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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 a nested family of additive subgroups compatible with multiplication. In the exhaustive increasing convention, $F_{i}R\subseteq F_{i+1}R$, ${\bigcup }_i{F}_{i}R=R$, $1\in F_{0}R$ and $F_{i}R{F}_{j}R\subseteq F_{i+j}R$. These are the conditions in filtration of a ring.

The inclusions are part of the definition; a collection of unrelated subgroups satisfying a product condition is insufficient. For example, the I-adic filtration can be put in this convention by setting $F_{-n}R=I^n$ and $F_{i}R=R$ for $i\ge 0$. Its associated graded ring records successive orders of vanishing.

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

A Feynman parameter combines propagator denominators; for example,After a momentum shift, this often converts a loop integral into a rotationally symmetric one.

At zero temperature, a two-dimensional free Fermi gas satisfies

For nonrelativistic electrons in area $A$, with spin degeneracy $g_s=2$ and energy $E={\hslash }^2{k}^2/(2m)$, the density of one-particle states is independent of energy:

Pauli paramagnetism is the weak positive magnetic response produced when a magnetic field shifts the two spin Fermi surfaces of a degenerate electron gas. At zero temperature, if $g(E_F)$ includes both spin states, its total magnetic moment and susceptibility are

Degeneracy pressure is the pressure associated with filling fermionic states under the Pauli exclusion principle, including at zero temperature. Compressing a fixed number into a smaller volume forces the highest occupied momenta upward, raising the ground-state energy.

For a three-dimensional nonrelativistic ideal Fermi gas, $U=(3/5)N{E}_F$ and the volume dependence gives $P=(2/5)n{E}_F$, where $n=N/V$. This pressure does not require thermal collisions to maintain a nonzero temperature. It contributes to the support of dense stellar matter, with relativistic motion and additional interactions changing the equation of state at sufficiently high density. Exclusion constrains occupation; the pressure follows from the energy cost of that constrained filling, rather than being a separate universal pairwise repulsive force.

For spin-one-half particles in three dimensions with dispersion $\epsilon =pc$, the density of states isAt zero temperature,