma distribution prior of shape $A$ and rate $B$ therefore updates to shape $A+nr$ and rate $B+{\sum }_i{x}_i$. When $A+nr>1$, the posterior mean of the conditional claim mean $r/\Theta$ is $r(B+{\sum }_i{x}_i)/(A+nr-1)$. For $A=rk+1$ and $B=k\mu$, this is a credibility estimate with weight $n/(n+k)$. It is the reciprocal-rate version of gamma scale inverse-gamma conjugacy.

For reception at the origin with unit propagation direction $e$, the null geodesic condition fixes $v({\eta }_0)=(1+2{\Psi }_0)e$. The perturbative spatial acceleration is $x^{i\prime \prime }=2e^{i}d\Psi /d\eta -2({\delta }^{ij}-e^i{e}^j){\partial }_j\Psi$. Its integrated trajectory is $x^i=e^i(\eta -{\eta }_0)+2e^i{\int }_{{\eta }_0}^{\eta }\Psi d{\eta }^{\prime }-2{\int }_{{\eta }_0}^{\eta }(\eta -{\eta }^{\prime })({\delta }^{ij}-e^i{e}^j){\partial }_j\Psi d{\eta }^{\prime }$. Potentials in these already first-order integrals can be evaluated on the straight background ray.

A weak Condorcet winner is never defeated by a strict pairwise majority, allowing ties. For even electorates with single-peaked preferences, all alternatives between the two middle peaks have this property. A single-valued selection requires a tie convention; arbitrary tie selection is not automatically strategyproof.