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Varying the relativistic charged-particle action with fixed endpoints givesor the equivalent raised-index equation.

Under $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\chi$, the interaction changes byFixed-endpoint equations of motion are therefore gauge invariant.

The preimage of a regular value of a smooth map between manifolds is a smooth submanifold whose codimension equals the dimension of the target.

A regular octagon of side length s has area 2(1+sqrt(2))s^2.

This theorem characterizes a regular language by regular expressions and acceptance by a finite-state automaton. Kleene's theorem states that a language is denoted by a regular expression exactly when it is accepted by a finite automaton.

For a deterministic partition $R_1,\dots ,R_J$ andconditional independence givesIf the conditional response variance is at most ${\sigma }^2$, the prediction variance at an independent test point is at most ${\sigma }^{2}J/n$.

In a string register machine, ${+}_a(k,q)$ appends $a$ to register $k$; ${?}_a(k,p,q)$ branches according to whether its final letter is $a$; ${?}_{\epsilon }(k,p,q)$ tests emptiness; and $-(k,p,q)$ branches to $p$ when empty, or deletes the final letter and branches to $q$ otherwise.

The composition of reflections in two planes through the origin is a rotation about their line of intersection through twice the oriented angle between the planes.

For a nonzero normal vector $a$, reflection in $a\cdot x=0$ has matrixIts eigenvalues are $-1$ along $a$ and $1$ on $a^{\perp }$, so its determinant is $-1$.

For a negative discriminant $D\equiv 0$ or $1(\mathrm{mod}4)$, the class number $h(D)$ is the number of proper equivalence classes of primitive positive definite integral binary quadratic forms of discriminant $D$. Equivalently, it is the number of reduced primitive forms of discriminant $D$.

For a reduced form of negative discriminant $D$, the inequalities imply $3a^2\le |D|$. One can therefore enumerate all classes by checking the finite setand retaining integral $c=(b^2-D)/(4a)$ with $a\le c$ and the boundary sign convention.

The Bell states $|{\Phi }^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$ and $|{\Phi }^{-}⟩=(|00⟩-|11⟩)/\sqrt{2}$ both have reduced state $I/2$ on either qubit. No measurement on one qubit alone can distinguish their relative sign.

A bipartite density operator is entangled when it is not a separable quantum state. For a pure state, this is equivalent to not being a product state, or to either reduced density matrix having rank greater than one.

A bipartite density operator is a product state when it factors as ${\rho }_A\otimes {\rho }_B$. A pure product state factors as $|{\psi }_A⟩\otimes |{\psi }_B⟩$; its reduced density matrices have rank one and zero entanglement entropy. Mixed product states can have nonzero marginal Von Neumann entropy.

For a nonempty set $A$ of words, the following are equivalent: $A$ is computably enumerable, $A$ is the domain of a partial computable function, $A$ is the range of a partial computable function, and $A$ is the range of a total computable function. The nonempty hypothesis is needed only for the last formulation, since the range of a total function cannot be empty.

Doubling ${\mathbb{RP}}^3$ minus an open ball along its spherical boundary gives ${\mathbb{RP}}^3\#{\mathbb{RP}}^3$. Its fundamental group isand its universal cover is $S^2\times \mathbb{R}$.

The integral homology groups are

The complement of an open three-ball in ${\mathbb{RP}}^3$ is the mapping cylinder of the antipodal covering $S^2\to {\mathbb{RP}}^2$. It deformation retracts to ${\mathbb{RP}}^2$ and has boundary $S^2$.

The mod-two cohomology ring of real projective space is

A projective Cremona transformation is a birational self-map of projective space, regular on complementary dense open subsets together with its inverse.

Composition of these birational self-maps forms the Cremona group.