Dualize the Euler sequence on complex projective space and tensor by the hyperplane line bundle to obtain $0\to {\Omega }^1(1)\to {\mathcal{O}}^{n+1}\to \mathcal{O}(1)\to 0$. On global sections the last map sends $(c_i)$ to ${\sum }_i{c}_i{X}_i$, an isomorphism onto the linear forms. Left exactness of sheaf cohomology therefore gives the asserted vanishing. No higher-cohomology assumption is needed for this final kernel calculation.

On a Riemann surface, a meromorphic n-differential is a meromorphic section of the $n$th power of the canonical bundle. In a coordinate it is $F(z)(dz{)}^n$, with coefficients transforming by the $n$th power of the derivative. At a ramification coordinate $w=t^e$, its pullback has order $e{\mathrm{ord}}_w\omega +n(e-1)$. At a modular cusp, the exponential coordinate gives $dz=(h/(2\pi i))dq/q$, so a weight-$2n$ form of cusp order $v$ produces a differential of order $v-n$.

In conformal Newtonian gauge, a free scalar has no linear anisotropic stress. Eliminating its perturbation with the momentum constraint gives ${\Phi }_k^{\prime \prime }+3{\Phi }_k^{\prime }/\tau +k^2{\Phi }_k=0$. Substitution of ${\Phi }_k=(-\tau {)}^{-1}y(-k\tau )$ gives the order-one Bessel equation. The displayed Hankel function has positive incoming frequency $e^{-ik\tau }$ at early times.

A coordinate of a vector in a chosen basis is one entry of its coordinate vector. The corresponding coordinate functional in the dual basis extracts it. For an orthonormal basis, it can instead be read as an inner product with the basis vector, as in orthonormal eigenbasis. For a nonorthogonal basis, ordinary dot products need not give the coordinates.

A value of $g\ne 1$ is a convex lattice subgroup maximal among those omitting $g$. Zorn's lemma supplies a value: the union of a chain of convex lattice subgroups omitting $g$ still omits $g$. Such subgroups are prime. Their unique cover is the smallest convex lattice subgroup properly containing them.

A proper convex lattice subgroup $P$ is prime when $a,b\ge 1$ and $a\wedge b=1$ imply $a\in P$ or $b\in P$. Equivalently the ordered right cosets of $P$ form a totally ordered set. Prime refers to this order property, and does not require a normal subgroup or prime index.

For $u\ge 1$ in a lattice-ordered group, the displayed set is the smallest convex lattice subgroup containing $u$. Products of two such bounds are bounded by a larger power of $u$; inversion and the lattice operations preserve the bounds. This gives an explicit test for membership.

The convex lattice subgroups of a lattice-ordered group form a distributive lattice, with intersection as meet and the convex lattice subgroup generated by a union as join. A positive element of a join is bounded by a finite product of positive elements from the generating subgroups. Riesz decomposition in a lattice-ordered group splits such an element into positive factors from those subgroups. If the original element belongs to another convex subgroup $E$, every factor does too. Consequently $E\cap (C\vee D)=(E\cap C)\vee (E\cap D)$.

The displayed relation makes the right cosets into a lattice, with $Pg\vee Ph=P(g\vee h)$ and $Pg\wedge Ph=P(g\wedge h)$. Antisymmetry uses convexity: $g\le ph$ and $h\le qg$ give $1\le ph{g}^{-1}\le pq\in P$, hence $h{g}^{-1}\in P$. A prime convex lattice subgroup makes this order total even if the subgroup is not normal.

Assume $uv\le v^2{u}^2$ for all $u,v\ge 1$ in a lattice-ordered group. For a convex lattice subgroup $H$ and $a\ge 1$, the displayed bounds allow $h\in H$ with $h\ge 1$ and integers $n\ge 0$. To prove the first set is a subgroup, use $(h{a}^n)(k{a}^m)\le h{k}^2{a}^{2n+m}$. Products and inverse products are both bounded because $xy\le |x||y|$ and $(xy{)}^{-1}\le |y||x|$. Taking the larger exponent and a join of the two bounds in $H$ bounds their absolute value in a lattice-ordered group. Inversion preserves that absolute value; common bounds using $h\vee k$ prove lattice closure and convexity. The set contains $H$ and $a$, and any convex lattice subgroup containing them contains the entire set. The second equality follows by applying the proof to the opposite group, where the required inequality is the original one with the variables interchanged.

James's theorem says that a Banach space is reflexive exactly when every continuous linear functional attains its supremum on the closed unit ball. Its separation proof also yields the equivalent convex-block criterion: nonreflexivity produces a bounded sequence whose initial and tail convex hulls remain uniformly separated.

For convective contraction with power-law opacity $\kappa \propto \rho T^4$ at fixed stellar mass and opacity normalization, $T_e\propto {\mu }^{1/2}{R}^{1/8}$ and $L\propto {\mu }^2{R}^{5/2}$. Eliminating $R$ gives $L\propto {\mu }^{-8}{T}_e^{20}$. Thus a larger mean molecular weight moves the track to larger effective temperature at a given luminosity, with $T_e{|}_L\propto {\mu }^{2/5}$. Contraction decreases both temperature and luminosity for this particular opacity exponent.

Match an $n=3/2$ stellar polytrope to the grey-atmosphere convection onset with seventeenth-power opacity. At each instant the convective entropy coefficient obeys $P=K{T}^{5/2}$, $K\propto M^{-1/2}{R}^{-3/2}$ at fixed composition, while the boundary gives $K\propto g^{1/2}{T}_e^{-21/2}$. Thus $T_e^{21/2}\propto M{R}^{1/2}$ and the first displayed luminosity law follows. The stellar virial theorem for a monatomic gas gives total energy $-3G{M}^2/(7R)$, so Kelvin-Helmholtz contraction has $\dot{R}=-7L{R}^2/(3G{M}^2)$. Integrating at fixed mass gives $R^{-67/21}-R_0^{-67/21}\propto t$. The second displayed relation holds after the initial-radius term is negligible. The entropy coefficient is spatially constant but changes during contraction.

For a fixed-mass, fully convective monatomic gas protostar, write $P=K{T}^{5/2}$ and $K=K_0{\mu }^{-5/2}{M}^{-1/2}{R}^{-3/2}$. Matching the photosphere to $\kappa P=2g/3$ with $\kappa ={\kappa }_0\rho T^4$ gives $T_e^8=2G\mathcal{R}{\mu }^4{M}^{2}R/(3{\kappa }_0{K}_0^2)$. The Stefan–Boltzmann law gives $L=\pi ac[2G\mathcal{R}/(3{\kappa }_0){]}^{1/2}{K}_0^{-1}{\mu }^{2}M{R}^{5/2}$. The stellar virial theorem gives total energy $-3G{M}^2/(7R)$, so $L=-3G{M}^2\dot{R}/(7R^2)$. Integrating proves $C=(49\pi ac/6)[2\mathcal{R}/(3{\kappa }_{0}G){]}^{1/2}{K}_0^{-1}$. This holds before stellar nuclear fusion, changing opacity or electron degeneracy pressure invalidates the assumed contraction law.

Particles perform independent continuous-time random walks until they meet, after which they share one path. A finite initial particle set remains finite and its cardinality cannot increase. They arise as the common ancestors of sites in a voter model. On the square lattice every pair eventually coalesces, because the difference walk is recurrent Markov chain.

If $M$ is a continuous local martingale and $f\in C^2(\mathbb{R})$, then finite variation of $f(M)$ forces $f(M_t)=f(M_0)$. The Itô formula and the quadratic variation of a stochastic integral give ${\int }_0^t{f}^{\prime }(M_s{)}^{2}d[M{]}_s=0$. The occupation-times formula implies that $L_t^a(M)=0$ almost everywhere where $f^{\prime }(a)\ne 0$. On the zero set of $f^{\prime }$, its derivative $f^{\prime \prime }$ vanishes almost everywhere: zeros at which $f^{\prime \prime }\ne 0$ are isolated and hence countable. Thus ${\int }_0^t|f^{\prime \prime }(M_s)|d[M{]}_s=\int |f^{\prime \prime }(a)|L_t^a(M)da=0$. Both terms in the Itô formula vanish.

Every nonempty open subset of an irreducible space is connected. A locally constant function into a fixed abelian group $D$ is therefore constant on that open set. Restrictions between nonempty opens identify with the identity of $D$, and restrictions to the empty set are surjective onto zero. Thus the constant sheaf is flasque, with global sections $D$ and zero higher sheaf cohomology on a nonempty irreducible space. For an integral scheme its constant function-field sheaf has its natural structure-sheaf action by regular functions viewed in the function field.

Under the same box-model assumptions, $M_0={\rho }_p{\varphi }_{0}W$ is initial particle mass and $m_d$ is final deposited mass per downstream length. Integrate the vertical particle deposition flux across the instantaneous triangular-valley width $2\lambda h$ from arrival of the front at $x$ until settling is complete. This gives the displayed profile on $0\le x\le R$ and zero beyond the runout. Its integral is $M_0$, verifying particle mass conservation. Areal deposit density also depends on the lateral coordinate because the wetted valley width changes as the current thins.

// Target: geophysics.bigb

With specific heating $ϵ={ϵ}_0\rho T^{16}$, the constant-opacity radiative-zero giant envelope has $T\propto \mu \beta M_c/r$ and $\rho \propto \mu \beta M_c{T}^3/L$. Therefore the shell power $4\pi r^2\rho ϵ$ is proportional to $(\mu \beta M_c{)}^{24}{L}^{-2}{r}^{-20}$. Integrating from the core radius gives $L^3\propto (\mu \beta M_c{)}^{24}{R}_c^{-19}$. The steep radial dependence justifies a thin-shell estimate dominated by the base. Using the outer radiative profile throughout the emitting layer is an approximation; a resolved shell can change the numerical coefficient. Also $\beta =1-{\kappa }_{0}L/(4\pi cG{M}_c)$, so retaining radiation pressure makes the relation implicit in luminosity.

Let $U$ be an open subset of the real line, and let $F,G$ be antiderivatives of the same function on $U$. Each connected component $I_j$ is an open interval. On that component, $(G-F{)}^{\prime }=0$, so the mean value theorem gives $G-F=C_j$. The constants on different components are independent: there is no subinterval inside $U$ joining them. Conversely, adding any chosen constant on each component preserves the derivative everywhere on $U$.

For example, the domain of $g(x)=1/x$ is $U=\mathbb{R}\setminus \{0\}$. Every antiderivative on this domain has the formwith arbitrary real $C_{-},C_{+}$. The logarithm of the absolute value differentiates to $1/x$ on each side. The function with $C_{-}=2$ and $C_{+}=-1$ is therefore a valid antiderivative, although it cannot be written as $\log |x|+C$ with one global constant.

An initial condition at $x=1$ fixes $C_{+}$ and leaves $C_{-}$ free. Fixing both constants requires one reference value in each connected component, or additional conditions that couple the components. The locally constant function $F(x)-\log |x|$ captures this remaining freedom.