For integration against $\sin x$ on $[0,\pi ]$, the monic quadratic orthogonal polynomial is $x^2-\pi x+2$. Its two roots are the Gaussian quadrature nodes $c_{1,2}=\pi /2\mp \sqrt{{\pi }^2/4-2}$, with weights one. The rule integrates every cubic exactly: divide it by the orthogonal quadratic and integrate the degree-one remainder.

For a bipartite elliptope matrix $X$, put $t=\log (1+\sqrt{2})$ and apply $\sinh (tx)$ within the two diagonal blocks and $\sin (tx)$ across them. Matching absolute power series coefficients give a positive semidefinite matrix by coefficient-dominated entrywise positivity, while $\sinh t=1$ gives unit diagonal. Gaussian hyperplane rounding then turns each cross-block correlation coefficient into $(2/\pi )\arcsin (\sin (t{X}_{ij}))=(2t/\pi )X_{ij}$ because $t<\pi /2$. The zero diagonal blocks of the bipartite objective eliminate every other contribution.

Completing the square in the free scalar path integral gives a quadratic exponential in the source. If ${\Delta }_F$ includes the factor $i$ in the momentum-space Feynman propagator, the normalized Minkowski functional is $\exp [-\int J{\Delta }_{F}J/2]$. Two source derivatives with their $1/i^2$ factor recover ${\Delta }_F$, while higher derivatives give Wick contractions.

For a positive differentiable covariance $C_t$ and a sufficiently regular, integrable $W$, differentiating the normalized Gaussian integral and integrating by parts twice gives the displayed identity. An unnormalized integral has the additional field-independent term $\mathrm{Tr}(C_t^{-1}{\dot{C}}_t)/2$ times the integral. A finite-mode regulator makes the analogous Gaussian functional integral identity precise before taking a continuum limit.

For the free parabolic wave equation $E_x=i{E}_{zz}/(2k)$, input $E(0,z)=e^{-A{z}^2}$ with $\mathrm{Re}A>0$ propagates asThe branch is continuous from $q(0{)}^{-1/2}=1$. One-dimensional transverse Fresnel propagation or two Gaussian integrals prove the formula. With $A=D^{-2}+ik/(2F)$, the squared envelope is $|q{|}^{-1}\exp [-2z^2/(D^2|q{|}^2)]$. The amplitude power is $-1/2$ because there is only one transverse coordinate. A two-transverse-coordinate Gaussian beam instead has power $-1$ when both coordinates share the same $q$.

For the passive convention $\tilde{x}=x+\xi$, expanding the tensor coordinate-transformation law around a fixed background gives this Lie derivative formula. The background time shift and Jacobian terms must both be retained. On an FLRW metric, the scalar lapse, shift and spatial potentials transform under two scalar coordinate functions.

Density is a scalar field. Under a first-order time shift $\tilde{\tau }=\tau +T$, its background changes by ${\bar{\rho }}^{\prime }T$, so its perturbation changes by the negative of that amount. Spatial shifts act only at second order on a homogeneous background density.

An antisymmetric tensor potential $B$ has gauge transformation $B\mapsto B+d\alpha$, where $\alpha$ is a one-form. Its field strength is the three-form $H=dB$, invariant because $d^2=0$. The Kalb–Ramond field is the string-theory example.

An Abelian $p$-form gauge potential $A_p$ is an antisymmetric tensor field with gauge equivalence $A_p\mapsto A_p+d{\Lambda }_{p-1}$ and field strength $F_{p+1}=d{A}_p$. Its field strength obeys the Bianchi identity for an Abelian p-form. The potential can have global gauge data even when its local field strength vanishes.

The Gamma function recurrence and $\Gamma (z)\sim 1/z$ at zero imply a simple pole at every $z=-n$, $n\ge 0$, with residue $(-1{)}^n/n!$.

For a finite Galois extension $L/K$, multiplicative Hilbert theorem 90 says $H^1(\mathrm{Gal}(L/K),L^{\ast })=0$. A multiplicative cocycle has the form $c_{\sigma }=\sigma (b)/b$. One proof forms a nonzero sum $A={\sum }_{\sigma }{c}_{\sigma }\sigma (a)$, using linear independence of field automorphisms. Its transformation law is $\tau A=c_{\tau }^{-1}A$, so $b=A^{-1}$ gives $c_{\tau }=\tau (b)/b$. Passing to the algebraic closure gives $H^1(K,{\overline{K}}^{\ast })=0$, so the multiplicative Kummer sequence identifies $H^1(K,{\mu }_m)$ with $K^{\ast }/K^{\ast m}$.

A discrete Galois module over $K$ is an abelian group with a continuous action of the absolute Galois group $G_K$. Continuity means that each element has an open stabilizer. Examples are the algebraic points of an elliptic curve, its finite torsion module $E[m]$, and the group of roots of unity ${\mu }_m$. Galois cohomology takes continuous cocycles with values in such a module.

The galaxy stellar mass function counts galaxies per stellar-mass interval and volume, often using comoving volume. Its stellar-mass moment gives the cosmic stellar mass density, while its number moment counts galaxies only if the low-mass population is integrable or truncated. It differs from the initial mass function of individual stars.

A thin ring of mass $m$ and radius $a$ has interior relative potential $Gm[1+R^2/(4a^2)+\cdots ]/a$. Its leading radial acceleration is outward, $GmR/(2a^3)$, unlike the zero interior force of a spherical shell. Exterior disc annuli therefore affect a galaxy rotation curve.

The spherical shell theorem yields $v_c^2=GM(<r)/r$ only in spherical geometry. A razor-thin axisymmetric disc instead requires $v_c^2=R{\partial }_R\Phi (R,0)$, including forces from exterior annuli. The interior gravitational force of an exterior thin ring gives a concrete counterexample with identical enclosed mass but different circular speeds.

Let a disk contain fraction $f_d$ of its host mass and let its edge specific angular momentum be $f_j{R}_h{v}_h$. With halo relation $v_h^2=G{M}_h/R_h$ and disk geometry coefficient $\kappa =v_d^2{R}_d/(G{M}_d)$, combining centrifugal balance and the edge angular momentum gives the displayed estimates. The coefficient records the radial mass profile and thickness: enclosed mass does not determine a disc rotation curve. Treating $\kappa =1$ as a monopole estimate must not be mistaken for an exact thin-disk theorem.

A starburst galaxy undergoes an episode of rapid star formation. An instantaneous-burst idealization places all newly formed stars at one stellar age, so their present evolutionary-state fractions follow from the initial mass function at that fixed age rather than from an average over formation times.

The differential Newtonian gravitational field of a galaxy across a smaller bound system. Its tidal tensor can slowly change the angular momentum of distant comets, changing their periapses and their access to planetary encounters.

This spherical gravitational potential, with $a,v_0>0$, has density $\rho =v_0^2(2a+r)/(4\pi Gr(a+r{)}^2)$ and circular speed $v_c=v_0\sqrt{r/(a+r)}$. It combines a central $1/r$ density cusp with an outer flat galaxy rotation curve, but its total mass is infinite. It differs from a logarithmic potential with a constant-density core.

The Oort constants describe local differential galactic rotation: $A=-R{\Omega }^{\prime }/2$ and $B=-\Omega -R{\Omega }^{\prime }/2$. Thus $\Omega =A-B$, the radial epicyclic frequency obeys ${\kappa }^2=-4\Omega B$, and the epicyclic ellipse has axis ratio $\sqrt{-B/(A-B)}$.