On the punctured Euclidean plane, the orthonormal polar frame isIts bracket is $[\hat{r},\hat{\theta }]=-\hat{\theta }/r$, so it is not a coordinate basis even though $({\partial }_r,{\partial }_{\theta })$ is one.

For the graph map $(1,f):M\to M\times M$, the cohomology class of the diagonal satisfiesIf $f$ is fixed-point-free, its graph misses the diagonal, so the pullback vanishes. This proves the Lefschetz fixed-point theorem for closed oriented manifolds.

A function $f:\beta \to \alpha$ between ordinals is cofinal when its range is unbounded in $\alpha$: for every $\gamma <\alpha$ some $\xi <\beta$ satisfies $\gamma \le f(\xi )$.

Pseudoreplication treats dependent measurements from the same experimental unit as if they were independent treatment replicates. It exaggerates the effective sample size and commonly makes standard errors too small. In the pot example of replication in experimental design, counting twelve leaves per group as twelve independent treatment replicates ignores the four containing pots. The variance of nested measurements quantifies the resulting understatement of uncertainty. Nested measurements themselves are useful; pseudoreplication is the error of using them as independent replicates for the wrong level of inference.^[ref]

Double-negation elimination is the inference from $\neg \neg A$ to $A$.

The law of excluded middle asserts that every proposition $A$ satisfies $A\vee \neg A$.

Let a compact smooth surface of revolution have exactly two poles, and let $c=f(v{)}^2\dot{u}=f(v)\cos \theta$ be the Clairaut first integral for a surface of revolution of a complete unit-speed geodesic. If $c\ne 0$, then $f(v)\ge |c|$, so the geodesic avoids open neighbourhoods of both poles. If $c=0$, then $u$ is constant away from the poles and the geodesic lies in a meridian of a surface of revolution, whose complement contains a nonempty open set. Thus no complete geodesic has dense image.

A photon sphere is a spherical set of circular null geodesics. In Schwarzschild spacetime it occurs at $r=3M$ and is unstable.

Pi is the ratio of a Euclidean circle's circumference to its diameter.

For every $g,k\ge 3$, there is a finite graph with no cycle of length at most $g$ and with chromatic number at least $k$. A probabilistic proof samples a sparse binomial random graph, observes that it has few short cycles and no large independent set, and deletes one vertex from every short cycle.

Forwrite dots and primes for $t$- and $r$-derivatives. Up to symmetry in the lower indices, the nonzero connection coefficients are

For a subgroup $H\le G$, the restriction of a $G$-character $\chi$ to $H$ is the $H$-character obtained by evaluating the same function only on elements of $H$.

An irreducible character $\phi$ is a constituent of a character $\chi$ when its multiplicity $⟨\chi ,\phi ⟩$ is positive.

Suppose $\mathbb{P}$ has the $\kappa$-chain condition and $\dot{f}$ is forced to be a function from an ordinal $\mu$ to an ordinal $\lambda$. For each $\xi <\mu$, a maximal antichain in a forcing order deciding $\dot{f}(\xi )$ has size below $\kappa$. Hence the ground model has a set $B_{\xi }\subseteq \lambda$ of size below $\kappa$ containing every possible value of $\dot{f}(\xi )$.

A chain subcomplex $D_{\bullet }\subseteq C_{\bullet }$ consists of subgroups $D_i\le C_i$ preserved by the boundary operator: $\partial (D_i)\subseteq D_{i-1}$.

The chain group $C_i$ is the group in degree $i$ of a chain complex. Its elements are $i$-chains.

The Lindeberg condition requires the contribution of unusually large summands in a triangular array to become negligible compared with the array's total variance. Together with convergence of that variance, it gives the Lindeberg central limit theorem.

For every irreducible complex character $\chi$ of a finite group $G$,Indeed, row orthogonality expresses $|G|/\chi (1)$ as a sum of products of algebraic integers:It is a rational algebraic integer and therefore an integer.

For $X\sim \mathrm{Cauchy}(x_0,\gamma )$,In particular, the Standard Cauchy distribution has characteristic function $e^{-|t|}$.

The standard Cauchy distribution is $\mathrm{Cauchy}(0,1)$, with density $1/[\pi (1+x^2)]$ and characteristic function $e^{-|t|}$.