The commutant of $A\subseteq \mathrm{End}(T)$ is $A\prime =\{b:ba=ab\text{ for all }a\in A\}$. This centralizer is an operator algebra. The double-centralizer theorem for semisimple operator algebras and Schur–Weyl duality use this meaning, which is distinct from the centralizer of a single element of a group.

A central simple algebra $A$ has dimension $d^2$ over its center. Its degree is this integer $d$. Over a splitting field it becomes $M_d$, so degree records the matrix size independently of the chosen splitting.

An extension $E/K$ splits a central simple algebra $A$ of degree $d$ when $A{\otimes }_{K}E\cong M_d(E)$. A finite separable splitting field exists. Splitting is preserved by further scalar extension and is equivalent to the vanishing of the restricted Brauer group class.

Every $L$-algebra automorphism of $M_d(L)$ is conjugation by an invertible matrix. For matrix units, the vectors $h(E_{i1})w$ with $0\ne w\in \mathrm{im}h(E_{11})$ form a basis in which $h$ acts in the usual way on all matrix units. Scalar conjugating matrices act trivially, giving ${\mathrm{Aut}}_L(M_d(L))={\mathrm{PGL}}_d(L)$.

The reduced norm of an element of a central simple algebra of degree $d$ is its determinant after any splitting isomorphism to $M_d$. Inner automorphisms of a matrix algebra preserve that determinant, while Galois invariance descends its polynomial coefficients to the center. It is a multiplicative homogeneous degree-$d$ polynomial, vanishing exactly on noninvertible elements. On a central division algebra it is anisotropic.

A continuous map between CW complexes is homotopic to a cellular map. If the map is already cellular on a subcomplex, the homotopy can be taken relative to that subcomplex. In particular, a map from a $d$-dimensional CW complex can be deformed into the $d$-skeleton of the target. This permits finite cellular chain complexes to compute induced homology maps, and turns maps to Complex projective space into maps to an appropriate finite-dimensional projective skeleton.

A real sequence converges if and only if the displayed condition holds. A Cauchy sequence makes all future differences uniformly small, so the forward implication is immediate. Conversely, failure of the Cauchy sequence criterion produces a fixed tolerance and, for each $n$, a future term differing from $a_n$ by at least half that tolerance. Choosing the least such positive lag defines one function that contradicts the displayed condition. The completeness of the real numbers then gives convergence. Testing only every fixed integer lag is weaker: $a_n=\log n$ passes those tests but diverges.

A regression factor is a categorical variable encoded by indicator columns or treatment contrasts in a statistical model. A factor with $k$ observed levels normally contributes $k-1$ statistical degrees of freedom when an intercept is present. The reference level in a regression factor has coefficient zero under treatment contrasts. Coding a year as a factor permits arbitrary year effects rather than imposing a linear trend.

A Cartesian formula is built from atoms and equality using truth, finite conjunction and uniquely witnessed existential quantification. Uniqueness is proved relative to the theory before admitting the quantifier. In a finite-limit category, such a quantified relation projects monomorphically into the remaining context, so its semantics needs no arbitrary image operation.

Objects are Cartesian formulas-in-context modulo provable equivalence, and arrows are provably total single-valued Cartesian relations. Composition quantifies the unique intermediate tuple. Context concatenation, conjunction and equality give finite limits. Its canonical model interprets a formula as its own context subobject, so factorization of antecedent through consequent is exactly sequent derivability.

In the positive finite-limit convention, a Horn theory uses atomic finite-conjunction sequents; uniquely witnessed existential extensions remain Cartesian. Its models are the finite-limit-preserving functors out of its Cartesian syntactic category. This convention does not insert arbitrary existential witnesses or falsity as Cartesian constructors. Its classifier is the presheaf classifier of a Horn theory.

For capillary residual trapping of an initial triangular current, the method of characteristics translates its leading and trailing faces at $v_A=u/\varphi$ and $v_R=u/[\varphi (1-s)]$. Their intersection has height $a[L-(v_R-v_A)t/2]$. Extinction occurs at $t_{\ast }=2L/(v_R-v_A)$ and position $x_{\ast }=2L/s$. These formulas require $0<s<1$; at zero residual saturation the triangle translates indefinitely.

Given a continuous surjection $h:\Omega \to K$ from Cantor space to a nonempty compact metric space, the pullback $f\mapsto f\circ h$ is a unital isometric embedding of $C(K)$ into $C(\Omega )$. Transport a normalized positive linear functional to its image, take a positive extension from a unital subspace of C(K), and represent that extension on Cantor space. The pushforward measure $h_{\ast }P$ then represents the original functional.

A Cantor cylinder fixes finitely many of the zero-or-two ternary digits in the standard Cantor set. These sets form finite clopen partitions of diameter at most $3^{-n}$ at level $n$, and a base for the relative topology. They permit uniform approximation of continuous functions by locally constant functions.

Spatial integration of the free Hamiltonian density cancels particle-antiparticle pair terms through $E_p^2-{\mathbf{p}}^2-m^2=0$. The raw Hamiltonian is $\int d{\Pi }_p{E}_p(a^{†}a+b{b}^{†})$. Normal ordering removes the field-independent vacuum energy and gives $H=\int d{\Pi }_p{E}_p(a^{†}a+b^{†}b)$. Both creation operators increase the energy by $E_p$, so a negative-frequency field mode represents a positive-energy antiparticle, not a negative-energy state.

For the global phase transformation $\delta \varphi =-i\alpha \varphi$, the Noether current is $j^{\mu }=i({\varphi }^{†}{\partial }^{\mu }\varphi -{\partial }^{\mu }{\varphi }^{†}\varphi )$. The Klein-Gordon equation gives ${\partial }_{\mu }{j}^{\mu }=0$. With zero vacuum charge, its normal-ordered charge is $Q=\int d{\Pi }_p(a^{†}a-b^{†}b)$. Thus $[Q,a^{†}]=a^{†}$, $[Q,b^{†}]=-b^{†}$ and $[Q,\varphi ]=-\varphi$. Particle and antiparticle creation carry opposite charges while both increase energy.

For a dimensionless two-point factor obeying $(-2r{\partial }_r+\beta {\partial }_g+2\gamma )C=0$, evolve $g^{\prime }=\beta (g)$ and accumulate the displayed anomalous-dimension multiplier. The flow property verifies the equation: its transport generator is $\beta {\partial }_g+2\gamma$. This fixes a frequent sign ambiguity between raising the physical momentum and changing the reference scale.

For $t=\log (|p|/\mu )$ and $d\bar{g}/dt=\beta (\bar{g})$, the displayed characteristic solution follows from $(-{\partial }_t+\beta {\partial }_g+2{\gamma }_{\varphi })d=0$. It separates the running coupling from the accumulated field-renormalization anomalous dimension. The exponent sign is determined by the precise Callan-Symanzik equation convention.

For a central division algebra $D$ of degree $d$ over a C2 field, the form $\mathrm{Nrd}(X)-a{T}^d$ has degree $d$ in $d^2+1$ variables. Its nontrivial zero must have $T\ne 0$, because the reduced norm of a division algebra is anisotropic. Dividing $X$ by $T$ gives norm $a$. Zero is the norm of zero.

The function field of a curve over an algebraically closed field is a C1 field. Consequently its Brauer group is zero. This is the input for cyclic splitting by extension of constants over a finite constant field.