For the endomorphism bundle connection, $({\tilde{\nabla }}_X\varphi )(s)={\nabla }_X(\varphi s)-\varphi ({\nabla }_{X}s)$. Expanding two derivatives and subtracting ${\tilde{\nabla }}_{[X,Y]}$ givesThus its curvature form of a connection is the commutator action of the original curvature.

The j-invariant is unchanged by admissible changes of Weierstrass equation of an elliptic curve. With $b_2=a_1^2+4a_2$ and $b_4=a_1{a}_3+2a_4$, set $c_4=b_2^2-24b_4$ and $j=c_4^3/\Delta$, where $\Delta$ is the elliptic-curve discriminant. For good reduction over a local field, an integral equation with unit discriminant shows that $j$ lies in the valuation ring. Thus a negative valuation of $j$ proves bad reduction, even if the displayed equation has not been proved minimal.

For the Higgs doublet vacuum $(0,v{)}^T/\sqrt{2}$, the neutral gauge-boson mass matrix in $(W^3,B)$ is $(v^2/4)\left(\begin{array}{cc}{g}^2& -g{g}^{\prime }\\ -g{g}^{\prime }& g^{\prime 2}\end{array}\right)$. Its zero eigenvector is proportional to $(g^{\prime },g)$, the photon; the perpendicular vector is the Z boson. The charged fields $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$ have the stated W boson mass. Thus $m_W=m_Z\cos {\theta }_W$, where ${\theta }_W$ is the Weinberg angle. The unbroken charge generator annihilates the vacuum and explains the zero eigenvalue.

For $T\ll m_e$ and negligible chemical potentials, the combined equilibrium density of electrons and positrons divided by the photon number density isThe exponential suppression eventually makes pair equilibrium incompatible with the residual electrons required by charge neutrality.

For noninteracting electrons at zero temperature, define $x=p_F/(m_{e}c)$ with Fermi momentum $p_F=\hslash (3{\pi }^2{n}_e{)}^{1/3}$. The isotropic momentum flux gives electron degeneracy pressureIts nonrelativistic and ultrarelativistic limits arerespectively. In a fully ionized stellar gas, $n_e=\rho /({\mu }_e{m}_u)$, using mean molecular weight per electron ${\mu }_e$. This neglects finite-temperature effects, interactions, and changes in composition.

A strictly decreasing demand function has a decreasing inverse on its range. The utility primitive $\int P_s(u)du$ is a concave function. A positive interior reference value avoids assuming that an improper integral from zero is finite.

A fixed-mass shell with ultraviolet opacity ${\kappa }_{\mathrm{UV}}$ has dust transparency radius $R_{\mathrm{UV}}=\sqrt{{\kappa }_{\mathrm{UV}}{M}_{\mathrm{sh}}/(4\pi )}$. For an initially expanding shell driven by constant ultraviolet luminosity, a sharp opaque-to-transparent approximation givesHere ${\Gamma }_{\mathrm{ss}}=L{R}_0^2/(GM{M}_{\mathrm{sh}}c)$ is the initial single-scattering force ratio and ${\Gamma }_{\mathrm{IR}}={\kappa }_{\mathrm{IR}}L/(4\pi GMc)$. The trajectory must actually reach $R_{\mathrm{UV}}$. Large initial ultraviolet optical depth enhances the radiative work at fixed initial single-scattering ratio; the same conclusion does not follow unconditionally at fixed electron-scattering luminosity.

A geometric drift condition requires a measurable $V\ge 1$, a small set $C$, constants $\lambda <1$ and $b<\infty$, andTogether with irreducibility and aperiodicity, it implies geometric ergodicity. A global Doeblin condition allows $C$ to be the entire state space and $V=1$.

Underdispersion means less variation than prescribed by a reference statistical model, corresponding to a working dispersion parameter below one. It is not, by itself, evidence that the reference mean function is improved.

For planar Gaussian free field theory, the Dirichlet energy space is the completion of real test functions $C_c^{\infty }(D)$ in the norm induced by the Dirichlet inner product,On a bounded domain, the zero-boundary Poincaré inequality identifies this with $H_0^1(D)$ with an equivalent Sobolev norm. On a general unbounded domain, it must be distinguished from completion in the inhomogeneous $H^1$ norm.

A conformal bijection identifies the two corresponding energy completions by conformal invariance of the planar Dirichlet inner product. In particular, if $D$ is a nonempty simply connected proper open subset of the complex plane, the Riemann mapping theorem supplies a conformal bijection with the unit disk. This unit-disk identification is not asserted for an arbitrary multiply connected domain or the whole complex plane.

For a conformal bijection $f:D\to \tilde{D}$ and smooth $u,v$ of compact support on $\tilde{D}$, $\boxed{(u\circ f,v\circ f{)}_{\nabla ,D}=(u,v{)}_{\nabla ,\tilde{D}}}$. The Jacobian matrix of $f$ is a rotation times $|f^{\prime }|$. Consequently the chain rule introduces $|f^{\prime }{|}^2$ into the gradient pairing, while the change of variables formula introduces precisely the same factor into area. They cancel. The identity extends by completion to the Dirichlet energy spaces; it does not preserve the $L^2$ term of an inhomogeneous Sobolev norm.

Let $j:{\mathbb{A}}_k^1\hookrightarrow {\mathbb{P}}_k^1$ be a standard affine chart and $\mathcal{G}=\tilde{M}$ a quasi-coherent sheaf there. Its direct image is quasi-coherent because $j$ is an affine morphism. The standard two-chart acyclic cover has Čech cochain complex $M\oplus M_x\to M_x$ in degrees zero and one, with differential $(m,n)\mapsto n-m/1$. Surjectivity gives $H^i({\mathbb{P}}_k^1,j_{\ast }\mathcal{G})=0$ for $i>0$. This does not assert that the direct image itself is flasque.

In the weak-forcing steady response about rest, diabatically forced quasi-geostrophic potential vorticity gives $\beta {\psi }_x=(f_0/N^2)R_z$. For $\beta \ne 0$, an eastern zero-flow condition fixes the horizontal circulation through $\psi =-(f_0/\beta N^2){\int }_x^{\infty }{R}_{z}dX$, up to a horizontally uniform pressure gauge. The thermodynamic equation gives $w=R/N^2$. A western zonal wake can remain even though the heating and vertical velocity vanish there. This is a formal steady interior solution: specified initial boundary buoyancy can prevent it from being the attained steady state. For example, under a rigid lid where the heating vanishes, initially uniform boundary buoyancy remains uniform at linear order.

If each level of one categorical variable belongs to exactly one level of another, indicators for the coarser levels are sums of indicators for the finer levels. Including unrestricted fixed effects for both therefore gives a rank-deficient ordinary least squares: the separate effects lack identifiability without additional constraints.

In the massless-target convention, the Callan-Gross relation sets this coefficient to zero at leading order in the parton model.

On a compact connected oriented $n$-dimensional smooth manifold without boundary, integration is a linear isomorphism $H_{\mathrm{dR}}^n(M)\to \mathbb{R}$. Thus two top-degree differential forms have the same de Rham cohomology class exactly when their integrals agree.

For a free smooth action of a finite group $G$ on a smooth manifold $M$, the quotient projection inducesInvariant differential forms descend uniquely through local inverse branches of the covering map. Averaging by $|G{|}^{-1}{\sum }_{g\in G}{g}^{\ast }$ commutes with the exterior derivative, produces invariant representatives of invariant classes, and produces an invariant primitive for every invariant exact form. Hence taking invariant forms and taking invariant cohomology give the same result over the real numbers.

Define $E_0(z)={\int }_0^z{e}^{-t^2}{\int }_0^t{e}^{u^2}dudt$ and $E_1(z)={\int }_0^z{e}^{-t^2}{\int }_0^t{e}^{u^2}\mathrm{erf}ududt$. They satisfy $(D^2+2zD)E_0=1$ and $(D^2+2zD)E_1=\mathrm{erf}z$. The first is an even function, the second an odd function, and their real tails are respectively $\frac{1}{2}\log |z|+C_1+o(1)$ and $\mathrm{sgn}z[\frac{1}{2}\log |z|+C_2+o(1)]$. Their derivatives have inverse-linear tails, explaining the logarithmic matching constants in a singular perturbation.

For the Levi-Civita connection, the curvature form of a connection is the endomorphism-valued two-formIt has tensoriality in all three arguments. Its first Bianchi identity follows from torsion-freeness: the cyclic sum is ${\sum }_{\mathrm{cyc}}({\nabla }_X[Y,Z]-{\nabla }_{[Y,Z]}X)={\sum }_{\mathrm{cyc}}[X,[Y,Z]]=0$ by the Jacobi identity.

An Edgeworth expansion refines the central limit theorem by adding corrections determined by higher cumulants. Let $X_1,X_2,\dots$ be independent and identically distributed with mean zero, variance ${\sigma }^2>0$ and skewness ${\gamma }_1=\mathbb{E}{X}_1^3/{\sigma }^3$. For the standardized sum $Z_n=(X_1+\cdots +X_n)/(\sigma \sqrt{n})$, the first density correction isHere $\varphi$ is the standard normal density. This is an asymptotic statement for a sequence, not a universal density formula for one arbitrary random variable. Sufficient smoothness and moment conditions are needed; one strong sufficient setting is that the common probability density function is a non-degenerate Schwartz function. In that setting the displayed remainder is valid for each fixed $z$. Lattice-valued variables need additional corrections rather than this density formula.^[ref]

To identify the coefficient, expand the log characteristic function of the standardized sum:for fixed $t$. Exponentiation and Fourier inversion turn $(it{)}^3{e}^{-t^2/2}$ into the Hermite polynomial ${\mathrm{He}}_3(z)=z^3-3z$ times $\varphi (z)$. The smoothness assumptions justify inversion beyond this formal calculation. Higher-order terms use further cumulants and Hermite polynomials.

The approximation improves central asymptotics, but it need not be a nonnegative normalized density after truncation. In particular the cubic correction can become negative in a far tail, where a fixed-point expansion does not provide a small relative error. Inverting a distribution-function Edgeworth expansion produces the Cornish-Fisher expansion for fixed quantiles.