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A field configuration or identity is on shell when the equations of motion hold.

If every face of a connected planar graph has size at least $g>2$, double-counting edge-face incidences and using Euler's formula gives

A finite simple graph that is a connected graph and a planar graph, with $n\ge 3$ vertices and $m$ edges, satisfies $m\le 3n-6$. Choose a plane graph: each facial boundary walk has length at least three, so $3f\le 2m$. Substitute $f=2-n+m$ from the Euler formula for a connected planar graph.^[ref]

The complete graph $K_4$ attains equality: $6=3\cdot 4-6$. The bound also holds for a disconnected finite simple graph that is a planar graph with $n\ge 3$: place its connected components of a graph in disjoint regions and join them with new bridges in a graph, then apply the connected bound.

For an integer matrix $A$ and prime $p$, the commuting-matrix binomial theorem and divisibility of $\left(\genfrac{}{}{0ex}{}{p}{j}\right)$ for $0<j<p$ give

For every odd prime,

Euler's criterion at $a=-1$ gives

This is a ball for the Euclidean metric.

The closed Euclidean ball of radius $R$ about $a\in {\mathbb{R}}^n$ is $\{x:\Vert x-a{\Vert }_2\le R\}$; replacing $\le$ by $<$ gives the open ball.

A proper Euclidean motion has the form $E(x)=Ax+b$ with $A\in SO(3)$. It preserves lengths, dot products, cross products, orientation, curvature, and torsion.

This is the orientation-preserving subgroup of the three-dimensional Euclidean group.

In $d$ dimensions, the conformal Killing equation is

Attouch--Wets convergence of closed sets is locally uniform convergence of their distance functions on bounded subsets. Decreasing closed sets converge to their nonempty intersection in this topology.

Spectral pollution is convergence of finite-section eigenvalues to points outside the spectrum of the limiting operator. For bounded operators, such pollution lies in the essential numerical range.

For a measure-preserving transformation $T$ and $f\in L^1(\mu )$, the ergodic averagesconverge almost everywhere to $\mathbb{E}[f\mid \mathcal{I}]$, the conditional expectation on the invariant sigma-algebra. The limit has the same integral as $f$.

A measurable transformation $T:X\to X$ preserves a measure $\mu$ when $\mu (T^{-1}A)=\mu (A)$ for every measurable set $A$.

If a fixed graph $H$ has $v$ vertices, $e$ edges, and $N_H(n)$ unlabelled copies in $K_n$, then the number $X_H$ of its copies in $G(n,p)$ satisfiesThis follows by writing $X_H$ as a sum of indicator random variables.

For every fixed $\epsilon >0$,whereas the probability tends to zero when $p\le (1-\epsilon )\log n/n$. Above the threshold a union bound excludes every component of size at most $n/2$; below it isolated vertices remain with high probability.

Let $N$ be the number of isolated vertices in $G(n,p)$. Indicator variables giveand, because a specified pair is simultaneously isolated precisely when the $2n-3$ incident edges are absent,

Topological equivalence does not determine uniform convergence. For the usual and arctangent metrics on $\mathbb{R}$, maps that change the value $n$ to $2n$ at a single domain point have arctangent sup-error tending to zero while their usual sup-error diverges.

On $\mathbb{R}$, the usual metric and $|\arctan x-\arctan y|$ induce the same topology, but their ratio along $(0,n)$ tends to zero. Thus no positive global lower comparison constant exists.

An adiabatic equation of state relates pressure and volume along a process with no heat transfer. It commonly has the form $p{V}^{\gamma }=\text{constant}$.

An isotropic ideal gas whose particles obey the ultrarelativistic dispersion $\epsilon =pc$ satisfies