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Under standard regularity conditions, $I_n(\theta )=-{\mathbb{E}}_{\theta }[{\nabla }_{\theta }^2{\ell }_n(\theta )]$.

Forwhere the ${\epsilon }_i$ are independent $N(0,1)$ and $0\le \gamma <1$,The information tensorizes exactly when $\gamma =0$.

Fisher information tensorizes when the information in an $n$-observation model is the sum of the information contributions from its observations. For identically distributed observations this meansIndependence and the mean-zero score identity make the cross terms vanish.

Deleting any countable initial segment from ${\omega }_1$ leaves a well-order of type ${\omega }_1$.

The mean recurrence time of $i$ is ${\mathbb{E}}_i{T}_i^{+}$. In a finite irreducible Markov chain with stationary distribution $\pi$, it equals $1/{\pi }_i$.

A universal theory can be axiomatized by sentences whose quantifiers are all universal. Such sentences often forbid specified finite configurations.

Every formal proof is a finite string and therefore uses only finitely many of the available assumptions.

A theory is consistent when it does not derive a contradiction. By first-order completeness, this is equivalent to having a model.

A class of structures is first-order axiomatizable when it is exactly the class of models of some first-order theory. If extra constants are allowed, the corresponding class of reducts consists of structures that admit interpretations of those constants satisfying the theory.

For every one-variable arithmetical formula $\theta (x)$, a sufficiently strong theory of arithmetic has a sentence $\gamma$ satisfying $\gamma \leftrightarrow \theta (┌\gamma ┐)$.

Peano arithmetic is the first-order theory of the natural numbers with zero, successor, addition, multiplication, order, and the induction scheme.

An automorphism is a bijection from a structure to itself that preserves every operation and relation in its language.

A structure $M$ is a model of $T$ when $M\models \sigma$ for every sentence $\sigma \in T$.

A function $f:[a,b]\to \mathbb{R}$ is absolutely continuous exactly when there is a function $g\in L^1(a,b)$ such thatThen $f$ is differentiable almost everywhere and $f^{\prime }=g$ almost everywhere.

A Sobolev extension operator is a bounded linear map $E:W^{1,p}(U)\to W^{1,p}({\mathbb{R}}^n)$ whose restriction to $U$ is the original function. For a half-space, reflection across the boundary gives such an operator.

For every open set $U\subseteq {\mathbb{R}}^n$, smooth functions in $U$ are dense in $W^{1,p}(U)$ for $1\le p<\infty$. Near a flat boundary, one may first translate the function into the domain and then apply a mollifier.

Every element of $W^{1,p}(a,b)$ for $1\le p<\infty$ has an absolutely continuous representative. Its ordinary derivative exists almost everywhere, equals its weak derivative, and belongs to $L^p(a,b)$.

A finite set of formulas mentions only finitely many nonlogical symbols. This observation lets compactness reduce arguments in a language indexed by a large set to a finite sublanguage.

The physical-process first law derives the area change caused by a small flux of matter through an initially and finally stationary horizon. Linearized Null Raychaudhuri equation evolution converts the Killing-energy flux into $\Delta M=\kappa \Delta A/(8\pi )$.

With unit normal $(-Z^{\prime }\cos v,-Z^{\prime }\sin v,Y^{\prime })$, the principal-form coefficients give