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If $0<\mathbb{E}{\xi }_1=1/\lambda <\infty$, then the strong law $S_n/n\to 1/\lambda$ and the inequalities $S_{N_t}\le t<S_{N_t+1}$ imply $N_t/t\to \lambda$ almost surely.

For independent identically distributed regenerative cycles with $0<\mathbb{E}{\xi }_1<\infty$ and $\mathbb{E}|R_1|<\infty$, the long-run reward rate isalmost surely, provided the unfinished-cycle reward is negligible after division by $t$.

Under a constant force $F$ with $p(0)=0$, momentum is $p=Ft$ andThe speed approaches $c$ without reaching it at finite time.

For $\mathbf{E}=(0,0,E)$, $\mathbf{B}=0$, and initial momentum $(p_0,0,0)$,With the initial position at the origin,

In an isolated relativistic collision, total four-momentum is conserved; its time component states conservation of total relativistic energy.

This is the relativistic form of conservation of energy, expressed through the conserved four-momentum.

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 polygon with eight sides, each of length $s>0$, has areaJoin its center to all eight vertices of a polygon. The resulting congruent isosceles triangles each have base $s$ and central angle $\pi /4$. Bisecting one gives a right triangle with opposite side $s/2$ and adjacent side $h$, the perpendicular center-to-side distance. HenceAdding the eight triangle areas gives $A=8(sh/2)=4sh$, proving the formula. For side length one, the area is $2+2\sqrt{2}$.

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 of a binary quadratic form $D\equiv 0$ or $1(\mathrm{mod}4)$, the class number of a negative discriminant $h(D)$ counts proper equivalence classes of primitive binary quadratic forms that are positive definite and have discriminant $D$.

Uniqueness of the reduced positive definite binary quadratic form in each class makes this the number of reduced primitive forms. Enumeration of reduced binary quadratic forms proves finiteness and gives an explicit procedure. For example, $h(-4)=1$, represented by $[1,0,1]$, while $h(-39)=4$, with the four forms listed under reduced positive definite binary quadratic forms of discriminant minus thirty-nine.

Imprimitive forms do not enter $h(D)$. At $D=-32$, the reduced form $[2,0,4]$ has content two, so the three reduced forms listed under reduced forms of discriminant minus thirty-two contribute only two primitive classes. If $D$ is a fundamental discriminant, this form class number agrees with the class number of the imaginary quadratic field of discriminant $D$. For nonfundamental $D$, the associated order in a number field need not be the full ring of integers, so that distinction must be retained.

Let $D<0$ with $D\equiv 0$ or $1(\mathrm{mod}4)$. A reduced positive definite binary quadratic form $[a,b,c]$ of discriminant of a binary quadratic form $D$ satisfiesConsequently every reduced form can be found by a finite search:

To count the class number of a negative discriminant, additionally retain only primitive binary quadratic forms, those with $\gcd (a,b,c)=1$. Otherwise the search also lists imprimitive forms. For example, $D=-12$ gives $[1,0,3]$ and $[2,2,2]$, but only the first is primitive, so $h(-12)=1$. This distinction is essential when turning an enumeration into a class-number claim.

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.