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A finite family of nonempty connected subgraphs of a tree with pairwise nonempty intersections has a common vertex. For a finite ambient tree, prove this by deleting a leaf of a tree $v$. If one member is $\{v\}$, pairwise intersection proves the result immediately. Otherwise remove $v$ from every member containing it. Each such member remains nonempty and connected and contains the unique neighbour of $v$. Hence every pair still intersects, even if its original intersection consisted only of $v$, and mathematical induction applies. This is the Helly property specialized to trees.

Two disjoint connected sets can have a connected union when one contains a contact point of the other's closure. To prove the displayed criterion, suppose the union had a separation into disjoint nonempty relatively open sets $P,Q$. Each connected set lies wholly on one side; if both lie on the same side, the other side is empty. Otherwise assume $A\subset P$ and $B\subset Q$. Choose $b\in B\cap \overline{A}$. Since $Q$ is relatively open in the union and contains $b$, some ambient open set containing $b$ meets the union only in $Q$. It must nevertheless meet $A$ by the definition of closure, a contradiction.

For example, the open unit disc centred at zero and the closed unit disc centred at two are disjoint, but their union contains the common contact point one through the second disc and is connected. In this example it is even path-connected: the real segment from zero to one lies in the first disc except for its endpoint, and that endpoint belongs to the second disc. Omitting the contact point from both discs instead gives a disconnected union.

The quasicomponent of a point is the intersection of all its clopen neighborhoods. Every connected set containing the point lies in its quasicomponent, since a clopen set cannot split a connected set. In a compact Hausdorff space, quasicomponents equal connected components by the Šura-Bura component lemma.

The quasi-component of a point is the intersection of its clopen neighborhoods. Its connected component is always contained in it: a connected set meeting a clopen set cannot also meet its complement. Equality need not hold in noncompact Hausdorff spaces. The Šura-Bura lemma proves equality in compact Hausdorff spaces.

Every connected component $C$ of a nonempty compact proper subset $D$ of a connected Hausdorff space meets its boundary. If not, the Šura-Bura lemma identifies $C$ as a quasi-component in the compact Hausdorff subspace $D$. Finitely many clopen neighborhoods separate it from the compact boundary, giving a nonempty relatively clopen $H\subseteq \mathrm{int}D$. Such $H$ is open in the ambient space, and is also closed there because $D$ is closed. It is proper because $D$ is proper. This contradicts connectedness.

For a nonsingular conic with a point $p$ and a projective line $L$ not containing $p$, send $q\in L$ to the second intersection of $pq$ with the conic, counted with multiplicity. The tangent direction gives $p$ itself. The inverse sends any other conic point to its joining line's intersection with $L$, and sends $p$ to the tangent's intersection. The Bézout theorem makes these maps inverse. With data over a field, the construction preserves rationality: a known rational root of the restricted quadratic equation leaves its other root rational too. A conic with a rational point is therefore rationally parametrized by a projective line.

Two complex plane conics meeting in four distinct points can be diagonalized by the same invertible change of homogeneous coordinates. Their intersection points form a projective frame: three on a line would force that line to be a common component by Bézout theorem, contradicting a finite intersection. Move the four points to the coordinate points and $[1:1:1]$. Each equation becomes $ayz+bzx+cxy=0$ with $a+b+c=0$. Substituting $x=v+w,y=w+u,z=u+v$ gives $a{u}^2+b{v}^2+c{w}^2$; the substitution matrix has determinant two. Composing this with the frame change yields the displayed simultaneous congruence of the symmetric coefficient matrices. This is congruence of quadratic forms, not a claim of simultaneous similarity of arbitrary matrices.

A complete residue system modulo a positive integer $m$ contains exactly one representative of every congruence class modulo $m$. Both $\{0,\dots ,m-1\}$ and $\{1,\dots ,m\}$ are examples. Counting the representatives that are coprime to $m$ gives Euler's totient function; those representatives form a reduced residue system.

The intersection of the unit disks centered at $1/2\pm i\sqrt{3}/2$ has endpoints zero and one. The Möbius transformation $w=z/(1-z)$ maps it onto the sector $|\arg w|<\pi /6$: the two disk conditions become $\mathrm{Re}(e^{\pm i\pi /3}w)>0$. Cubing gives the right half-plane, and the Cayley transform between the half-plane and disk gives the displayed holomorphic bijection to the unit disk. The sector width ensures that cubing is injective.

A directed graph with a vertex for each database transaction and an edge from the earlier to the later participant in each conflicting pair of operations. A database schedule is a conflict-serializable database schedule exactly when this directed graph has no directed cycle.

For a symmetric interval of the form $[L,U]=[\hat{\theta }-c\mathrm{SE}(\hat{\theta }),\hat{\theta }+c\mathrm{SE}(\hat{\theta })]$, where $c\ge 0$ and the standard error is nonnegative, adding and subtracting the endpoints proves the displayed identities. Thus the endpoints must be ordered and their midpoint must equal the point estimate. Rounding should be assessed using intervals of possible unrounded values before declaring a numerical contradiction.

When this confidence interval is obtained by inverting a matching two-sided test of $\theta ={\theta }_0$, exclusion of ${\theta }_0$ agrees with rejection at the corresponding significance level, with the usual convention at boundary equality. This is a check for a specified symmetric interval procedure, such as the usual t interval for a mean difference; it does not require asymmetric intervals or intervals on transformed scales to have their midpoint at the estimate on the original scale.

Invert a family of hypothesis tests by including each parameter value in the confidence set exactly when its test does not reject. For any true value $\theta$, the events $\{\theta \notin C(X)\}$ and $\{\text{reject }{H}_{\theta }\}$ are identical. Thus size at most $\alpha$ gives coverage at least $1-\alpha$, and exact size gives exact coverage. When the accepted set is an interval, this is a confidence interval.

For the normal sample mean with unknown variance, the central Student t-distribution interval excludes zero exactly when $|\sqrt{n}\overline{X}/S|$ exceeds the same two-sided critical quantile. A two-sided p-value strictly below $\alpha$ therefore implies exclusion of zero from the matching closed interval. Boundary conventions agree up to events of probability zero in this continuous model.

For a count $s>0$ from a Poisson distribution, equal-tail inversion uses the inclusive upper-tail event $S\ge s$ for the lower endpoint and the inclusive lower-tail event $S\le s$ for the upper endpoint. Their cumulative distribution indices differ by one, as displayed. If $s=0$, the lower endpoint is zero. For $n$ observations of mean $\mu$, apply this to their total of mean $\lambda =n\mu$ and divide endpoints by $n$. The construction has at least the nominal frequentist coverage; discreteness can make it conservative.

A box contains one certainly red ball and $n\ge 1$ independently coloured extra balls, each red with probability $1/2$. Conditional on the first draw being red, a second draw without replacement is red with probability $(n+3)/(2(n+2))$. To see the effect of using a conditional probability, let $R=1+B$, where $B$ has a binomial distribution with parameters $n,1/2$. The conditional probability is $\mathbb{E}[R(R-1)]/(n\mathbb{E}[R])$: histories with more red balls are more likely to produce the first red draw and therefore receive greater weight.

Constructive solid geometry describes a solid by combining primitive regions with Boolean set union, set intersection and difference. With primitive defining functions nonpositive inside, minimum implements union and maximum implements intersection. These formulas encode membership rather than signed distance functions. Solid modellers often regularize Boolean operations to remove isolated lower-dimensional remnants.

A fillet rounds a corner or edge by inserting a tangent transition. In a planar circular construction, intersect normal lines through the prescribed tangent points to find the circle centre. Choosing the retained side and local clipping region is part of the design: intersecting the entire original solid with the fillet disk generally removes unrelated geometry. Tangent joining need not make curvature continuous.

Joined regular parametric curves have order-$k$ geometric continuity when their local images admit compatible orientation-preserving regular parameter choices with matching derivatives through order $k$. At order one the nonzero one-sided tangent vectors must be positive scalar multiples. At order two the oriented bending must also match: equality of unsigned curvature alone does not suffice if the curves bend to opposite sides.

Joined pieces have order-$k$ parametric continuity when their positions and one-sided derivatives through order $k$ agree in the actual common parameter. Changing parameter-interval lengths changes the required derivative scale by the chain rule. For regular curves, parametric continuity implies the corresponding geometric continuity.

A boundary representation specifies a solid through its oriented bounding faces and their edge and vertex of a polyhedron incidence. Supporting planes and coordinates supply geometry; face loops and adjacency supply topology. A valid closed manifold model requires compatible orientation, two incident faces per edge and a single local fan at each vertex of a polyhedron, together with an embedded boundary.

A two-dimensional point sink of strength $Q>0$ removes volume flux $Q$ per unit depth at $z_0$. Its complex potential of a two-dimensional potential flow is the displayed expression, so $u-iv=-Q/[2\pi (z-z_0)]$. The radial velocity is $u_r=-Q/(2\pi r)$: integrating over a circle gives outward flux $2\pi r{u}_r=-Q$. This is the planar cross-section of a line sink. A passive particle satisfies $d(r^2)/dt=-Q/\pi$ and reaches the ideal point singularity in finite time. This is a flow singularity, distinct from a stable sink equilibrium of a regular dynamical system.