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To remove an internal state $k$, replace each surviving label by $R_{ij}\cup R_{ik}(R_{kk}{)}^{\ast }{R}_{kj}$. The added term accounts for entering $k$, traversing its loop any number of times, and leaving it. Fresh initial and final states allow all old states to be eliminated, producing a regular expression for the same language.

Successively decompose an irreducible representation along a subgroup chain with multiplicity-free restriction. A complete path selects a one-dimensional subspace; choosing one nonzero vector on each path gives a Gelfand–Tsetlin basis. For a symmetric group over the complex numbers, paths are standard Young tableaux, and the Young–Jucys–Murphy elements act diagonally with their cell contents.

For a nonzero right module $M$ that is a finitely generated module, choose a finite-dimensional generating vector subspace $W$ and define ${\mathrm{GKdim}}_{A}M={\mathrm{limsup}}_{n\to \infty }\log ({\dim }_{k}W{V}^n)/\log n$, where $V$ generates the algebra and contains $1$. For a left module use $V^{n}W$. It is independent of these choices and measures polynomial-order growth of a generating filtration of a module.

This covariance function is periodic with period $P>0$. Map time to the circle $u(t)=(\cos (2\pi t/P),\sin (2\pi t/P))$ and pull back a Gaussian kernel; since $\Vert u(t)-u(t^{\prime }){\Vert }^2=4{\sin }^2(\pi (t-t^{\prime })/P)$, this proves positive semidefiniteness. A Gaussian process with this kernel has $\mathrm{Var}(f(t+P)-f(t))=0$ and therefore repeats at every fixed phase. Replacing the inner $\pi$ by $2\pi$ changes the actual period to $P/2$.

After integrating a free-field momentum shell, restore the cutoff with $x^{\prime }=x/b$ and ${\varphi }^{\prime }(x^{\prime })=b^{(D-2)/2}{\varphi }_{<}(b{x}^{\prime })$. The gradient coefficient is invariant, the mass changes as $r^{\prime }=b^{2}r$, and a uniform conjugate field changes as $h^{\prime }=b^{(D+2)/2}h$. This follows by counting measure, derivative and field factors in the quadratic Hamiltonian. A nonuniform source instead obeys $h^{\prime }(x^{\prime })=b^{(D+2)/2}h(b{x}^{\prime })$, with the source on the right projected onto the retained Fourier modes. The field has engineering dimension $(D-2)/2$ and zero anomalous dimension.

For a primitive Dirichlet character, $S_{\chi }(r)=\overline{\chi }(r)g(\chi )$ for every integer $r$, with zero on nonunits. For a unit $r$ this follows by substitution. If a prime $p$ divides both $r$ and $N$, primitivity supplies a unit $u\equiv 1(\mathrm{mod}N/p)$ with $\chi (u)\ne 1$; substitution by $u$ fixes the additive exponential and forces $S_{\chi }(r)=0$. For an imprimitive Dirichlet character this vanishing can fail.

Put $d=c-a-b\notin \mathbb{Z}$ and choose a branch of $(1-z{)}^d$. The solution regular at zero isThe two characteristic exponents at a regular singular point are $0,d$. The Euler integral for the hypergeometric function determines the first coefficient by a finite endpoint limit and the second by scaling the singular endpoint; analytic continuation extends the parameter ranges.

The invariant measure for the Gauss continued-fraction map with density $1/[(\log 2)(1+x)]$ on $[0,1]$. Inverse branches $x=1/(n+y)$ give transformed density $(\log 2{)}^{-1}{\sum }_{n\ge 1}[(n+y{)}^{-1}-(n+y+1{)}^{-1}]$, which telescopes to the same density.

A gauge group representation specifies how a matter field transforms under a gauge group. It fixes the gauge charges and whether gauge-invariant mass terms can pair a field with a field in the conjugate group representation.

For an Abelian gauge theory with $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, a nonlinear gauge functional $F[A]$ has Faddeev-Popov determinant $\det (\delta F[A+\partial \alpha ]/\delta \alpha )$. This operator can depend on $A$ even though the Adjoint representation of $U(1)$ is trivial. Consequently Faddeev-Popov ghost fields need not decouple in a nonlinear Abelian gauge, unlike a field-independent linear gauge condition.

With $D_{\mu }={\partial }_{\mu }+R(A_{\mu })$ and ${\delta }_X{A}_{\mu }=-ϵ{\partial }_{\mu }X+ϵ[X,A_{\mu }]$, the derivative terms in ${\delta }_X(D_{\mu }\varphi )$ cancel. The remaining terms combine using $R([X,A_{\mu }])=[R(X),R(A_{\mu })]$, so the covariant derivative transforms in the same Lie algebra representation as the scalar. This convention absorbs the gauge coupling into the connection.

With canonically normalized gauge field kinetic terms, a gauge-boson mass matrix is the real symmetric matrix of coefficients in the displayed quadratic mass term. Its eigenvalues are squared physical gauge boson masses. For a complex scalar field vacuum ${\Phi }_0$ in the Higgs mechanism, $(M^2{)}_{ab}=2\mathrm{Re}[(g_a{T}_a{\Phi }_0{)}^{†}(g_b{T}_b{\Phi }_0)]$. Thus for real $x_a$, $x^T{M}^{2}x=2\Vert {\sum }_a{x}_a{g}_a{T}_a{\Phi }_0{\Vert }^2\ge 0$: it is a Gram matrix and a positive semidefinite matrix. An unbroken generator annihilates the vacuum expectation value and lies in its null space. Diagonalizing the matrix identifies the massive and massless field combinations.

With signature $(+,-,-,-)$, one standard representation is ${\tilde{\gamma }}^0=\mathrm{diag}(I_2,-I_2)$ and ${\tilde{\gamma }}^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$. Block multiplication and the Pauli matrix multiplication law give time-time anticommutator $2I_4$, mixed anticommutators zero, and spatial anticommutators $-2{\delta }^{ij}{I}_4$. Thus these gamma matrices represent the spacetime Clifford algebra.

With metric $(+---)$, the chirality matrix has $({\gamma }^5{)}^2=1$ and anticommutes with each gamma matrix. Its eigenspaces define chirality; $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ are the corresponding projectors. These identities follow directly by moving a gamma matrix through the ordered product and using the Clifford algebra.

With metric $(+---)$, ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and ${ϵ}^{0123}=+1$, odd ordinary gamma matrix traces vanish, $\mathrm{Tr}({\gamma }^{\mu }{\gamma }^{\nu })=4g^{\mu \nu }$, andThe first follows by repeated Clifford algebra anticommutation and cyclicity of the trace. The second is an antisymmetric tensor, whose normalization is fixed by the $0123$ component.

Tidal capture converts sufficient relative orbital energy into internal tidal heating to leave two initially unbound galaxies on a bound orbit. For identical spherical galaxies of mass $m_g$, a distant rapid flyby of impact parameter $p$ and relative speed $v$ gives the approximate capture condition $pv<[(32/3)G^2{m}_g^2⟨r^2⟩{]}^{1/4}$. This requires the impulse approximation and weak gravitational focusing; it is not a general criterion for subsequent coalescence.

The gas fraction of a galaxy is commonly $\mu =M_g/(M_g+M_s)$, using the gas and stellar masses remaining in the system. In a leaky-box model of galactic chemical evolution, it differs from the fraction $g=M_g/M_{g,0}$ of the initial gas reservoir. For an initially star-free box with outflow rate $\alpha {\dot{M}}_s$, $\mu =(1+\alpha )g/(1+\alpha g)$. These different fractions give different-looking metallicity formulae and effective yields.

If a squarefree positive $d$ has a prime divisor $p\equiv 3(\mathrm{mod}4)$, a unit of norm $-1$ in $\mathbb{Q}(\sqrt{d})$ would give $a^2-d{b}^2=-4$ for integers $a,b$. Modulo $p$ this makes $-1$ a quadratic residue, which is impossible. The obstruction is sufficient, not a full criterion for negative-norm units.

An equivalence consists of functors $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ with natural transformation isomorphisms $GF\cong 1_{\mathcal{C}}$ and $FG\cong 1_{\mathcal{D}}$. A full and faithful functor which is essentially surjective, meaning every target object is isomorphic to an image object, is part of an equivalence under the usual choice convention. An isomorphism of categories is stronger: it has a strictly inverse functor.