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.

A UMVCUE has minimum conditional variance among all estimators unbiased in the selected experiment for every parameter. For independent Gaussian stage estimates $A,V$ with information $I_1,J$, continuation $A\ge c$, total information $I_2=I_1+J$ and pooled estimate $T$, Rao-Blackwellization of the fresh estimate gives $T-(I_{1}s/J)\varphi ((T-c)/s)/\Phi ((T-c)/s)$, where $s^2=1/I_1-1/I_2$. The conditional family has complete sufficient statistic $T$, so the Lehmann–Scheffé theorem establishes the optimum. A conditional-likelihood bias correction does not automatically have this unbiasedness property.

Consider independent stage means $X\sim N(\delta ,1/I_1)$ and $Y\sim N(\delta ,1/I_2)$, with positive known information $I_1,I_2$. Continue to stage two only if $X\sqrt{I_1}\ge f$. The pooled mean is $M=wX+(1-w)Y$, where $w=I_1/(I_1+I_2)$. Write $m$ for its observed value. Its conditional mean under a candidate parameter $d$ isHere $\varphi$ and $\Phi$ are the standard normal probability density and cumulative distribution functions. The Conditional bias-corrected normal mean estimate solves $g(d)=m$. This inverts the selection-shifted mean instead of ignoring the fact that the first-stage mean exceeded a threshold.

For the upper-tail inverse Mills ratio $\lambda (a)=\varphi (a)/(1-\Phi (a))$, $g^{\prime }(d)=1-w{\lambda }^{\prime }(f-d\sqrt{I_1})$. The variance of a truncated standard normal variable gives $0<{\lambda }^{\prime }(a)<1$, hence $1-w<g^{\prime }(d)<1$. Since $0<w<1$, $g$ is strictly increasing and ranges over the whole real line; the root exists uniquely. Differentiating the joint normal likelihood conditional on continuation shows that the same root maximizes that conditional likelihood. Inverting a conditional mean does not itself make an estimator conditionally unbiased. The separate Rao-Blackwell estimator after interim selection provides that property in the equal-information setting.

For a nonempty decidable binary tree of finite strings with no terminal nodes, begin at the empty string and repeatedly choose the first immediate successor that belongs to the tree. Each finite decision terminates and some successor always exists. This produces a total computable function giving an infinite path. In particular, a nonempty decidable tree in which every node has two incompatible extensions cannot have only noncomputable paths.

For an effective enumeration $({\phi }_e)$ of unary partial computable functions, admit a length-$n$ string $\sigma$ if it disagrees with every binary value of ${\phi }_e(e)$ observed within $n$ steps for $e<n$. The bounded halting predicate makes membership decidable, and the tests are prefix compatible. Its paths are exactly the binary diagonally noncomputable functions. The path space is nonempty and perfect because infinitely many indices of divergent programs leave arbitrarily late bits free; no path is computable. Dead ends in this decidable presentation are essential.

For independent Poisson processes with rates $r_{+},r_{-}\ge 0$, their difference is a Lévy process with characteristic function $\exp (t[r_{+}(e^{iu}-1)+r_{-}(e^{-iu}-1)])$. When the combined rate is positive, its jumps occur at rate $r_{+}+r_{-}$, with direction probabilities proportional to the two rates. Equal unit rates give $\exp (2t(\cos u-1))$ and a rate-two Compound Poisson process with Rademacher distribution jumps. Its paths are integer-valued, càdlàg, and of finite variation on compact intervals.

Applying a measurable per-claim retention $g$ to independent claim sizes preserves the compound Poisson distribution form. The transformed severity is the pushforward measure of the original severity law. With count parameter $\lambda$, the retained aggregate has expected value $\lambda \mathbb{E}g(X)$ and variance $\lambda \mathbb{E}[g(X{)}^2]$. Zero retained marks may be removed by Poisson thinning.

For a finite-dimensional real Lie algebra $\mathfrak{g}$, the solvable radical of its complexification of a Lie algebra is invariant under complex conjugation. Its real fixed subspace is a solvable ideal of $\mathfrak{g}$, and complexifying that fixed subspace recovers the whole radical. Conversely the complexification of any real solvable ideal is solvable. These inclusions prove the displayed identity; in particular $\mathfrak{g}$ is semisimple if and only if ${\mathfrak{g}}_{\mathbb{C}}$ is semisimple.