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A nonunit Euclidean dilation sends straight-line geodesics to straight-line geodesics but scales their tangent inner products. Preservation of geodesics alone therefore does not imply local isometry.

An elliptic curve over a field $K$ is a smooth projective curve of genus one equipped with a specified $K$-rational point $O$. The chord-and-tangent law makes its rational points an abelian group with identity $O$.

The forcing theorem identifies truth in a generic extension with the forcing relation: a formula is true in $M[G]$ exactly when some condition in $G$ forces it.

Likelihood-ratio tests of different null subspaces can have crossing rejection regions; comparing their quadratic-form geometry exhibits observations accepted by one and rejected by another.

For independent unit-variance samples, testing equal means uses $Z=\sqrt{mn/(m+n)}(\bar{X}-\bar{Y})$ and rejects for large $|Z|$.

The deviance of a fitted generalized linear model isFor regular nested models, the deviance reduction $D_0-D_1=2({\ell }_1-{\ell }_0)$ is asymptotically chi-squared with degrees of freedom equal to the difference in fitted dimensions.

Whitening multiplies a model by a square root of its precision matrix so that the transformed errors have identity covariance.

For a linear operator $f$, the generalized eigenspacesbelonging to distinct eigenvalues are linearly independent. On the generalized $\alpha$-eigenspace, $f-\beta I=(\alpha -\beta )I+N$ is invertible for $\beta \ne \alpha$ because the nilpotent part $N$ has a finite geometric-series inverse.

The Gaussian free field is a centered Gaussian random field whose covariance is the Green kernel of a Laplace operator. It has discrete versions on graphs and continuum versions as a random generalized function on a domain.

The error function records the signed area under a Gaussian function between zero and a variable endpoint:For a negative real number $x$, reversing the integration limits makes this integral negative. The normalization uses the Gaussian integral: the full area of $e^{-t^2}$ is $\sqrt{\pi }$, so its positive half contributes one after multiplication by $2/\sqrt{\pi }$. The same definition extends to a complex number $z$ along any path from zero to $z$, giving an entire function.^[ref]

For example, $\mathrm{erf}(0)=0$, $\mathrm{erf}(1)\approx 0.842701$, and $\mathrm{erf}(-1)\approx -0.842701$. A finite positive endpoint captures only part of the positive Gaussian integral; $\mathrm{erf}(x)$ approaches one as $x\to +\infty$ but never reaches it at a finite real $x$.

The useful idea is to name a recurring integral once instead of carrying it through every calculation. The error function and the normal distribution express probabilities and quantiles through this area, while an error-function diffusion front describes how diffusion smooths a sharp jump. The complementary error function retains the small area left in the tail.

The fundamental theorem of calculus givesThus the real-axis error function is strictly increasing, with values in $(-1,1)$. Its integrand is an even function, so the error function is an odd function, $\mathrm{erf}(-x)=-\mathrm{erf}(x)$. The derivative also gives ${\mathrm{erf}}^{\prime \prime }(x)=-4x{e}^{-x^2}/\sqrt{\pi }$: the curve is concave for $x>0$ and convex for $x<0$. These real-axis properties do not imply boundedness throughout the complex plane; the existing quadrant asymptotics of the error function describe a different behavior in other directions.

Every regular cone parametrized by $X(r,t)=rc(t)$ has zero Gaussian curvature away from its vertex. Its unit normal is independent of $r$, whileis tangent, so two coefficients of the second fundamental form vanish and $\det II=0$.

Every rational function in $\mathbb{C}(X,Y)$ fixed by exchanging $X$ and $Y$ is a quotient of coprime symmetric polynomials. In a coprime presentation $p/q$, symmetry givesUnique factorization implies $p^{\ast }=\lambda p$ and $q^{\ast }=\lambda q$. The alternative $\lambda =-1$ would make both divisible by $X-Y$, contradicting coprimality, so $\lambda =1$.

If $R$ is a unique factorization domain with fraction field $F$, then $R[X]$ is a unique factorization domain. A primitive polynomial irreducible in $R[X]$ remains irreducible, hence prime, in the principal ideal domain $F[X]$; Gauss lemma for polynomials brings divisibility back to $R[X]$.

For a two-plane $\sigma =\mathrm{span}(X,Y)$ in a Riemannian tangent space,

If $m$ is a positive integer and $U_1,\dots ,U_m$ are independent uniform variables, thenhas the gamma distribution with shape $m$ and rate $r$, because each summand is exponential with rate $r$.

The sum of $n$ independent exponential variables of rate $r$ has the Erlang distribution with densityConvolving with one more exponential proves the formula inductively.

If $X_0$ embeds compactly into $X$ and $X$ embeds continuously into $X_1$, then boundedness in $L^p(0,T;X_0)$ together with a suitable time-derivative bound in $L^q(0,T;X_1)$ makes a family relatively compact in $L^p(0,T;X)$. A standard case is

Attach Möbius bands to a torus along the classes $ab$ and $a^2{b}^3$. If their core generators are $x,y$, the resulting presentation isBecause the exponent vectors $(1,1)$ and $(2,3)$ form a unimodular basis of ${\mathbb{Z}}^2$, eliminating $a,b$ givesThis group maps onto $S_3$ by $x\mapsto (12)$ and $y\mapsto (23)$, so it is nonabelian.

A group presentation gives a two-dimensional cell complex with one vertex, one oriented loop for each generator, and one 2-cell attached along each relator. Its fundamental group is the presented group.

This equation characterizes a minimal surface presented as a graph.

The graph of $h$ is minimal exactly whenEquivalently, it is the Euler--Lagrange equationfor the area functional $\int \sqrt{1+|\nabla h{|}^2}$.