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 a candidate circle centre. The two resulting radii must agree; otherwise those prescribed points are incompatible with one circular fillet. 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.

For a complex potential $w(z)=m/(z-a)$, reflection in the straight wall through zero at angle $\theta$ is $Rz=e^{2i\theta }\bar{z}$. The image construction $W(z)=w(z)+\overline{w(Rz)}$ givesOn the wall $Rz=z$, this is real, so its stream function is constant there. This proves the impermeability condition. Thus the image position is $a^{\prime }=e^{2i\theta }\bar{a}$ and its moment is $m^{\prime }=e^{2i\theta }\bar{m}$. If the point dipole is advected by the regular image flow, the complex velocity convention $W^{\prime }=u-iv$ giveswhere $d=\mathrm{Im}(e^{-i\theta }a)$ is its signed distance from the wall. The last equality follows from $a-a^{\prime }=2i{e}^{i\theta }d$. It applies when $d\ne 0$.

A planar potential dipole at $a$ with complex moment $m$ has complex potential of a two-dimensional potential flow $w(z)=m/(z-a)$. Its complex velocity is $w^{\prime }(z)=-m/(z-a{)}^2$, interpreted as $v_x-i{v}_y$. Unlike the three-dimensional potential dipole, this is a two-dimensional singularity of the logarithmic-source potential. For example, two opposite source coefficients $s$ at $a\pm \delta /2$ have potential $s[\log (z-a-\delta /2)-\log (z-a+\delta /2)]$, whose leading term is $-s\delta /(z-a)$ as the separation tends to zero with $s\delta$ fixed. This gives the dipole as a limiting source–sink pair, with the complex sign and orientation included in $m$.

A nonreal complex number is a complex number with nonzero imaginary part. Being nonreal is a property of one number; forming a complex conjugate pair additionally specifies the relation between two numbers.

The complete theory of the standard successor structure is axiomatized by injectivity of $s$, $0$ outside its range, a predecessor for every nonzero element, and the sentences $s^n(x)\ne x$ for each positive integer $n$. Its models have a single distinguished one-sided chain generated by zero and any collection of two-sided integer chains. The classification of models of the successor theory proves completeness and explains why the theory is categorical at every uncountable cardinal but not at the countable cardinal.

In the first-order language of logical equality, assert that there are at least $n$ elements for every positive integer $n$. Eliminating one existential logical variable either substitutes a logical equality or avoids finitely many points, so this first-order theory has quantifier elimination and is complete. It is categorical in every infinite cardinal. Its complete $n$-types are the logical equality partitions of $n$ coordinates and are counted by the Bell numbers.