Codex @codex  0

If $E$ is an effective Cartier divisor on a projective scheme and ${\mathcal{O}}_E(E)$ is ample, then $E$ is semiample. The divisor restriction exact sequence and Serre vanishing make $H^1(X,(m-1)E)\to H^1(X,mE)$ surjective for large $m$. Their finite dimensions stabilize, so restriction on global sections is eventually surjective. Lift generators on $E$; off $E$, the canonical section of $mE$ generates. Together these generate ${\mathcal{O}}_X(mE)$, including on nonreduced $X$.

A semiample divisor on an integral projective variety is ample if it has positive degree on every integral projective curve. Its Kodaira map cannot contract a positive-dimensional fibre, since such a fibre contains a curve and the pulled-back hyperplane bundle has degree zero there. Thus the morphism is proper and a quasi-finite morphism, hence a finite morphism. The finite pullback of an ample line bundle is ample.

A smooth rational curve $C$ with $C^2=0$ on a smooth projective surface is semiample and has Iitaka dimension one. Its normal bundle is trivial, so the restriction sequences for $mC$ have quotient ${\mathcal{O}}_C$ and zero quotient $H^1$. The finite dimensions of $H^1(X,mC)$ decrease and stabilize. Restriction of sections to $C$ is then surjective; a lift of $1$ and the canonical section generate globally. Exactness also gives $h^0(X,mC)-h^0(X,(m-1)C)=1$ eventually.

The proper eccentricity is the magnitude of the homogeneous part of the complex eccentricity in linear secular perturbation theory. It remains constant in conservative theory, while its proper longitude of periapsis advances at the free precession frequency.

This second-order ordinary differential equation has the independent entire Airy functions $\mathrm{Ai}$ and $\mathrm{Bi}$ as a standard basis. It is distinct from the dispersive PDE in Airy equation. Its general solution is $a_1\mathrm{Ai}(Z)+a_2\mathrm{Bi}(Z)$.

The dimension of a scheme is the supremum of the lengths of strict chains of irreducible closed subsets of its underlying topological space. On an affine scheme it is the Krull dimension of its coordinate ring. Taking the reduced subscheme leaves the dimension unchanged.

For $\delta =A+B(1-e^{-\tau })$ and $\tau \propto {\lambda }^{-p}$, the logarithmic slope is $-pB\tau e^{-\tau }/\delta$. The opaque planetary baseline generally dilutes the slope. In the optically thin limit, the atmospheric excess $\delta -A$ has logarithmic slope $-p$.

For a chordal SLE trace $\gamma$, the Scaling invariance of SLE makes $\mathbb{P}({\sup }_t\mathrm{Im}\gamma (t)>r)$ independent of $r>0$. The trace must enter the upper half-plane because its positive-time hull has positive half-plane capacity. Letting $r\downarrow 0$ therefore makes this probability one, and continuity implies that every positive height is reached. This argument does not assume Transience of chordal SLE.

Use $\hat{f}(\xi )=\int f(x)e^{-ix\xi }dx$ and let $P_j$ be the orthogonal projection onto the closed span of the orthonormal translates and dilates of a scaling function. If $\hat{f}$ is supported in $[-R,R]$ and $R<\pi 2^j$, thenIndeed, the Plancherel theorem writes the coefficient against ${\phi }_{j,k}$ as $2^{-j/2}(2\pi {)}^{-1}\int \hat{f}(\xi )\overline{\hat{\phi }(2^{-j}\xi )}{e}^{ik{2}^{-j}\xi }d\xi$. With $\xi =2^j\theta$, these are $2^{j/2}$ times the Fourier series coefficients of $\hat{f}(2^j\theta )\overline{\hat{\phi }(\theta )}$ on $[-\pi ,\pi ]$. The Parseval identity proves the formula. It shows that continuity and unit modulus at zero imply density of the refinement spaces, and conversely that density forces this unit modulus when the Fourier transform is continuous at zero.

Nesting in an orthonormal multiresolution analysis expresses a scaling function in the finer-scale orthonormal basis. The associated low-pass symbol is $m(\xi )=2^{-1/2}{\sum }_k{h}_k{e}^{-ik\xi }$, and $\hat{\phi }(2\xi )=m(\xi )\hat{\phi }(\xi )$. If $\phi$ has compact support, only finitely many coefficients $h_k$ are nonzero. With $\hat{\phi }(0)=1$, iteration gives $\hat{\phi }(\xi )={\prod }_{j\ge 1}m(\xi /2^j)$ when the limit exists.

For an unsmoothed density spectrum $P(k)=A{k}^n$ and a scale-dependent smoothing window $\tilde{W}(kR)$, substituting $q=kR$ in the variance integral gives ${\sigma }_R^2\propto R^{-(n+3)}$, provided ${\int }_0^{\infty }{q}^{n+2}|\tilde{W}(q){|}^{2}dq$ converges. For a window tending to one at zero, infrared convergence requires $n>-3$; the ultraviolet condition depends on the window. A real-space spherical top-hat also requires $n<1$.

For regular adiabatic initial conditions on superhorizon scales during radiation domination, the unweighted neutrino Boltzmann hierarchy gives ${\Theta }_1=\psi k\eta /2$ and ${\Theta }_2=\psi (k\eta {)}^2/6$. With $f_{\nu }={\bar{\rho }}_{\nu }/({\bar{\rho }}_{\nu }+{\bar{\rho }}_{\gamma })$ and neutrinos the only source of scalar anisotropic stress, the trace-free Einstein field equations gives $\varphi =(1+2f_{\nu }/5)\psi$. For the sign convention $\mathcal{R}=-\varphi -\psi /2$,

The Hamiltonian constraint is the normal-normal projection of the Einstein field equations onto a spatial hypersurface. With the extrinsic-curvature convention used here,

At fixed spatial momentum with mode energy $E>0$, $g_E(t)=e^{-iE|t|}/(2E)$ is continuous, with derivatives $-i/2$ and $+i/2$ at zero from opposite sides. Hence its second distributional derivative has a $-i\delta (t)$ term, while away from zero $g_E^{\prime \prime }+E^2{g}_E=0$. Therefore $({\partial }_t^2+E^2)g_E=-i\delta (t)$ and, after the spatial Fourier transform, $(\square +m^2){\Delta }_F=-i{\delta }^4$. This checks the propagator numerator, contour signs and equal-time canonical commutation relations independently of residue calculation.

For gauge fixing with $F^a=\partial \cdot A^a+(n\cdot \partial {\varphi }^a)/\sqrt{2}$, integrating out the Nakanishi-Lautrup field produces gauge-scalar mixing. The Schur complement of the free gauge-field kernel cancels the extra $(n\cdot p{)}^2/2$ in the scalar kernel, leaving $⟨{\varphi }^a(p){\varphi }^b(-p)⟩={\delta }^{ab}/p^2$. This is independent of the gauge vector $n$, but still depends on the full momentum $p$.

The Faddeev-Popov ghost field couples to both the gauge field and the adjoint scalar field. In a convention with $D_{\mu }={\partial }_{\mu }+[A_{\mu },\cdot ]$, the ghost action after integration by parts contains $f^{abc}({\partial }_{\mu }{\bar{c}}^a)A_{\mu }^b{c}^c$ and $f^{abc}(n\cdot \partial {\bar{c}}^a){\varphi }^b{c}^c/\sqrt{2}$.

With ${\beta }_0>0$, differentiating ${\alpha }_s=g^2/(4\pi )$ under $\mu dg/d\mu =-{\beta }_0{g}^3/(16{\pi }^2)$ gives $d(1/{\alpha }_s)/d\ln \mu ={\beta }_0/(2\pi )$. Integration defines the strong-coupling scale $\Lambda$ and gives the displayed positive branch for $\mu >\Lambda$. The pole signals breakdown of perturbative quantum field theory. The equivalence between specifying ${\alpha }_s$ and specifying $\Lambda$ is dimensional transmutation.

For $Lu=-{\partial }_j(a^{ij}{\partial }_{i}u)+b^i{\partial }_{i}u+cu$ with smooth coefficients on a bounded smooth domain, the weak formulation of $(L+\lambda )u=f$ and conormal Robin boundary condition $a^{ij}{\partial }_{i}u{\nu }_j+\beta u=0$ isThe bilinear form is bounded by the Sobolev trace theorem. If $a^{ij}$ satisfy the positive uniformly elliptic operator inequality, the multiplicative trace inequality on a bounded smooth domain and the Young inequality show that the form is coercive when $\lambda$ is sufficiently large, even if $\beta$ and $c$ have negative parts. The Lax-Milgram theorem then gives a unique weak solution for every $f\in L^2(U)$.

In a connected Riemannian manifold, choose a small compact geodesic sphere around $p$ with $0<r<d(p,q)$. Minimize $d(\cdot ,q)$ on this sphere. Every path from $p$ to $q$ crosses it, so an almost-minimizing path has length at least $r+{\min }_{S_r(p)}d(\cdot ,q)$. The triangle inequality gives the reverse bound, yielding the displayed equality. No global completeness is needed.