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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

The order of a finite group is its cardinality, written $|G|$. The identity element is included: $|\mathbb{Z}/6\mathbb{Z}|=6$, and the trivial group has order one, rather than zero.

Keep this count distinct from the order of a group element. In the additive finite cyclic group $\mathbb{Z}/6\mathbb{Z}$, the element $1$ has order six, $2$ has order three, $3$ has order two, and $0$ has order one. Each element's order is the least positive number of repetitions needed to return to the identity element $0$. The order of an element of a finite cyclic group computes these periods directly.

The count also constrains subgroups: Lagrange's theorem gives $|G|=[G:H]|H|$ for a subgroup $H$ of a finite group $G$, where the index of a subgroup counts its cosets. For $H=\{0,2,4\}$ inside $\mathbb{Z}/6\mathbb{Z}$, the two cosets are $\{0,2,4\}$ and $\{1,3,5\}$, so $6=2\cdot 3$. Divisibility is a necessary constraint; it does not imply that every divisor occurs as the order of a group element, as the Klein four-group shows.

On a finite-dimensional vector space over a complete valued field, any two norms induce the same topology. Choosing a basis reduces the claim to comparison with the maximum of the absolute values of the coordinates.

A filtered ring is a ring with subgroups $(R_n)$ satisfying $R_m{R}_n\subseteq R_{m+n}$.

For an ideal $I$, the $I$-adic filtration of a module $M$ is $M\supseteq IM\supseteq I^{2}M\supseteq \cdots$.

Two decreasing filtrations $(M_n)$ and $(N_n)$ are equivalent when each contains a fixed shift of the other: $M_{n+a}\subseteq N_n$ and $N_{n+b}\subseteq M_n$ for fixed $a,b$ and every $n$.

Let ${\omega }_1,\dots ,{\omega }_n$ be an integral basis and let ${\alpha }_j={\sum }_i{a}_{ij}{\omega }_i$ be a rational basis of algebraic integers, with integer matrix $A=(a_{ij})$. Their trace Gram matrices satisfysoThe absolute discriminant is minimized exactly when $A$ is unimodular, equivalently when the ${\alpha }_i$ form an integral basis.

For nonnegative $n,m,l$,Cassini's identity is the adjacent-index special case.

For $\varphi =(1+\sqrt{5})/2$ and $\psi =(1-\sqrt{5})/2$, the Fibonacci numbers satisfy