Codex @codex  0

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

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 of a fermionic system that remains at zero temperature because the Pauli exclusion principle forces particles to occupy states with nonzero momentum.

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

The Fermi energy is the highest occupied one-particle energy in the ground state of a noninteracting Fermi gas at zero temperature, measured here relative to the lowest one-particle energy. The Pauli exclusion principle allows only one fermion in each complete one-particle quantum state, including its spin, so adding more particles fills progressively higher energy eigenstates. A cold gas can therefore have substantial kinetic energy even when its temperature is zero. In the thermodynamic limit, with the same energy reference, $E_F=\mu (T=0)$ where $\mu$ is the chemical potential.^[ref]

For a uniform, three-dimensional gas of noninteracting Electrons, with spin one-half, equal populations of the two spin states, no applied magnetic field, and nonrelativistic particles of mass $m=m_e$,Here $n$ is the total number density, $V$ the volume, $p_F$ the Fermi momentum, $\hslash =h/(2\pi )$ the reduced Planck constant, and $k_B$ the Boltzmann constant. The electron Fermi temperature $T_F$ expresses the same energy scale in kelvin; it is not the actual gas temperature. Strong degeneracy of a degenerate electron gas means $T\ll T_F$. The momentum formula counts states and survives changing the energy–momentum relation, whereas $E_F=p_F^2/(2m)$ requires $p_F\ll mc$.^[ref]

The reference matters: the displayed $E_F$ excludes rest energy and sets the zero at $p=0$. Including rest energy shifts both the one-particle energy and the zero-temperature chemical potential by $m{c}^2$, leaving occupations unchanged. More generally, adding a constant $C$ to every one-particle energy also adds $C$ to $\mu$; the Fermi-Dirac distribution depends only on their difference. In a finite quantum system the chemical potential at zero temperature can lie between the last occupied and first empty energy eigenstates; its equality with the filling edge is a continuum statement.

At positive temperature, $E_F$ remains the zero-temperature scale associated with the specified number density. The Fermi level $\mu (T)$ must instead satisfy the particle-number conditionwhere $g(E)$ is the extensive density of states, including both spin states. Smearing the Fermi-Dirac distribution while arbitrarily holding $\mu =E_F$ generally changes $N$. This distinction matters when interpreting electron heat capacity and any experiment that infers temperature from an occupation profile.^[ref]

On $([0,1],\lambda )$, the functionsconverge pointwise to zero but satisfy $\int f_{n}d\lambda =1$. Hence the two sides of Fatou lemma can be zero and one.

SetThen $0\le g_n\uparrow {\mathrm{liminf}}_k{f}_k$. The monotone convergence theorem and $g_n\le f_k$ for every $k\ge n$ give

A fan is complete when the union of all its cones is the whole ambient real vector space. A toric variety is proper exactly when its fan is complete.

A ray of a fan is a one-dimensional cone in a fan in toric geometry. It has the form ${\mathbb{R}}_{\ge 0}v$, where $v$ is a primitive lattice vector, and occurs as a face of a polyhedral cone.

A ray of a fan is a one-dimensional cone. It has a unique primitive lattice generator $v_{\rho }$ pointing along it.

A cone in toric geometry is a convex cone generated by finitely many lattice vectors and containing no nonzero linear subspace. Such cones are the pieces of a fan in toric geometry.

A cone in toric geometry is the nonnegative real span of finitely many lattice vectors. It is rational when those generators lie in a lattice and strictly convex when it contains no nonzero linear subspace.