For numbered states of a finite-state automaton, $R_{ij}^{(k)}$ is a regular expression describing paths from $i$ to $j$ whose internal states have numbers at most $k$. Splitting a path at its visits to state $k$ gives $R_{ij}^{(k)}=R_{ij}^{(k-1)}+R_{ik}^{(k-1)}(R_{kk}^{(k-1)}{)}^{\ast }{R}_{kj}^{(k-1)}$. At stage zero use individual edge labels and an empty-word option when the endpoints coincide. This proves the automaton-to-expression direction of Kleene theorem.

For an invertible finite-mode Bogoliubov transformation $a=A{a}^{\prime }+B{a}^{\prime }{}^{†}$, the symmetric squeezing matrix is $M=-A^{-1}B$. The canonical commutation relations make its singular values strictly less than one. The normalized state has $|C|=\det (I-M{M}^{†}{)}^{1/4}$. Infinitely many modes additionally require bosonic mode mixing implementability.

If $e=ir$ and $ri=1$, then $r$ is the coequalizer of $(e,1)$: an arrow $h$ with $he=h$ factors uniquely as $(hi)r$. Conversely a coequalizer $q$ makes $e$ factor as $iq=e$; epimorphic cancellation gives $qi=1$. Thus the splitting of an idempotent morphism is equivalent to this particular coequalizer, without assuming arbitrary coequalizers exist.

For distinct interpolation sites and an invertible B-spline collocation matrix, the interpolant is $P=B{A}^{-1}R$, where $R$ samples data and $Ba={\sum }_i{a}_i{N}_i$. Sampling and basis synthesis have operator norms at most one in the supremum norm. For the reverse inequality, realize the signs of a maximal absolute row sum of $A^{-1}$ as values of a continuous function of norm one, and apply uniform-norm stability of a B-spline basis. This yields both displayed bounds.

In a real-generator convention, $M^{\rho \sigma }=\frac{1}{4}[{\gamma }^{\rho },{\gamma }^{\sigma }]$ obeys $[M^{\rho \sigma },{\gamma }^{\mu }]=g^{\sigma \mu }{\gamma }^{\rho }-g^{\rho \mu }{\gamma }^{\sigma }$. Applying this identity to the commutator of two gamma matrices proves the Lorentz algebra relations. With the matching vector generators and real antisymmetric parameters, $S=\exp (\frac{1}{2}{\Omega }_{\rho \sigma }{M}^{\rho \sigma })$ obeys $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$. A convention using Hermitian Lorentz generators instead must consistently insert factors of $i$.

The generators in the Spinor representation of the Lorentz group satisfy $M^{†}{\gamma }^0+{\gamma }^{0}M=0$, so the finite transformation preserves the indefinite form $S^{†}{\gamma }^{0}S={\gamma }^0$. This makes $\bar{\psi }\psi$ a Lorentz scalar through the Dirac adjoint. Boost generators are Hermitian rather than anti-Hermitian: a boost has eigenvalues $e^{\pm \eta /2}$ and is not unitary in the ordinary positive-definite component norm. Rotations are unitary. This component representation is distinct from the unitary action on the physical state space.

With $\{{\gamma }_{\mu },{\gamma }_{\nu }\}=2{\eta }_{\mu \nu }$, $\Omega ={\omega }^{\mu \nu }{\gamma }_{\mu \nu }/4$, and $\Lambda =e^{\omega }$, the Clifford algebra gives $[\Omega ,{\gamma }^{\rho }]=-{\omega }^{\rho }{}_{\mu }{\gamma }^{\mu }$. Exponentiation proves the inverse Lorentz action displayed above, equivalently $S^{-1}{\gamma }^{\rho }S={\Lambda }^{\rho }{}_{\mu }{\gamma }^{\mu }$. This second order is what combines with $\psi \mapsto S\psi$ and the Dirac adjoint action $\bar{\psi }\mapsto \bar{\psi }{S}^{-1}$ to make the gamma bilinears transform as ordinary contravariant tensors.

A spin structure is a consistent choice of lifting oriented orthonormal frame changes to their spin double cover, permitting globally defined spinor fields. For a spatial circle in a worldsheet, the two choices give periodic and antiperiodic fermions, hence the Ramond sector and Neveu–Schwarz sector. Local field equations do not choose the global spin structure.

For a finite singular Borel measure, choose compact subsets of a Lebesgue-null carrier whose omitted mass is arbitrarily small. Outside any such compact subset, sufficiently small balls see only the omitted measure. The uncentered maximal weak-type inequality bounds the volume of points where the omitted measure has large density. Letting its total mass tend to zero shows that the upper spherical density is zero almost everywhere. This argument does not require the null carrier itself to be closed or separated from every point outside it.

For an oriented rank-four real vector bundle $E\to S^4$, its sphere bundle has fibre $S^3$. Let $m=⟨e(E),[S^4]⟩$ be its Euler class evaluated on the orientation class. The Gysin sequence gives integral homology $\mathbb{Z}$ in degrees $0,7$, $\mathbb{Z}/m\mathbb{Z}$ in degree $3$, and a copy of $\mathbb{Z}$ in degree $4$ exactly when $m=0$, with all other groups zero. Here $\mathbb{Z}/0\mathbb{Z}=\mathbb{Z}$. A transition function $v\mapsto q^{k}v{q}^l$ on the unit quaternions has Euler number $k+l$, up to an overall orientation sign: evaluating it at a fixed unit vector gives the power map $q\mapsto q^{k+l}$, whose mapping degree is $k+l$. The same homology calculation follows by applying the Mayer–Vietoris sequence to the two trivializations over the hemispheres.

A weakly stationary process has a time-series spectral density precisely when its spectral measure of a stationary time series is absolutely continuous with respect to Lebesgue measure. Absolute summability of its autocovariances is sufficient and yields a continuous Fourier-series density. It is not necessary for existence; for a general integrable density, Fejér sums give an $L^1$ recovery formula.

For a real weakly stationary process, restricting its even angular-frequency density $f$ to $[0,\pi ]$ gives $\gamma (k)=2{\int }_0^{\pi }f(\omega )\cos (k\omega )d\omega$. Folding both frequency halves into one density gives $g=2f$ and removes that factor two. Consequently white noise of variance $v$ has restricted density $v/(2\pi )$ and folded density $v/\pi$.

For real stationary processes, the one-sided density on $[0,\pi ]$ is twice the usual two-sided density. Its integral is the variance and ${\gamma }_k={\int }_0^{\pi }f(\omega )\cos (k\omega )d\omega$. When covariances are absolutely summable, $f(\omega )={\pi }^{-1}({\gamma }_0+2{\sum }_{k\ge 1}{\gamma }_k\cos (k\omega ))$.

The SU(3) group consists of unitary complex three-by-three matrices of determinant one. Its defining action on ${\mathbb{C}}^3$ preserves the Hermitian norm. It contains block-diagonal copies of SU(2) and the real SO(3) group. Its Lie algebra consists of traceless anti-Hermitian matrices.

For $n\ge 2$, the special unitary group acts transitively on the complex unit sphere. Extend any unit vector to an orthonormal basis, then adjust the phase of a remaining column to give determinant one. The stabilizer subgroup of the first vector consists of special unitary transformations of its orthogonal complement. Thus the sphere is a homogeneous quotient, not just a set with a norm-preserving action.

The four-by-four unitary matrices of determinant one form this special unitary group. Its two-index antisymmetric representation has complex dimension six and the real structure of the SU4 exterior square realizes it on a six-dimensional real space.

A matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ acts on the projective line by the Möbius transformation $x\mapsto (ax+b)/(cx+d)$, with the pole sent to infinity and infinity sent to $a/c$ when $c\ne 0$. The action is transitive because $\left(\begin{array}{cc}t& -1\\ 1& 0\end{array}\right)$ sends infinity to any finite $t$. The stabilizer subgroup of infinity consists of $\left(\begin{array}{cc}a& b\\ 0& a^{-1}\end{array}\right)$ with $a\ne 0$. The orbit-stabilizer theorem gives $|{\mathrm{SL}}_2({\mathbb{F}}_q)|=q(q^2-1)$. Its kernel is the scalar matrices of determinant one: a transformation fixing infinity has $c=0$, fixing zero then forces $b=0$, and fixing one gives $a=d$. Thus the kernel is $\{I,-I\}$, with these matrices coinciding in characteristic two.

For a finite field of odd characteristic and nonzero $a,b$, let $U_a=\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)$. An arbitrary determinant-one matrix $P=\left(\begin{array}{cc}r& s\\ t& u\end{array}\right)$ satisfies $P{U}_a=U_{b}P$ only if $t=0$ and $ar=bu$. The determinant condition $ru=1$ then gives $b/a=r^2$. Conversely $\mathrm{diag}(r,r^{-1})U_a\mathrm{diag}(r,r^{-1}{)}^{-1}=U_{a{r}^2}$. Thus nonzero upper unipotent matrices fall into two conjugacy classes under this square-class criterion. For example, $U_1$ and $U_3$ are conjugate over ${\mathbb{F}}_{11}$ because $5^2=3$, but not over ${\mathbb{F}}_5$. The criterion compares these particular upper unipotent representatives; it does not classify all determinant-one matrices.

A functor $G:\mathcal{C}\to \mathcal{D}$ from a locally small category that is complete and well-powered, with a small cogenerating family, to a locally small category has a left adjoint exactly when it preserves small categorical limits. For sufficiency, the cogenerator bound for comma-category solution sets produces a weakly initial set in each complete comma category $(B\downarrow G)$. The initial-object lemma for complete categories with a weakly initial set supplies its initial object, a universal arrow from an object to a functor. Necessity is limit preservation by a right adjoint.

With normal modes $e^{ikx-i\omega t}$, a finite upper bound on $\mathrm{Im}\omega$ allows the inverse temporal Laplace transform contour to start above all temporal singularities. An unbounded growth rate at high real wavenumber prevents the standard causal Green function construction used in the Briggs-Bers criterion. For $\mathrm{Im}\omega =\alpha k^2-\beta k^4-\gamma$ with nonzero real $\beta$, the bound holds exactly when $\beta >0$.