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A contraction mapping $T:X\to X$ on a nonempty complete metric space has exactly one fixed point $x_{\ast }$, and the sequence $x_{n+1}=T(x_n)$ converges to $x_{\ast }$ from every $x_0\in X$. The same constant $0\le L<1$ must satisfy $d(Tx,Ty)\le Ld(x,y)$ for all $x,y\in X$. The statement applies to metric spaces, including closed subsets of a Banach space; neither linearity nor a vector space structure is required.^[ref]

The force of the result is that existence, uniqueness, and a practical approximation procedure arrive together. Repeated application produces corrections bounded by a geometric series, hence a Cauchy sequence; completeness supplies its limit. The Lipschitz continuity of $T$ makes that limit a fixed point. Two different fixed points would have their positive distance multiplied by at most $L<1$ while remaining unchanged, which is impossible.

A topological space has the fixed-point property when every continuous self-map has a fixed point.

Every monotone self-map of a complete lattice has a complete lattice of fixed points. Its least fixed point is the meet of its prefixed points, and its greatest fixed point is the join of its postfixed points.

Let $T$ be a compact continuous self-map of a Banach space. If the set of solutions of $u=tT(u)$ for $0\le t\le 1$ is bounded, then $T$ has a fixed point. The theorem converts a priori estimates for a homotopy of equations into existence.

An imaginary-time-ordered correlation function orders operators along the Euclidean thermal circle. Bosonic correlators are periodic with period $\beta$.

For a bosonic operator $A$, the retarded Green function is $G_R(t)=-i\theta (t)⟨[A(t),A(0)]⟩$. Causality makes its Fourier transform analytic in the upper half frequency plane.

The nonzero elements of every finite field form a cyclic group. For a field with $q$ elements, combine elements realizing each prime-power factor of the group exponent to obtain an element of exponent order; the root bound for $X^m-1$ then forces that order to be $q-1$.

Primordial non-Gaussianity is any departure of the primordial perturbations from Gaussian random fields. Cubic interactions generate a nonzero three-point correlation function at leading order.

This is non-Gaussianity of primordial cosmological perturbations rather than non-Gaussian statistics of an arbitrary physical measurement.

A $P(X,\varphi )$ scalar theory has Lagrangian density $P$ depending on the field and $X=-g^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi /2$. It includes canonical scalar theory and models with noncanonical kinetic interactions.

A k-essence model uses noncanonical kinetic terms for cosmological acceleration; a general P(X, phi) scalar field theory need not have that application.

If $L/K$ and $K/F$ are algebraic field extensions, then $L/F$ is algebraic. In particular, an element algebraic over an algebraic extension of $F$ is algebraic over $F$.

This is a closure property of algebraic extensions, not an alternative name for an extension.

A quadratic extension is a field extension of degree two. It is normal because the other root of the minimal polynomial $T^2+bT+c$ of any non-base-field element is $-b-\alpha$ and lies in the same extension.

For a field extension $L/K$, an intermediate field is a field $M$ satisfying $K\subseteq M\subseteq L$.

An algebraic closure of a field $K$ is an algebraic extension $\overline{K}/K$ for which $\overline{K}$ is algebraically closed.

Inside a common overfield, the compositum $L_1{L}_2$ is the smallest field containing both $L_1$ and $L_2$.

The multiplication image of the tensor product of fields in a common overfield generates the compositum as a field. For algebraic extensions that image is already a field; in general it need not be. The compositum is therefore distinct from the tensor product algebra.

For finite field extensions $K\subseteq L\subseteq M$, the degree of a field extension satisfiesThis tower law is a statement about the dimension of the underlying vector spaces.

Choose a basis $u_1,\dots ,u_r$ of $L$ over $K$ and a basis $v_1,\dots ,v_s$ of $M$ over $L$. The products $u_i{v}_j$ span $M$ over $K$: first expand an element using the $v_j$, then expand each coefficient using the $u_i$.

They are also linearly independent. If ${\sum }_{i,j}{a}_{ij}{u}_i{v}_j=0$ with $a_{ij}\in K$, independence of the $v_j$ gives ${\sum }_i{a}_{ij}{u}_i=0$ for every $j$; independence of the $u_i$ then gives every $a_{ij}=0$. There are $rs$ products, proving the formula.

The characteristic of a field $F$ is the least positive integer $n$ for which $n\cdot 1_F=0$, or zero if no such integer exists.

A field is algebraically closed when every nonconstant polynomial over it has a root in it, equivalently when every such polynomial splits into linear factors.

The lowering operator sends the occupied state to the vacuum and kills the vacuum; its adjoint performs the reverse and kills the occupied state.

The fermionic occupation Hamiltonian satisfies $H^2=H$, so its spectrum lies in $\{0,1\}$.

Let $N>2$ be an odd integer that is a composite number and write $N-1=2^{s}m$ with $m$ odd. For a base $b$ coprime to $N$, it is a strong pseudoprime whenThese are exactly the passing outcomes of the Miller-Rabin primality test for that base. The test computes $b^m$ by modular exponentiation and repeatedly squares it, looking for the permitted value $-1$ before reaching exponent $N-1$.

Every strong pseudoprime to a base is a Fermat pseudoprime to that base, by squaring a permitted congruence sufficiently many times. The converse fails. For $N=341$, one has $N-1=4\cdot 85$ and $2^{85}\equiv 32$, $2^{170}\equiv 1(\mathrm{mod}341)$. Neither tested residue is $-1$, and the first is not $1$, so base two detects compositeness even though $2^{340}\equiv 1(\mathrm{mod}341)$.